Advertisement

Topological spaces of persistence modules and their properties

  • Peter Bubenik
  • Tane Vergili
Article
  • 18 Downloads

Abstract

Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules, including many of those that have been previously studied, and describe the relationships between them. In the cases where these classes are sets, interleaving distance induces a topology. We undertake a systematic study the resulting topological spaces and their basic topological properties.

Keywords

Persistent homology Persistence modules Interleaving distance 

Mathematics Subject Classification

55N99 54D99 54E99 18A25 

1 Introduction

A standard tool in topological data analysis is persistent homology (Ghrist 2008, 2018; Carlsson 2009; Chazal and Michel 2017; Wasserman 2016). It is often applied as follows. One starts with some data, constructs an increasing family of complexes or spaces, and applies homology with coefficients in some fixed field to obtain a persistence module. Next, one computes a summary [e.g. barcode (Collins et al. 2004), persistence diagram (Cohen-Steiner et al. 2007), or persistence landscape (Bubenik 2015; Bubenik and Dlotko 2017)] which determines this persistence module up to isomorphism. In practice, one computes these summaries directly from the increasing family of complexes or spaces. Nevertheless, the persistence module is the central algebraic object in this pipeline, and has been a focus of research.

A key discovery in the study of persistence modules is the notion of interleaving (Chazal et al. 2009) which provides a way of measuring the distance between persistence modules. For many persistence modules, this distance equals the bottleneck distance (Cohen-Steiner et al. 2007) between the corresponding persistence diagrams (Lesnick 2015; Bubenik and Scott 2014). Interleavings and the resulting interleaving distance have been extensively studied both for the persistence modules considered here (Lesnick 2015; Bubenik and Scott 2014; Bauer and Lesnick 2015, 2016; Chazal et al. 2016b; Bubenik et al. 2017a; Blumberg and Lesnick 2017), for Reeb graphs (de Silva et al. 2016; Munch and Wang 2015), for zig-zag persistence modules (Botnan and Lesnick 2018), for multiparameter persistence modules (Lesnick 2015), and for more general persistence modules (Bubenik et al. 2015, 2017b; de Silva et al. 2017; Bjerkevik and Botnan 2017; Meehan and Meyer 2017a, b).

For sets of persistence modules, the interleaving distance induces a topology. The main goal of the research reported here is to study the basic topological properties of the resulting topological spaces.

Unfortunately, this research program runs into an immediate difficulty: the collection of persistence modules is not a set, but a proper class. While it is possible to consider this class with the interleaving distance (Bubenik et al. 2015, 2017a, b), here we want to work with actual topological spaces.

So to start, we consider various classes of persistence modules. These include classes that have been previously considered in theoretical work, such as pointwise finite-dimensional persistence modules (Crawley-Boevey 2015), q-tame persistence modules (Chazal et al. 2014), interval-decomposable persistence modules, ephemeral persistence modules (Chazal et al. 2016a), and constructible persistence modules (Patel 2018; Curry 2014), as well as classes of persistence modules that arise in applications, such as those decomposable into finitely many interval modules, where each interval lies in some fixed bounded closed interval.

We determine various relationships between these classes, such as inclusion (Fig. 1). We also identify pairs of classes where for each element of one, there is an element of the other that has interleaving distance 0 from the first (Sect. 3.3). We define and calculate an asymmetric distance we call enveloping distance that measures how far one needs to expand a given class to include another (Sect. 3.4). These two results are summarized in Fig. 2.

Next, we determine which of these classes are sets and which are proper classes. We show that the classes of interval-decomposable persistence modules and q-tame persistence modules are not sets (Corollary 5), though the classes of pointwise finite-dimensional persistence modules and persistence modules decomposable into countable-many interval modules are sets (Propositions 5 and 6). We introduce a set of persistence modules containing these two sets that consists of persistence modules decomposable into a set of interval modules with cardinality of the continuum (Definition 8 and Proposition 8).

For the remainder, we restrict ourselves to the identified sets of persistence modules and the topologies induced by the interleaving distance (Fig. 3). We identify which of the inclusions in Fig. 3 are inclusions of open sets (Proposition 9).

We show that these topological spaces are large and poorly behaved in the following ways. They do not have the \(T_0\) or Kolmogorov property (Corollary 6), they are not locally compact (Corollary 7), and their topological dimension is infinite (Corollary 10). In fact, we prove the following.

Cube Theorem

(Theorem 9) Let \(N \ge 1\). There exists an \(\varepsilon >0\) such that there is an isometric embedding of the cube \([0,\varepsilon ]^N\) with the \(L^{\infty }\) distance into each of our topological spaces of persistence modules.

On the other hand, our topological spaces of persistence modules do have the following nice properties. They are paracompact (Lemma 12), first countable (Lemma 15), and are compactly generated (Lemma 16).

We determine which of these topological spaces are separable (Theorems 4 and 5), as well as second countable and Lindelöf (Lemma 17). We show that the space of pointwise finite-dimensional persistence modules is not complete (Theorem 6), but that the space of persistence modules that are both q-tame and that decompose into countably-many intervals is complete (Theorem 7). We prove a Baire category theorem for complete extended pseudometric spaces (Theorem 8) that implies that this space is also a Baire space (Corollary 9).

We also identify the path components of the zero module in our topological spaces (Propositions 13 and 14), and show that they are contractible (Proposition 15).

Along the way, we observe the following mild strengthening of the structure theorem for persistent homology (Chazal et al. 2016a), which may be of independent interest.

Structure Theorem

(Theorem 3) The radical of a q-tame persistence module is a countable direct sum of interval modules.

1.1 Persistence modules and persistence diagrams

Topological data analysis tends to focus on persistence diagrams (Cohen-Steiner et al. 2007) rather than persistence modules. Readers more familiar with persistence diagrams may wonder why we work with persistence modules and what our results imply for persistence diagrams.

Let us present three responses. First, persistent homology produces persistence modules. In many but not all cases, these persistence modules may be represented by a persistence diagram. Mathematically, persistence modules are the fundamental object of study. Second, one of our main motivations was to develop a theory that could be extended to multiparameter persistence modules (Carlsson and Zomorodian 2009; Lesnick 2015) and generalized persistence modules (Bubenik et al. 2015, 2017b; de Silva et al. 2017). In this more general setting there is no hope for an analog of the persistence diagram. Third, our results for persistence modules may be used to obtain results for persistence diagrams as corollaries.

To be more precise, consider persistence modules that are pointwise finite-dimensional (see Sect. 3.1) with the interleaving distance. This forms an extended pseudometric space that we label \(\mathrm {(pfd)}\). If we take the quotient obtained by identifying persistence modules with zero interleaving distance, then we obtain an extended metric space that is isometric with a space of persistence diagrams with the bottleneck distance (Cohen-Steiner et al. 2007). This is the celebrated isometry theorem (Chazal et al. 2009, 2016b; Lesnick 2015; Bauer and Lesnick 2015; Bubenik and Scott 2014). Call this extended metric space \(\mathrm {(pd)}\).

Now \(\mathrm {(pd)}\) inherits many of the properties of \(\mathrm {(pfd)}\). Specifically, it is not totally bounded, any element of \(\mathrm {(pd)}\) does not have a compact neighborhood, it is not path connected, the path component of the empty persistence diagram consists of persistence diagrams without points with infinite persistence, and this path component is contractible. Furthermore, \(\mathrm {(pd)}\) is not separable and is not complete. In addition, for each N there is an \(\varepsilon >0\) such that there is an isometric embedding of the N-cube with diameter \(\varepsilon \) and the \(L^{\infty }\) distance into \(\mathrm {(pd)}\). So the topological dimension of \(\mathrm {(pd)}\) is infinite.

1.2 For the data scientist

For the reader primarily interested in topological data analysis, we would summarize our results by stating that the extended metric space of persistence diagrams with the bottleneck distance is “big”. Say we fix \(c<d\) and restrict ourselves to persistence diagrams with finitely many points \((a_i,b_i)\) each of which satisfies \(c \le a_i < b_i \le d\). This is a metric space. However, every neighborhood of every persistence diagram in this metric space is not compact. Also, the topological dimension of this metric space is infinite.

In order to apply certain statistical and machine learning tools, one may be tempted to start with a compact set of persistence diagrams. In light of these results, this is a drastic step.

1.3 Extended pseudometric spaces

The results presented here for extended pseudometric spaces are straight-forward extensions of the standard results for metric spaces (Lemmas 1215, 16, and 17 and Theorem 8). However, in order to keep the material accessible to applied mathematicians without a background in point-set topology, we include the proofs.

1.4 Related work

Mileyko et al. (2011) consider the set of persistence diagrams with countably many points in \({\mathbb {R}}^2\) together with the topology induced by the p-Wasserstein distance for \(1 \le p < \infty \). They show that the subspace consisting of persistence diagrams with finite distance to the empty persistence diagram is complete and separable. We show the corresponding space for the bottleneck distance \((p=\infty )\) is complete (Theorem 7) but not separable (Theorem 5). In a subsequent paper with Turner et al. (2014) they study geometric properties of the same set with a slightly different metric.

Blumberg et al. (2014) show that the set of persistence diagrams with finitely many points with the bottleneck distance is separable and that its Cauchy completion is separable. This completion is the set of persistence diagrams with the property that for every \(\varepsilon > 0\) there are only finitely many points with persistence at least \(\varepsilon \).

The authors have been informed of related work that is in preparation. Perea et al. (personal communication) have characterized (pre)compact sets of persistence diagrams with the bottleneck distance. Their results imply that compact sets have empty interior. Cruz (personal communication) has results on metric properties for generalized persistence diagrams with interleaving distance.

1.5 Organization of the paper

In Sect. 2, we provide background on persistence modules, indecomposable modules, interleaving distance, and pseudometric spaces. In Sect. 3, we define the classes of persistence modules that we consider, study the relationships between them, and identify which of them are sets. In Sect. 4, we study the basic topological properties of our topological spaces of persistence modules. Throughout, most of our arguments are elementary, except our proof of completeness which uses basic ideas from category theory. We also provide an appendix where we examine interleavings of interval modules.

2 Background

In this section we define persistence modules and interleaving distance, giving examples and basic properties. We also define extended pseudometric spaces and their induced topological spaces.

2.1 Persistence modules

Let \({\mathbf {k}}\) be a fixed field. A persistence module M is a set of \({\mathbf {k}}\)-vector spaces \(\{M(a) \ | \ a \in {\mathbb {R}}\}\) together with \({\mathbf {k}}\)-linear maps \(\{v_a^b: M(a) \rightarrow M(b) \ | \ a\le b \}\) such that
  1. (i)

    for all a, \(v_a^a : M(a) \rightarrow M(a)\) is the identity map, and

     
  2. (ii)

    if \(a \le b \le c\) then \(v_a^c=v_b^c\circ v_a^b\).

     
Equivalently, a persistence module is a functor \(M: \underline{\mathrm{R}}\rightarrow {\underline{\mathrm{Vect}}}_{\mathbf {k}}\), where \(\underline{\mathrm{R}}\) is the category whose set of objects is \({\mathbb {R}}\) and whose morphisms are the inequalities \(a \le b\), and \({\underline{\mathrm{Vect}}}_{\mathbf {k}}\) is the category of \({\mathbf {k}}\)-vector spaces and \({\mathbf {k}}\)-linear maps.

Example 1

Let X be a topological space and \(f: X \rightarrow {\mathbb {R}}\) be a function. For each \(a\in {\mathbb {R}}\) the subset
$$\begin{aligned} F_a:=\{x\in X \ | \ f(x)\le a\} \subset X \end{aligned}$$
is called a sublevel set. Note that \(a\le b\) implies \(F_a\subset F_b\) so that we have an inclusion map \(i_a^b: F_a \hookrightarrow F_b\) for all \(a\le b\). This inclusion map induces a linear map
$$\begin{aligned} H_n\left( i_a^b\right) : H_n(F_a;{\mathbf {k}}) \rightarrow H_n(F_b;{\mathbf {k}}) \end{aligned}$$
on singular homology groups with a coefficients in \({\mathbf {k}}\) of degree \(n\ge 0\). We thus have a persistence module \(HF: \underline{\mathrm{R}}\rightarrow {\underline{\mathrm{Vect}}}_{\mathbf {k}}\) given by \(HF(a) = H_n(F_a;{\mathbf {k}})\) and \(HF(a\le b) = H_n(i_a^b)\).

Example 2

Consider the half open interval [0, 2) in \({\mathbb {R}}\) and define the persistence module \(\chi : \underline{\mathrm{R}}\rightarrow {\underline{\mathrm{Vect}}}_{\mathbf {k}}\) given by
$$\begin{aligned} \chi (a) = {\left\{ \begin{array}{ll} {\mathbf {k}}&{} a \in [0,2) \\ 0 &{} \text {otherwise} \end{array}\right. } \quad \text {and} \quad \chi (a\le b) = {\left\{ \begin{array}{ll} 1&{} a,b \in [0,2) \\ 0 &{} \text {otherwise} \end{array}\right. } \end{aligned}$$
where \(1\) is the identity map on \({\mathbf {k}}\). For simplicity, we will abuse notation and denote this persistence module by [0, 2).

Example 3

Replacing [0, 2) in the above with an arbitrary interval \(J \subset {\mathbb {R}}\) we obtain a persistence module that we call an interval module and we will also denote by J.

Example 4

A trivial but important example is the zero module, denoted \(0\), that has \(0(a) = 0\) for all a.

A morphism of persistence modules M and N is a collection of linear maps \(\{\varphi _a: M(a) \rightarrow N(a) \ | \ a \in {\mathbb {R}}\}\) such that the following diagram commutes for each pair \(a\le b\).
Equivalently, a morphism of persistence modules is a natural transformation \(\varphi :M \Rightarrow N\). We will often denote a morphism of persistence modules as \(\varphi :M \rightarrow N\). Such a morphism is an isomorphism if and only if each linear map \(\varphi _a\) is an isomorphism.

Example 5

It is a good exercise to check that because of the constraints due to the commutative squares in (2.1), there is a nonzero morphism from the interval module [ab) to the interval module [cd) only if \(c \le a \le d \le b\).

In the appendix, we present a more thorough discussion of interval modules (Section A.1) and maps between them (Section A.2).

2.2 Indecomposables

Given two persistence modules M and N, their direct sum is the persistence module \(M \oplus N\) given by \((M \oplus N)(a) = M(a) \oplus N(a)\) and \((M \oplus N)(a \le b) = M(a \le b) \oplus N(a \le b)\). In the same way we can define the direct sum of a collection of persistence modules indexed by an arbitrary set.

A persistence module is said to be indecomposable if it is not isomorphic to a nontrivial direct sum. For example, interval modules are indecomposable. However, not all indecomposable persistence modules are interval modules [see Chazal et al. (2016b), Theorem 2.5, Remark 2.6; for a discussion of examples due to do (Webb 1985), Lesnick, and Crawley-Boevey].

A special case of the following theorem follows from work of Gabriel (1972), but the general case was proved by Crawley-Boevey (2015).

Theorem 1

(Structure Theorem) Let \(M: \underline{\mathrm{R}}\rightarrow {\underline{\mathrm{Vect}}}_{\mathbf {k}}\) be a persistence module. If M(a) is finite dimensional for each \(a\in {\mathbb {R}}\), then M is isomorphic to a direct sum of interval modules.

2.3 Interleaving distance

Interleaving distance was introduced in Chazal et al. (2009) and further studied in the context of multiparameter persistence in Lesnick (2015). Here we also adopt the categorical point of view from Bubenik and Scott (2014).

Definition 1

Let \(\varepsilon \ge 0\). An \(\varepsilon \)-interleaving between persistence modules M and N consists of morphisms \(\varphi _a: M(a) \rightarrow N(a + \varepsilon )\) and \(\psi _a:N(a)\rightarrow M(a+\varepsilon )\) for all a such that the following four diagrams commute for all \(a \le b\), where the horizontal maps are given by the respective persistence modules.
Equivalently, we may describe this in terms of natural transformations. First, for \(x \in {\mathbb {R}}\) let \(T_{x}: \underline{\mathrm{R}}\rightarrow \underline{\mathrm{R}}\) denote the functor given by \(T_x(a) = a + x\). Next if \(x \ge 0\), let \(\eta _x: 1_{\underline{\mathrm{R}}} \Rightarrow T_{x}\) denote the natural transformation from the identity functor on \(\underline{\mathrm{R}}\) to \(T_x\) that has components \((\eta _x)_a: a \le a+x\). Then an \(\varepsilon \)-interleaving consists of natural transformations \(\varphi : M \Rightarrow N T_{\varepsilon }\) and \(\psi : N \Rightarrow M T_{\varepsilon }\) such that \((\psi T_{\varepsilon }) \varphi = M \eta _{2\varepsilon }\) and \((\varphi T_{\varepsilon }) \psi = N \eta _{2\varepsilon }\). See Bubenik and Scott (2014), Section 3 for more details. We say M and N are \(\varepsilon \)-interleaved.

Remark 1

Two persistence modules are 0-interleaved if and only if they are isomorphic. If persistence modules M and N are \(\varepsilon \)-interleaved and N and P are \(\delta \)-interleaved then M and P are \((\varepsilon +\delta )\)-interleaved.

Definition 2

Let M and N be two persistence modules. Then the interleaving distance \(d_I(M,N)\) between M and N is defined as
$$\begin{aligned} d_I(M,N):= \inf \big (\varepsilon \in [0,\infty ) \ | \ M \ \text {and} \ N \ \text {are} \ \ \varepsilon \text {-interleaved} \ \big ) \end{aligned}$$
If no such \(\varepsilon \) exists, then \(d_I(M,N)=\infty \).

Example 6

The interval modules [0, 2] and (0, 2) are not 0-interleaved. In fact, there are no nonzero maps between [0, 2] and (0, 2). However they are \(\varepsilon \)-interleaved for all \(\varepsilon >0\). Thus, \(d_I([0,2],(0,2)) = 0\).

Example 7

The interval modules \(M=[0,1)\) and \(N=[0,\infty )\) are not \(\varepsilon \)-interleaved for any \(\varepsilon \ge 0\). Indeed, assume \(\varphi \) and \(\psi \) provide such an interleaving. Consider the following trapezoid.
It decomposes into a commutative parallelogram and commutative triangle from (2.2) and (2.3) in two different ways. In either case, this diagram commutes. Furthermore, the bottom horizontal arrow is the identity on \({\mathbf {k}}\) and the top horizontal arrow is 0, which is a contradiction.

In the “Appendix”, we give a careful study of interleavings of interval modules (Section A.3).

We will make use of the following lemma without reference.

Lemma 1

(Converse Algebraic Stability Theorem (Lesnick 2015, Theorem 3.4)) Let \(\varepsilon \ge 0\). If for all \(\alpha \in A\), the persistence modules \(I_{\alpha }\) and \(J_{\alpha }\) are \(\varepsilon \)-interleaved, then \(\bigoplus _{\alpha \in A} I_{\alpha }\) and \(\bigoplus _{\alpha \in A} J_{\alpha }\) are \(\varepsilon \)-interleaved. Thus \(d_I(\bigoplus _{\alpha \in A} I_{\alpha },\bigoplus _{\alpha \in A} J_{\alpha }) \le \sup _{\alpha \in A}d_I(I_{\alpha },J_{\alpha })\).

Proof

For \(\alpha \in A\), let \(\varphi _{\alpha }\) and \(\psi _{\alpha }\) be maps giving an \(\varepsilon \)-interleaving of \(I_{\alpha }\) and \(J_{\alpha }\). Then \(\bigoplus \varphi _{\alpha }\) and \(\bigoplus \psi _{\alpha }\) provide the desired \(\varepsilon \)-interleaving. \(\square \)

2.4 Pseudometric spaces

Definition 3

A pseudometric on a set X is a map \(d: X\times X \rightarrow [0,\infty )\) that satisfies
  1. M1)

    \(d(x,x)=0\),

     
  2. M2)

    \(d(x,y)=d(y,x)\), and

     
  3. M3)

    \(d(x,y)\le d(x,z)+d(z,y)\)

     
for all \(x,y,z \in X\). Note that we have omitted the condition \(d(x,y)=0\) implies \(x=y\) required of metric. More generally, an extended pseudometric on X is a map \(d: X \times X \rightarrow [0,\infty ]\) satisfying the same three axioms. We call a set with an (extended) pseudometric an (extended) pseudometric space.

Theorem 2

((Chazal et al. 2009; Lesnick 2015; Bubenik and Scott 2014)) The interleaving distance is an extended pseudometric on any set of (isomorphism classes of) persistence modules.

Remark 2

A proper class of persistence modules with the interleaving distance is not an extended pseudometric space since it is not a set. However it is a symmetric Lawvere space (Bubenik et al. 2015, 2017a, b).

In an extended (pseudo)metric space, the condition \(d(x,y)<\infty \) defines an equivalence relation. As a result, such a space has a natural partition into (pseudo)metric spaces.

In an (extended) pseudometric space one can consider equivalence classes of the equivalence relation \(x \sim y\) if \(d(x,y)=0\) to obtain an (extended) metric space. However, for persistence modules, one may be interested in distinguishing nonisomorphic modules with zero interleaving distance, so we will not apply this simplification.

Any extended pseudometric on a set induces a topology on it. Indeed, for any \(x\in X\) and a real number \(r>0\) consider the open ball \(B_r(x)\) centered at x with radius r,
$$\begin{aligned} B_r(x):=\{y\in X \ | \ d(x,y)<r \}. \end{aligned}$$
We call a set O open in X if for each \(x\in O\), there exists \(r>0\) such that \(B_r(x)\subset O\). Then it is easy to check that the collection of all open sets is a topology on X.

Note that each open ball \(B_r(x)\) is also an open set in X and the collection of all open balls forms a base for this topology X since each open set O in X can be written as a union of open balls.

Example 8

Consider the interval module [0, 5) and let \(\varepsilon >1\). Then the ball \(B_{\varepsilon }([0,5))\) contains the interval modules \([-\,1,6]\) and (1, 4).

In the appendix, we study the interval modules in an \(\varepsilon \)-neighborhood of an interval module (Section A.4).

A sequence \((x_n)_{n \ge 1}\) in an extended pseudometric space X is said to converge to \(x \in X\) if for all \(\varepsilon > 0\) there exists \(N>0\) such that for all \(n \ge N\), \(d(x_n,x) < \varepsilon \). The point x is called a limit of the sequence. Note that in an extended pseudometric space we no longer have unique limits, but we do have that if x and \(x'\) are limits, then by the triangle inequality \(d(x,x') = 0\).

A sequence \((x_n)_{n \ge 1}\) in an extended pseudometric space is a Cauchy sequence if for all \(\varepsilon >0\) there exists an \(N>0\) such that for all \(n,m \ge N\), \(d(x_n,x_m)< \varepsilon \). If a subsequence of a Cauchy sequence has a limit x, then by the triangle inequality, x is also a limit of the Cauchy sequence.

3 Sets and classes of persistence modules

In this section we define classes of persistence modules that contain many of the persistence modules considered in the literature. We study the relationships between these classes and determine which of them are in fact sets.

For the remainder of the paper, we will only consider isomorphism classes of persistence modules. That is, whenever we say ‘persistence module’, we really mean ‘isomorphism class of persistence modules’. This is standard when discussing both vector spaces and persistence modules.

3.1 Classes of persistence modules

In this section, we consider the classes of persistence modules in Fig. 1, which we now describe.
Fig. 1

Hasse diagram of sets and classes of persistence modules

  • \(\mathrm {(pm)}\) is the class of persistence modules.

  • \(\mathrm {(id)}\) is the class of interval-decomposable persistence modules: those isomorphic to \(\bigoplus _{\alpha \in A} I_{\alpha }\), where A is some indexing set, and each \(I_{\alpha }\) is an interval module.

  • \(\mathrm {(cid)}\), the countably interval-decomposable persistence modules, is the subclass of \(\mathrm {(id)}\) where the index set A is countable.

  • \(\mathrm {(cfid)}\), the countably finite-interval decomposable persistence modules, is the subclass of \(\mathrm {(cid)}\) in which each interval \(I_{\alpha }\) is finite.

  • \(\mathrm {(fid)}\), the finitely interval-decomposable persistence modules, is the class of persistence modules isomorphic to \(\bigoplus _{k=1}^N I_k\) for some N, where each \(I_k\) is an interval module.

  • \(\mathrm {(ffid)}\), the finitely finite-interval decomposable persistence modules, is the subclass of \(\mathrm {(fid)}\) in which each \(I_k\) is a finite interval.

  • Given \(c<d\), \(\mathrm {(ffid^{[c,d]})}\) is the subclass of \(\mathrm {(ffid)}\) in which each \(I_k \subset [c,d]\).

  • \(\mathrm {(pfd)}\), the pointwise finite dimensional persistence modules, is the class of all persistence modules M with each M(a) finite dimensional.

  • \(\mathrm {(qtame)}\), the q-tame persistence modules, is the class of all persistence modules M where each \(a<b\) the linear map \(v_a^b: M(a) \rightarrow M(b)\) has a finite rank.

  • \(\mathrm {(eph)}\), the ephemeral persistence modules, is the class of all persistence modules M where for each \(a<b\) the linear map \(v_a^b: M(a) \rightarrow M(b)\) is zero.

  • \(\mathrm {(0)}\) is the class consisting of only the zero persistence module.

Remark 3

The class \(\mathrm {(fid)}\) is a slight generalization of the class of constructible persistence modules. A persistence module M is said to be constructible (Patel 2018) if there exists a finite subset \(A=\{a_1,\ldots ,a_n\}\) of \({\mathbb {R}}\) such that
  • for \(t< a_1\), \(M(t)=0\),

  • for \(a_i \le s \le t < a_{i+1}\), \(M(s\le t)\) is an isomorphism where \(i\in \{1,\ldots ,n-1\}\) , and

  • for \(a_n\le s\le t\), \(M(s\le t)\) is an isomorphism.

A constructible module M, satisfies \(M \cong \bigoplus _{k=1}^N I_k\) where each \(I_k\) is of the form \([a_i,a_j)\) or \([a_i,\infty )\).1

3.2 Inclusions

Lemma 2

Let M be an ephemeral module. Then \(M \cong \bigoplus _{\alpha \in A} M_{\alpha }\), where each \(M_{\alpha } \cong [r,r]\) for some \(r \in {\mathbb {R}}\).

Proof

Let \(M \in \mathrm {(eph)}\). For \(r \in {\mathbb {R}}\), let \(M_r\) be the persistence module with \(M_r(x) = M(r)\) if \(x=r\) and otherwise \(M_r(x)=0\). Then \(M \cong \oplus _{r \in {\mathbb {R}}} M_r\). Furthermore each M(r) has a basis, so \(M_r\) decomposes over this basis into [rr] interval modules. \(\square \)

Proposition 1

The diagram in Fig. 1 is a Hasse diagram for the poset structure of these classes of persistence modules under the inclusion order.

Proof

By Theorem 1, \(\mathrm {(pfd)}\) is in \(\mathrm {(id)}\). By Lemma 2, \(\mathrm {(eph)}\subset \mathrm {(id)}\). It is easy to check that all of the other arrows indicated in the diagram are inclusions and that in fact all of the inclusions are proper. With the observation that if \(A \subset B\), \(C \subset D\) and \(A \not \subset D\) then \(B \not \subset C\), it remains to check the following cases.
  1. 1.

    \(\mathrm {(eph)}\not \subset \mathrm {(pfd)}\): \(\bigoplus _{k=1}^{\infty } [0,0]\) is in \(\mathrm {(eph)}\) but not in \(\mathrm {(pfd)}\).

     
  2. 2.

    \(\mathrm {(eph)}\not \subset \mathrm {(cid)}\): \(\bigoplus _{r \in {\mathbb {R}}} [0,0]\) is in \(\mathrm {(eph)}\) but not in \(\mathrm {(cid)}\).

     
  3. 3.

    \(\mathrm {(ffid^{[c,d]})}\not \subset \mathrm {(eph)}\): [cd] is in \(\mathrm {(ffid^{[c,d]})}\) but is not in \(\mathrm {(eph)}\).

     
  4. 4.

    \(\mathrm {(fid)}\not \subset \mathrm {(cfid)}\): \([0,\infty )\) is in \(\mathrm {(fid)}\) but is not in \(\mathrm {(cfid)}\).

     
  5. 5.

    \(\mathrm {(pfd)}\not \subset \mathrm {(cid)}\): \(\bigoplus _{r \in {\mathbb {R}}} [r,r]\) is in \(\mathrm {(pfd)}\) but is not in \(\mathrm {(cid)}\).

     
  6. 6.

    \(\mathrm {(cfid)}\not \subset \mathrm {(qtame)}\): \(\bigoplus _{k=1}^{\infty }[0,1)\) is in \(\mathrm {(cfid)}\) but is not in \(\mathrm {(qtame)}\).

     
  7. 7.

    \(\mathrm {(qtame)}\not \subset \mathrm {(id)}\): \(\prod _{k=1}^{\infty }[0,\frac{1}{k})\) is in \(\mathrm {(qtame)}\) but is not in \(\mathrm {(id)}\) (Chazal et al. 2016a).

     
\(\square \)

3.3 Almost inclusions

Definition 4

Say that a class of persistence modules \({\mathcal {A}}\) almost includes in a class of persistence modules \({\mathcal {B}}\) if for each \(A \in {\mathcal {A}}\) there exists an element \(B \in {\mathcal {B}}\) such that \(d_I(A,B) = 0\).

Lemma 3

A finite sequence of inclusions and almost inclusions is an almost inclusion.

Proof

This follows from the triangle inequality. \(\square \)

Lemma 4

M is an ephemeral persistence module if and only if \(d_I(M,0)=0\). That is, \(\mathrm {(eph)}\) almost includes in \(\mathrm {(0)}\).

Proof

Let M be an ephemeral persistence module. Then M and \(0\) are \(\varepsilon \)-interleaved for all \(\varepsilon >0\) by the zero maps.

Next assume \(d_I(M,0)=0\). Consider \(a<b\). Let \(\varepsilon = \frac{b-a}{2}\). Since M and \(0\) are \(\varepsilon \)-interleaved, the map \(M(a<b)\) factors through 0, and is thus the zero map.
Therefore M is an ephemeral persistence module. \(\square \)

For a persistence module M, define the radical of M by \(({{\mathrm{rad}}}M)(a) = \sum _{c< a} {{\mathrm{im}}}M(c<a)\) (Chazal et al. 2016a). Note that \({{\mathrm{rad}}}M \subset M\) and inherits the structure of a persistence module.

Proposition 2

Let M be a persistence module. Then \(d_I(M,{{\mathrm{rad}}}M) = 0\).

Proof

Let \(\varepsilon >0\). For all \(a \in {\mathbb {R}}\), let \(\varphi _a = M(a<a+\varepsilon ): ({{\mathrm{rad}}}M)(a) \rightarrow M(a+\varepsilon )\), and let \(\psi _a = M(a<a+\varepsilon ): M(a) \rightarrow ({{\mathrm{rad}}}M)(a+\varepsilon )\). Then by the functoriality of M, this is an \(\varepsilon \)-interleaving of \({{\mathrm{rad}}}M\) and M. Therefore \(d_I({{\mathrm{rad}}}M, M) = 0\). \(\square \)

Theorem 3

Let \(M \in \mathrm {(qtame)}\). Then \({{\mathrm{rad}}}M \in \mathrm {(qtame)}\) and \({{\mathrm{rad}}}M \in \mathrm {(cid)}\).

Proof

Let \(M \in \mathrm {(qtame)}\). Since \({{\mathrm{rad}}}M\) is a submodule of M, it follows that \({{\mathrm{rad}}}M \in \mathrm {(qtame)}\) as well. By (Chazal et al. 2016a, Corollary 3.6), \({{\mathrm{rad}}}M \in \mathrm {(id)}\). We will strengthen this to show that \({{\mathrm{rad}}}M \in \mathrm {(cid)}\).

Since \({{\mathrm{rad}}}M \in \mathrm {(id)}\), \({{\mathrm{rad}}}M \cong \bigoplus _{\alpha \in A} I_{\alpha }\). For \(q,r \in {\mathbb {Q}}\) with \(q<r\), let \(A_{q,r} = \{ \alpha \in A \mid q,r \in I_{\alpha }\}\), and let \(A' = \bigcup _{q<r \in {\mathbb {Q}}} A_{q,r}\). Since \({{\mathrm{rad}}}M \in \mathrm {(qtame)}\), for each \(q<r \in {\mathbb {Q}}\), \(|A_{q,r}| < \infty \). Therefore \(A'\) is countable.

Furthermore, by definition, for each \(a \in {\mathbb {R}}\) and for each \(x \in ({{\mathrm{rad}}}M)(a)\) there exists \(c<a\) and \(y \in M(c)\) such that \(M(c\le a)(y) = x\). Choose \(b \in (c,a)\). Then \(z:= M(c\le b)(y) \in ({{\mathrm{rad}}}M)(b)\) and \(({{\mathrm{rad}}}M)(b\le a)(z) = x\). Hence the interval decomposition of \({{\mathrm{rad}}}M\) does not contain any one-point intervals, and thus \(A = A'\). Therefore \({{\mathrm{rad}}}M \in \mathrm {(cid)}\). \(\square \)

Combining the previous two results we have the following.

Corollary 1

Let \(M \in \mathrm {(qtame)}\). Then there exists \(N \in \mathrm {(cid)}\) such that \(d_I(M,N) = 0\). That is, \(\mathrm {(qtame)}\) almost includes in \(\mathrm {(cid)}\).

3.4 Enveloping distance

In this section, we define a non-symmetric distance between classes of persistence modules and calculate its value for all of the pairs in Fig. 1.

Definition 5

Let \({\mathcal {A}}\) and \({\mathcal {B}}\) be classes of persistence modules. We define the enveloping distance from \({\mathcal {A}}\) to \({\mathcal {B}}\) as follows.
$$\begin{aligned} E({\mathcal {A}},{\mathcal {B}}) = \inf (r \mid \forall B \in {\mathcal {B}}\text { and } s > r, \exists A \in {\mathcal {A}}\text { such that } A,B \text { are { s}-interleaved}) \end{aligned}$$
If there is no such r, we set \(E({\mathcal {A}},{\mathcal {B}}) = \infty \).

For example, as we will demonstrate later in this section, \(E(\mathrm {(0)},\mathrm {(ffid^{[c,d]})}) = \frac{d-c}{2}\) and \(E(\mathrm {(ffid^{[c,d]})},\mathrm {(0)}) = 0\).

We will use the following basic fact about interleavings.

Lemma 5

(Lesnick 2015; Bubenik and Scott 2014) If persistence modules A and B are s-interleaved and persistence modules B and C are t-interleaved, then A and C are \((s+t)\)-interleaved.

The enveloping distance has the following properties.

Lemma 6

\(E({\mathcal {A}},{\mathcal {A}}) = 0\) and \(E({\mathcal {A}},\mathcal {C}) \le E({\mathcal {A}},{\mathcal {B}}) + E({\mathcal {B}},\mathcal {C})\).

Proof

For reflexivity, each persistence module is s-interleaved with itself for all \(s \ge 0\). The triangle inequality follows from Lemma 5. \(\square \)

Definition 6

In the case that \(E({\mathcal {A}},{\mathcal {B}}) = \infty \), we write that \(E({\mathcal {A}},{\mathcal {B}}) = \infty ^{-}\) if \(\forall B \in {\mathcal {B}}\ \exists s\) and \(A \in {\mathcal {A}}\) such that AB are s-interleaved. From now on we reserve \(E({\mathcal {A}},{\mathcal {B}})=\infty \) for the case that this condition is not satisfied.

Lemma 7

If \({\mathcal {A}}\) (almost) includes in \({\mathcal {B}}\) then \(E({\mathcal {B}},{\mathcal {A}}) = 0\).

Proof

This follows immediately from the definitions. \(\square \)

Corollary 2

\(E(\mathrm {(0)},\mathrm {(eph)}) = 0\) and \(E(\mathrm {(eph)},\mathrm {(0)}) = 0\).

Lemma 8

If \({\mathcal {A}}\) (almost) includes in \({\mathcal {B}}\), \(E({\mathcal {B}},\mathcal {C}) = \infty \), and \(\mathcal {C}\) (almost) includes in \({\mathcal {D}}\), then \(E({\mathcal {A}},{\mathcal {D}}) = \infty \).

Proof

Assume \(E({\mathcal {A}},{\mathcal {D}}) < \infty \). Then there is some \(s \ge 0\) such that for all \(D \in {\mathcal {D}}\) there exists an \(A \in {\mathcal {A}}\) such that D and A are s-interleaved.

Let \(\varepsilon >0\). Let \(C \in \mathcal {C}\). Since \(\mathcal {C}\) (almost) includes in \({\mathcal {D}}\), there is a \(D \in {\mathcal {D}}\) such that C and D are \(\varepsilon \)-interleaved. By our first observation, there is an \(A \in {\mathcal {A}}\) such that D and A are s-interleaved. Since \({\mathcal {A}}\) (almost) includes in \({\mathcal {B}}\), there is a \(B \in {\mathcal {B}}\) such that A and B are \(\varepsilon \)-interleaved. Therefore by Remark 1, C and B are \((s+2\varepsilon )\)-interleaved. So for all \(C \in \mathcal {C}\) there is a \(B \in B\) such that C and B are \((s+2\varepsilon )\)-interleaved. Thus \(E({\mathcal {B}},\mathcal {C})<\infty \). \(\square \)

Proposition 3

  1. 1.

    We have the following enveloping distances: \(E(\mathrm {(0)},\mathrm {(ffid^{[c,d]})}) = \frac{d-c}{2}\) and \(E(\mathrm {(ffid^{[c,d]})},\mathrm {(ffid)}) = \infty ^-\). Also, \(E(\mathrm {(0)},\mathrm {(ffid)}) = \infty ^-\), \(E(\mathrm {(eph)},\mathrm {(ffid)}) = \infty ^-\) and \(E(\mathrm {(eph)},\mathrm {(ffid^{[c,d]})}) = \frac{d-c}{2}\).

     
  2. 2.

    In addition, \(E(\mathrm {(cfid)},\mathrm {(fid)})=\infty \) and \(E(\mathrm {(qtame)},\mathrm {(cfid)})=\infty \).

     
  3. 3.

    Furthermore, \(E(\mathrm {(pfd)},\mathrm {(qtame)})=0\).

     
  4. 4.

    With the exception of \(\mathrm {(0)}\subset \mathrm {(eph)}\), \(\mathrm {(0)}\subset \mathrm {(ffid^{[c,d]})}\), and \(\mathrm {(ffid^{[c,d]})}\subset \mathrm {(ffid)}\), all of the other inclusions \({\mathcal {A}}\subset {\mathcal {B}}\) in Fig. 1 have enveloping distance \(E({\mathcal {A}},{\mathcal {B}})=\infty \). Also \(E(\mathrm {(qtame)},\mathrm {(cid)}) = \infty \).

     

Proof

  1. 1.
    • \(\mathrm {(0)}\subset \mathrm {(ffid^{[c,d]})}\): \(d_I([c,d],0) = \frac{d-c}{2}\) and for all \(M \in \mathrm {(ffid^{[c,d]})}\), \(d_I(M,0) \le \frac{d-c}{2}\).

    • \(\mathrm {(ffid^{[c,d]})}\subset \mathrm {(ffid)}\): For all \(M \in \mathrm {(ffid^{[c,d]})}\) and \(N \in \mathrm {(ffid)}\), \(d_I(M,N) \le d_I(M,0) + d_I(0,N) < \infty \). Let \(z\ge 0\). For all \(M \in \mathrm {(ffid^{[c,d]})}\), there are no nontrivial maps from M to \((d,d+2z]\). Thus \(d_I(M,(d,d+2z]) \ge d_I((d,d+2z],0) \ge z\).

    • The other three cases follow from the same arguments.

     
  2. 2.
    • \(\mathrm {(cfid)}\) to \(\mathrm {(fid)}\): Consider \([0,\infty )\).

    • \(\mathrm {(qtame)}\) to \(\mathrm {(cfid)}\): Consider \(\bigoplus _{k=1}^{\infty } [0,k)\).

     
  3. 3.

    This case is more interesting and its proof will occupy the remainder of this subsection below.

     
  4. 4.
    • \(\mathrm {(ffid)}\subset \mathrm {(fid)}\): Consider \([0,\infty )\).

    • \(\mathrm {(ffid)}\subset \mathrm {(cfid)}\): Consider \(\bigoplus _{k=1}^{\infty } [0,k)\).

    • \(\mathrm {(cfid)}\subset \mathrm {(cid)}\): Consider \([0,\infty )\).

    • \(\mathrm {(fid)}\subset \mathrm {(cid)}\): Consider \(\bigoplus _{k=1}^{\infty } [0,\infty )\).

    • \(\mathrm {(fid)}\subset \mathrm {(pfd)}\): Consider \(\bigoplus _{k=0}^{\infty } [2^k,2^{k+1})\).

    • \(\mathrm {(cid)}\subset \mathrm {(id)}\): Consider \(\bigoplus _{r \in {\mathbb {R}}} [0,\infty )\).

    • \(\mathrm {(id)}\subset \mathrm {(pm)}\): Consider \(\prod _{k=1}^{\infty } [0,\infty )\).

    • \(\mathrm {(eph)}\subset \mathrm {(id)}\), \(\mathrm {(eph)}\subset \mathrm {(qtame)}\), \(\mathrm {(pfd)}\subset \mathrm {(id)}\), \(\mathrm {(qtame)}\subset \mathrm {(cid)}\), and \(\mathrm {(qtame)}\subset \mathrm {(pm)}\) follow from Lemma 8. \(\square \)

     

Remark 4

Together with Corollary 2, Lemmas 7 and 8, this proposition implies all of the pairwise enveloping distances between the sets and classes of persistence modules in Fig. 2. For example, \(E(\mathrm {(id)},\mathrm {(qtame)})=0\), \(E(\mathrm {(cid)},\mathrm {(pfd)})=0\), and \(E(\mathrm {(cid)},\mathrm {(qtame)})=0\) by Lemmas 7 and 3, and \(E(\mathrm {(fid)},\mathrm {(cfid)})=\infty \) by Lemma 8.

We end this section by showing that \(E(\mathrm {(pfd)},\mathrm {(qtame)})=0\). First we give a definition.

Definition 7

Let M be a persistence module. Let \(p\ge 0\). We define the p-persistent submodule of M by
$$\begin{aligned} M^{(p)}(a) = {{\mathrm{im}}}M(a-p \le a). \end{aligned}$$
For \(a \le b\), there is an induced map between objects \(M^{(p)}(a)\) and \(M^{(p)}(b)\) given by \(M(a \le b)\). Since M is a persistence module, so is \(M^{(p)}\), and since \(M^{(p)}(a)\) is a sub-vector space of M(a) for all a, \(M^{(p)}\) is a submodule of M.

Proposition 4

Let M be a persistence module and let \(p\ge 0\). Then M and \(M^{(p)}\) are p-interleaved.

Proof

For \(a \in \underline{\mathrm{R}}\), define \(\varphi _a:M(a) \rightarrow M^{(p)}(a+p)\) by \(\varphi _a = M(a\le a+p)\), and \(\psi _a:M^{(p)}(a) \rightarrow M(a+p)\) by \(\psi _a = M(a \le a+p)\). Then all the arrows in diagrams (2.2) and (2.3) are maps in M and hence commute. \(\square \)

Corollary 3

\(E(\mathrm {(pfd)},\mathrm {(qtame)})=0\).

Proof

Let M be a q-tame persistence module. Let \(p>0\). Then by definition, \(M^{(p)}\) is a pointwise finite-dimensional persistence module. By Proposition 4, M and \(M^{(p)}\) are p-interleaved. Thus, by definition, \(E(\mathrm {(pfd)},\mathrm {(qtame)})=0\). \(\square \)

Fig. 2

Diagram of sets and classes of persistence modules. Solid arrows indicate inclusions, dashed arrows indicate almost inclusions, and dotted arrows do not indicate any relationship. Annotations of arrows indicate enveloping distance from the source to the target, given in Definitions 5 and 6

3.5 Sets of persistence modules

Next we consider whether the classes defined above are sets or proper classes. We will use the following notation. Let \(\overline{{\mathbb {R}}}:={\mathbb {R}}\cup \{\pm \infty \}\) and \(\overline{{\mathbb {N}}}:={\mathbb {N}}\cup \{\infty \}\). Given a set X, let \({\mathcal {P}}(X)\) denote its power set. Let \({\mathbb {I}}\) be the set of all intervals in \({\mathbb {R}}\). We define a map, \(f: {\mathbb {I}} \longrightarrow \{1,2,3,4\}\) by
$$\begin{aligned} f(I) = {\left\{ \begin{array}{ll} 1, &{} \inf I \not \in I, \sup I \not \in I \\ 2, &{} \inf I \in I, \sup I \not \in I \\ 3, &{} \inf I \not \in I, \sup I \in I \\ 4, &{} \inf I \in I, \sup I \in I. \end{array}\right. } \end{aligned}$$

Proposition 5

The class \(\mathrm {(cid)}\) is a set.

Proof

Consider the map
$$\begin{aligned} \mathrm {(cid)}\longrightarrow {\mathcal {P}}(\overline{{\mathbb {R}}}^2\times \{1,2,3,4\} \times \overline{{\mathbb {N}}}) \end{aligned}$$
defined by
$$\begin{aligned} \bigoplus _{\alpha \in A} I_{\alpha } \longmapsto \bigcup _{\alpha \in A} \big [ \{(\inf I_{\alpha },\sup I_{\alpha })\} \times \{f(I_{\alpha })\} \times \{m(i)\} \big ] \end{aligned}$$
where m(i) denotes the multiplicity of the direct summand \(I_{\alpha }\). This map is an injection, hence \(\mathrm {(cid)}\) is a set. \(\square \)

Corollary 4

Therefore the classes \(\mathrm {(cfid)}\), \(\mathrm {(fid)}\), \(\mathrm {(ffid)}\), \(\mathrm {(ffid^{[c,d]})}\), and \(\mathrm {(0)}\) are also sets.

Lemma 9

Each interval appears only finitely many times in the direct-sum interval-module decomposition of a pointwise finite-dimensional persistence module.

Proof

For each interval I, \(\bigoplus _{k=1}^{\infty } I \not \in \mathrm {(pfd)}\). \(\square \)

Proposition 6

The class \(\mathrm {(pfd)}\) is a set.

Proof

Let \(M \in \mathrm {(pfd)}\). By Theorem 1, \(M \cong \bigoplus _{\alpha \in A} I_{\alpha }\) where \(I_{\alpha }\) is an interval and A is a set. By Lemma 9, we can define the following map.
$$\begin{aligned}&\mathrm {(pfd)}\longrightarrow {\mathcal {P}}(\overline{{\mathbb {R}}}^2\times \{1,2,3,4\} \times {\mathbb {N}})\\&\bigoplus _{\alpha \in A} I_{\alpha } \longmapsto \bigcup _{\alpha \in A} \big [ \{(\inf I_{\alpha },\sup I_{\alpha })\} \times \{f(I_{\alpha })\} \times \{m(i)\} \big ] \end{aligned}$$
where m(i) denotes the multiplicity of the direct summand \(I_{\alpha }\). This map is an injection, hence \(\mathrm {(pfd)}\) is a set. \(\square \)

Proposition 7

The class \(\mathrm {(eph)}\) is not a set.

Proof

For a cardinal c, let \(F_c = \bigoplus _{\alpha \in c}[0,0]\). That is, \(F_c\) is the \({\mathbf {k}}\)-vector space generated by c. For \(c \not \cong d\), \(F_c \not \cong F_d\). Thus we have an injection from the proper class of cardinals into \(\mathrm {(eph)}\). \(\square \)

Corollary 5

Since \(\mathrm {(eph)}\) is not a set, neither are \(\mathrm {(id)}\)  \(\mathrm {(qtame)}\) and \(\mathrm {(pm)}\).

3.6 Interval-decomposable persistence modules of arbitrary cardinality

Motivated by the desire to have a set of persistence modules that contains all of the sets of persistence modules in Sect. 3.5 and the proofs of Proposition 5 and 6, we make the following definition.

Definition 8

Given a cardinal \(\kappa \), let \({(\kappa -\mathrm{id}})\) denote the class of persistence modules isomorphic to \(\bigoplus _{\alpha \in A} I_{\alpha }\) where \(I_{\alpha }\) is an interval module and the cardinality of A is at most \(\kappa \). As a special case, and to avoid confusion with our previously defined notation, let \(\mathrm {(rid)}\) denote the class of interval-decomposable persistence modules with at most the cardinality of \({\mathbb {R}}\)-many summands.

By definition, \(\mathrm {(cid)}\subset \mathrm {(rid)}\) and by Lemma 9, \(\mathrm {(pfd)}\subset \mathrm {(rid)}\).

Proposition 8

For any cardinal \(\kappa \), the class \({(\kappa -\mathrm{id}})\) is a set.

Proof

The proof is the same as the proof of Proposition 5, replacing \(\overline{{\mathbb {N}}}\) with \(\kappa \). \(\square \)

4 Topological properties

Since we are interested in studying topological spaces of persistence modules, we will for the most part restrict ourselves to the sets in Fig. 3. We will consider the basic topological properties of these sets with the topology induced by the interleaving metric.
Fig. 3

Sets of metric spaces, each with the topology induced by the interleaving metric

4.1 Open subsets

In this section we consider which of the inclusion maps in Fig. 3 are inclusions of open subsets. Recall that in a pseudometric space X, a subset \(A\subset X\) is said to be open if for all \(a \in A\), there exists \(\varepsilon >0\) such that \(B_\varepsilon (a)\subset A\).

Proposition 9

Among the inclusion maps in Fig. 3, only the inclusions \(\mathrm {(ffid)}\hookrightarrow \mathrm {(fid)}\) and \(\mathrm {(cfid)}\hookrightarrow \mathrm {(cid)}\) are inclusions of open subsets.

Proof

Let \(M \in \mathrm {(ffid)}\) and \(N \in \mathrm {(fid)}\setminus \mathrm {(ffid)}\). Then N is isomorphic to a direct sum of interval modules, at least one of which is unbounded. It follows that \(d_I(M,N) = \infty \). Thus \(\mathrm {(ffid)}\) is an open subset of \(\mathrm {(fid)}\). The same argument shows that \(\mathrm {(cfid)}\) is an open subset of \(\mathrm {(cid)}\). For each of the following inclusions \({\mathcal {A}}\subset {\mathcal {B}}\) we show that for all \(M \in {\mathcal {A}}\) and for all \(\varepsilon > 0\), there is an \(N \in {\mathcal {B}}\setminus {\mathcal {A}}\) such that \(d_I(M,N) < \varepsilon \). Therefore \({\mathcal {A}}\) is not an open subset of \({\mathcal {B}}\).
  • \(\mathrm {(ffid^{[c,d]})}\subset \mathrm {(ffid)}\). Let \(N = M \oplus [d,d+2\varepsilon )\).

  • \(\mathrm {(ffid)}\subset \mathrm {(cfid)}\). Let \(N = M \oplus \bigoplus _{k=1}^{\infty } [0,2\varepsilon )\).

  • \(\mathrm {(fid)}\subset \mathrm {(cid)}\). Let \(N = M \oplus \bigoplus _{k=1}^{\infty } [0,2\varepsilon )\).

  • \(\mathrm {(fid)}\subset \mathrm {(pfd)}\). Let \(N = M \oplus \bigoplus _{k=1}^{\infty } [k,k+2\varepsilon )\).

  • \(\mathrm {(cid)}\subset \mathrm {(rid)}\). Let \(N = M \oplus \bigoplus _{{\mathbb {R}}}[0,2\varepsilon )\).

  • \(\mathrm {(pfd)}\subset \mathrm {(rid)}\). Let \(N = M \oplus \bigoplus _{k=1}^{\infty }[0,2\varepsilon )\). \(\square \)

Remark 5

While \(\mathrm {(ffid^{[c,d]})}\) is not an open subset of \(\mathrm {(ffid)}\), if we restrict \(\mathrm {(ffid)}\) to direct sums of interval modules whose intervals are contained in an open interval (cd), then we obtain an open subset of \(\mathrm {(ffid)}\).

4.2 Separation

Proposition 10

Any set of ephemeral persistence modules with the interleaving distance has the indiscrete topology.

Proof

Let S be a set of ephemeral persistence modules. By Lemma 4, each \(M \in \mathrm {(eph)}\) has \(d_I(M,0)=0\). So for \(M,N \in S\), by the triangle inequality, \(d_I(M,N)=0\). Thus for all \(M \in S\) and for all \(\varepsilon >0\), \(B_{\varepsilon }(M) \supseteq S\). \(\square \)

Lemma 10

Let M be a persistence module let \(r \in {\mathbb {R}}\). Then \(d_I(M,M \oplus [r,r]) = 0\).

A topological is said to be a \(T_0\)-space (or a Kolmogorov space), if for any pair of distinct elements in the space there exists at least one open set which contains one of them but not the other.

Proposition 11

Let \(c < d\). Then \(\mathrm {(ffid^{[c,d]})}\) is not a \(T_0\)-space.

Proof

Apply Lemma 10 to \(M=[a,b)\) where \(c \le a < b \le d\), and \(r = \frac{c+d}{2}\). Then \(M' = M \oplus [r,r] \in \mathrm {(ffid^{[c,d]})}\) and there does not exist an open neighborhood U of M that does not contain \(M'\) and vice versa. \(\square \)

Since \(\mathrm {(ffid^{[c,d]})}\) is a subspace of any the other spaces in Fig. 3, we obtain the following.

Corollary 6

None of the spaces in Fig. 3 are \(T_0\).

4.3 Compactness

Let X be an extended pseudometric space. Then a subset \(S\subset X\) is totally bounded if and only if for each \(\varepsilon >0\), there exists a finite subset \(F=\{x_1,x_2,\ldots ,x_n \} \subset X\) such that \(S\subset \cup _{i=1}^n B_\varepsilon (x_i)\). Such a union is called a finite \(\varepsilon \)-cover.

Lemma 11

The space \(\mathrm {(ffid^{[c,d]})}\) is not totally bounded.

Proof

Let \(\varepsilon < \frac{d-c}{2}\). For \(n\ge 0\) consider \(M_n = \bigoplus _{k=1}^n [c,d)\). Then for \(m\ne n\), \(d_I(M_m,M_n) = \frac{d-c}{2}\). Therefore \(\mathrm {(ffid^{[c,d]})}\) does not have a finite \(\varepsilon \)-cover. \(\square \)

An open cover of a topological space X is a collection of open sets \({\mathcal {O}}=\{O_i\}_{i \in I}\) of X such that \(\cup _{i\in I} O_i = X\). A topological spaces is compact if every open cover has a finite subcover. We say that a topological space is locally compact if each point has a compact neighborhood, where by a neighborhood of a point \(p \in X\) we mean a subset \(V \subset X\) such that there exists an open set \(p \in U \subset V\).

Proposition 12

Any of element in \(\mathrm {(ffid^{[c,d]})}\) does not have a compact neighborhood.

Proof

Let \(M \cong \bigoplus _{j=1}^q I_j\) with \(I_j \subset [c,d]\). Suppose that M has a compact neighborhood, K. Then there exists a real number \(\varepsilon > 0\) such that \(M \in B_\varepsilon (M) \subset K\).

Choose \(\delta >0\) such that \(\delta < \varepsilon \), \(\delta < d-c\) and \(\delta < \frac{1}{4} \min _{j} {{\mathrm{diam}}}I_j\). Choose an interval I of diameter \(\delta \) contained in [cd]. Consider for \(n \in {\mathbb {N}}\), the persistence modules \(M_n= M \oplus \bigoplus _{k=1}^n I\). Then for each n, \(d_I(M,M_n) \le \frac{\delta }{2}\) so that the set \(\{M_n\}_{n\in {\mathbb {N}}}\) is contained in \(B_\varepsilon (M)\), and hence in K.

Let \(M_0 = M\). Then by the algebraic stability theorem (Chazal et al. 2009), \(d_I(M_p,M_q) \ge \frac{\delta }{2}\) for all \(p>q \ge 0\). Now consider the open cover \(\{B_{\frac{\delta }{6}}(N) \mid N \in K\}\) of K. It does not have a finite subcover, since there does not exist a persistence module N such that \(B_{\frac{\delta }{6}}(N)\) contains \(M_n\) and \(M_m\) for \(m \ne n\). \(\square \)

Corollary 7

All of the spaces in Fig. 3 are not locally compact.

An open covering \({\mathcal {O}}=\{O_i\}_{i \in I}\) of X is locally finite if every \(x\in X\) has a neighborhood which has a nonempty intersection with only finitely many of the open sets \(\{O_i\}\). Given an open cover \(\{O_i\}_{i \in I}\) of X, another open cover \({\mathcal {V}}=\{V_j\}_{j\in J}\) is called a refinement of \({\mathcal {O}}\) if for each V in \({\mathcal {V}}\), there exists \(O \in {\mathcal {O}}\) such that \(V \subset O\). A topological space X is said to be a paracompact if every open covering admits a locally finite refinement.

Lemma 12

An extended pseudometric space is paracompact.

Proof

Let (Xd) be an extended pseudometric space. Let \(Y = X/{\sim }\) be the quotient space where the equivalence relation \(\sim \) is defined on X by \(x\sim y \ \Leftrightarrow \ d(x,y)=0\). So \((Y,\rho )\) is an extended metric space where \(\rho ([x],[y])=d(x,y)\). Let \(\pi : X \rightarrow Y\) denote the quotient map. Since \(\pi \) maps the open ball \(B_r(x)\) to the open ball \(B_r([x])\) for all \(x \in X\) and all \(r>0\), it is an open map.

Now the equivalence relation on Y given by \(x \sim y \Leftrightarrow d(x,y)< \infty \) partitions Y into a disjoint union of metric spaces, \(Y = \coprod Y_{\alpha }\). Given an open cover \({\mathcal {U}}\) of Y, each open set in \({\mathcal {U}}\) is a disjoint union of open sets, each of which is in one of the \(Y_{\alpha }\). This gives a refinement of \({\mathcal {U}}\) that is a disjoint union of open covers of each of the \(Y_{\alpha }\). Each of these metric spaces is paracompact (Munkres 1975, Theorem 41.4). Taking the disjoint union of the resulting locally finite refinements gives the desired locally finite refinement of Y.

Let \({\mathcal {U}}=\{U_i\}_{i \in I}\) be an open cover for X. Since \(\pi \) is an open map \(\{\pi (U_i)\}_{i\in I}\) forms an open cover for Y and since Y is paracompact there is a locally finite refinement \({\mathcal {V}}=\{V_j\}_{j\in J}\) for \(\{\pi (U_i)\}_{i\in I}\). Then the open cover \(\pi ^{-1}({\mathcal {V}})=\{\pi ^{-1}(V_j)\}_{j\in J}\) is a locally finite refinement for \({\mathcal {U}}=\{U_i\}_{i \in I}\). Hence X is paracompact. \(\square \)

4.4 Path connectedness

Lemma 13

Let S be an extended pseudometric space. Let \(a,b \in S\) with \(d(a,b)= \infty \). Then there does not exist a path in S from a to b.

Proof

Suppose there is a path \(\gamma \) from a to b in S. Then \(\gamma \) has a compact image. Therefore the cover \(\{B_{1}(x) \mid x \in \gamma \}\) should have a finite subcover, which by the triangle inequality contradicts \(d(a,b) = \infty \). \(\square \)

Corollary 8

The spaces of persistence modules \(\mathrm {(fid)}\), \(\mathrm {(cid)}\), \(\mathrm {(pfd)}\), \(\mathrm {(rid)}\), and \(\mathrm {(cfid)}\) are not path connected.

Proof

The first four of these sets contain both 0 and \([0,\infty )\) and \(d_I(0,[0,\infty ))= \infty \). The set \(\mathrm {(cfid)}\) contains 0 and \(\bigoplus _{k=1}^{\infty } [0,k)\) and \(d_I(\bigoplus _{k=1}^{\infty } [0,k),0) = \infty \). \(\square \)

Lemma 14

Let I be a finite interval. There exists a path in \(\mathrm {(ffid^{[c,d]})}\) from I to the zero module.

Proof

Let \(c = \inf I\) and \(d = \sup I\). Let \(M^{(0)} = I\) and \(M^{(1)} = 0\). For \(0< t < 1\), let \(M^{(t)} = [c + t\frac{d-c}{2}, d - t\frac{d-c}{2})\). Then for \(0 \le s \le t \le 1\), \(d_I(M^{(s)},M^{(t)}) = (t-s)\frac{d-c}{2}\). Thus \(\gamma (t) = M^{(t)}\) is a (continuous) path from I to 0. \(\square \)

With a similar argument we will show the following.

Proposition 13

The path component of the zero module in \(\mathrm {(fid)}\) is \(\mathrm {(ffid)}\).

Proof

By Lemma 13, the path component of \(0\) in \(\mathrm {(fid)}\) is contained in \(\mathrm {(ffid)}\). It remains to show that any \(M \in \mathrm {(ffid)}\) is path connected to \(0\).

Let \(M \cong \bigoplus _{k=1}^N I_k\), where \(I_k\) is a finite interval. For \(1 \le k \le N\), let \(c_k = \inf I_k\) and \(d_k = \sup I_k\). Let \(M^{(0)} = M\) and \(M^{(1)}=0\). For \(0<t<1\), let \(M^{(t)} = \bigoplus _{k=1}^N [c_k + t \frac{d_k-c_k}{2}, d - t \frac{d_k-c_k}{2})\). Then for \(0 \le s \le t \le 1\), \(d_I(M^{(s)},M^{(t)}) \le (t-s)\max _{1 \le k \le N} \frac{d_k-c_k}{2}\). So \(M^{(t)}\) is a continuous path from M to \(0\). \(\square \)

Remark 6

It is not the case that the path component of the zero module in \(\mathrm {(cid)}\) is \(\mathrm {(cfid)}\), since \(\mathrm {(cfid)}\) is not path connected. Since infinite intervals have infinite distance from the zero module, the path component of the zero module in \(\mathrm {(cid)}\) is the same as the path component of the zero module in \(\mathrm {(cfid)}\).

Proposition 14

The path component of  0 in \(\mathrm {(cfid)}\), \(\mathrm {(pfd)}\), and \(\mathrm {(rid)}\) consists of modules \(\bigoplus _{\alpha \in A} I_{\alpha }\), where \(\sup _{\alpha \in A} {{\mathrm{length}}}(I_{\alpha }) < \infty \).

Proof

Let \(M = \bigoplus _{\alpha \in A} I_{\alpha }\). If \(\sup _{\alpha \in A} {{\mathrm{length}}}(I_{\alpha }) = \infty \) then \(d_I(0,M) = \infty \) and M is not in the path component of 0. If \(\sup _{\alpha \in A} {{\mathrm{length}}}(I_{\alpha }) < \infty \) then the proof of Proposition 13 (replacing \(\max \) with \(\sup \)) shows that M is in the path component of 0.

\(\square \)

The paths in the previous proposition may be used to show that the following spaces are nullhomotopic.

Proposition 15

The spaces \(\mathrm {(ffid^{[c,d]})}\) and \(\mathrm {(ffid)}\) and the path component of  \(0\) of \(\mathrm {(cfid)}\), \(\mathrm {(pfd)}\) and \(\mathrm {(rid)}\) are contractible to the zero module.

Proof

Let S denote either \(\mathrm {(ffid^{[c,d]})}\), \(\mathrm {(ffid)}\) or the path component of \(0\) in \(\mathrm {(cfid)}\), \(\mathrm {(pfd)}\), or \(\mathrm {(rid)}\). Assume \(M \cong \bigoplus _{k \in A} I_k\), where A is countable. Let \(c_k = \inf I_k\), \(d_k = \sup I_k\) and let \(h_k = \frac{d_k-c_k}{2}\). Let \(M^{(0)} = M\), \(M^{(1)} = 0\) and for \(0<t<1\), \(M^{(t)} = \bigoplus _{k \in A} [c_k+th_k,d-th_k)\).

We will use these paths to construct a homotopy from the identity map on S to the constant map to the zero module. Define \(H: S \times [0,1] \rightarrow S\) by \((M,t) \mapsto M^{(t)}\). Let \(H_t = H(-,t)\). Then \(H_0 = 1_S\) and \(H_1 = 0\). It remains to show that H is continuous. Let \((M,t) \in S\times [0,1]\). Given \(\varepsilon >0\), choose \(\delta = \frac{\varepsilon }{1+d_I(M,0)}\). Let d denote the product metric on \(S\times [0,1]\). Whenever \((N,s) \in S\times [0,1]\) satisfies \(d((M,t),(N,s))< \delta \), \(d_I(M,N)< \delta \) and \(|t-s|<\delta \). Furthermore
$$\begin{aligned}&d_I(M^{(t)},N^{(s)}) \le d_I(M^{(t)},M^{(s)}) + d_I(M^{(s)},N^{(s)}) \\&\quad \le |t-s| d_I(M,0) + s d_I(M,N) \le \delta d_I(M,0) + \delta = \varepsilon \end{aligned}$$
which completes the proof. \(\square \)

4.5 Separability

A topological space is said to be separable if it has a countable dense subset.

Theorem 4

The spaces \(\mathrm {(fid)}\), \(\mathrm {(ffid)}\) and \(\mathrm {(ffid^{[c,d]})}\) are separable.

Proof

First we will show that \(\mathrm {(ffid)}\) is separable. Let
$$\begin{aligned} D_n =\left\{ \bigoplus _{i=1}^n (p_i,q_i) \in \mathrm {(fid)}\ | \ p_i, q_i \in {\mathbb {Q}}, \ p_i < q_i \right\} \end{aligned}$$
(4.1)
and then consider
$$\begin{aligned} D=\bigcup _{i=1}^\infty D_n. \end{aligned}$$
Then D is countable and D is dense in \(\mathrm {(ffid)}\) since every open ball of every persistence module in \(\mathrm {(ffid)}\) contains an element of D.

This proof also works for \(\mathrm {(fid)}\) if we allow the intervals in (4.1) to be infinite, and it works for \(\mathrm {(ffid^{[c,d]})}\) if we restrict the intervals in (4.1) to be subintervals of [cd]. \(\square \)

Theorem 5

The spaces \(\mathrm {(cfid)}\), \(\mathrm {(cid)}\), \(\mathrm {(pfd)}\), and \(\mathrm {(rid)}\) are not separable. The same is true for the subspace of \(\mathrm {(cid)}\) with finite distance to 0 (which equals the subspace of \(\mathrm {(cfid)}\) with finite distance to 0), and for \(\mathrm {(cid)}\cap \mathrm {(qtame)}\) and \(\mathrm {(cfid)}\cap \mathrm {(qtame)}\).

Proof

We assign to each binary sequence, \(\alpha =(\alpha _n)_{n\ge 1}\) where \(\alpha _n\in \{0,1\}\), a persistence module. See Fig. 4. For \(n \ge 1\), define
$$\begin{aligned} I_{n}^{(\alpha )} = {\left\{ \begin{array}{ll} {[}2n-1,2n+1), &{} \alpha _n=0 \\ {[}2(n-1),2n+2), &{} \alpha _n=1 \end{array}\right. } \end{aligned}$$
and let \(M_\alpha =\bigoplus _{n=1}^{\infty } I_n^{(\alpha )}\). Then \(M_{\alpha }\) is a persistence module in \(\mathrm {(cfid)}\), \(\mathrm {(cid)}\), \(\mathrm {(pfd)}\), \(\mathrm {(rid)}\), and \(\mathrm {(qtame)}\) and \(d_I(M_{\alpha },0) \le 2\).
Fig. 4

Persistence modules corresponding to binary sequences, which are used in the proof of Theorem 5

The set \(\{M_\alpha \ | \ \alpha \ \text {is a binary sequence} \}\) is uncountable and for all pairs of binary sequences \(\alpha \ne \beta \), we have \(d_I(M_\alpha ,M_\beta ) = 1\). Then any dense subset of \(\mathrm {(cfid)}\), \(\mathrm {(cid)}\), \(\mathrm {(pfd)}\), or \(\mathrm {(rid)}\), contains a point in an open ball centered at each \(M_\alpha \) of radius \(\frac{1}{2}\) and thus cannot be countable. The same is true for the subspace of \(\mathrm {(cid)}\) with finite distance to \(0\), and for \(\mathrm {(cid)}\cap \mathrm {(qtame)}\) and \(\mathrm {(cfid)}\cap \mathrm {(qtame)}\). \(\square \)

4.6 Countability

A topological space is said to be a first countable if it has a countable basis at each of its points.

Lemma 15

An extended pseudometric space is first countable.

Proof

Let x be a point in the space. Then the countable collection of open balls \(\{B_{\frac{1}{n}}(x) \ | \ n\in {\mathbb {N}}\}\) is the desired local base at x. \(\square \)

A space X is compactly generated if a set \(A \subset X\) is open if each \(A \cap C\) is open in C for each compact subspace \(C \subset X\). Equivalently, a set \(B \subset X\) is closed if each \(B \cap C\) is closed in C for each compact subspace \(C \subset X\). The following is well known.

Lemma 16

If a space is first countable then it is compactly generated.

Proof

For \(B \subset X\), assume that \(B \cap C\) is closed in C for each compact subspace \(C \subset X\). Let x be a limit point of B. That is, every neighborhood of x contains point of B other than x. Since X is first countable, there is a sequence of points \((x_i)\) converging to x. Now \((x_i) \cup \{x\}\) is compact, so by assumption \(B \cap ((x_i) \cup \{x\})\) is closed in \((x_i) \cup \{x\}\). Since \((x_i) \subset B\) it follows that \(x \in B\). Therefore B is closed. \(\square \)

A topological space is said to be second countable if it has a countable basis. A topological space X is said to be Lindelöf if every open cover of X admits a countable subcover.

Lemma 17

For an extended pseudometric space the following properties are equivalent:
  1. 1.

    second countable;

     
  2. 2.

    separable; and

     
  3. 3.

    Lindelöf.

     

Proof

Let X be an extended pseudometric space.

\((1) \Rightarrow (2)\): Assume that X has a countable basis \(\{B_i\}\). For each i, choose \(x_i \in B_i\). Then for each \(x \in X\) and \(r>0\), there exists i such that \(B_i \subset B_r(x)\). So \(\{x_i\}\) is a countable dense subset of X.

\((2) \Rightarrow (3)\): Assume that X has a countable dense subset \(\{x_i\}\). Let \({\mathcal {U}}\) be an open cover of X. For each i, choose \(U_i \in {\mathcal {U}}\) with \(x_i \in U_i\). Since \(U_i\) is open, \(U_i \supset B_{r_i}(x_i)\) for some \(r_i>0\). Since \(\{x_i\}\) is dense, \(\{U_i\}\) is a countable subcover.

\((3) \Rightarrow (1)\): Assume that X has the Lindelöf property. For each \(n \ge 1\), let \({\mathcal {U}}_n\) be a countable subcover of the open cover \(\{B_{\frac{1}{n}} (x) \ |?\ x\in X\}\). Then \({\mathcal {U}} := \cup _n {\mathcal {U}}_n\) is a countable basis for X. \(\square \)

4.7 Completeness

An extended pseudometric space is said to be complete if every Cauchy sequence converges (see the end of Sect. 2.4).

Theorem 6

The spaces \(\mathrm {(pfd)}\), \(\mathrm {(fid)}\), \(\mathrm {(ffid)}\) and \(\mathrm {(ffid^{[c,d]})}\) are not complete.

Proof

For \(n \ge 0\), let \(M_n = \bigoplus _{k=0}^n \left[ -\,\frac{1}{2^k},\frac{1}{2^k}\right) \). Then the sequence \((M_n) \subset \mathrm {(ffid)}\subset \mathrm {(fid)}\subset \mathrm {(pfd)}\), and \((M_n) \rightarrow M = \bigoplus _{k=0}^{\infty } \left[ -\,\frac{1}{2^k},\frac{1}{2^k}\right) \), which is not in \(\mathrm {(pfd)}\).

We claim that there is no \(N \in \mathrm {(pfd)}\) such that \(d_I(M,N)=0\). Assume \(N \in \mathrm {(pfd)}\). Then \({{\mathrm{rank}}}N(0) = R < \infty \). Thus for all \(\varepsilon > 0\), \({{\mathrm{rank}}}N(-\,\varepsilon \le \varepsilon ) \le R\). Now for all \(\varepsilon >0\), M and N are \(\varepsilon \)-interleaved, and thus \({{\mathrm{rank}}}M(-\,2\varepsilon \le 2\varepsilon ) \le {{\mathrm{rank}}}N(-\,\varepsilon \le \varepsilon ) \le R\), which is a contradiction.

If we adjust \(M_n\) to lie in [cd], then the same argument shows that \(\mathrm {(ffid^{[c,d]})}\) is not complete. \(\square \)

Theorem 7

In the class of persistence modules and the class of q-tame persistence modules, every Cauchy sequence has a limit. Furthermore, the space \(\mathrm {(cid)}\cap \mathrm {(qtame)}\) is complete, and so is \(\mathrm {(cfid)}\cap \mathrm {(qtame)}\).

Proof

Let \((M'_n)_{n\ge 1}\) be a Cauchy sequence of persistence modules. For each \(k \ge 0\), choose a natural number \(n_k\) so that \(d_I(M'_m,M'_n)< \frac{1}{2^{k}}\) for all \(m,n \ge n_k\). Let \(M_k\) denote \(M'_{n_k}\). Thus \((M_k)\) is a subsequence of \((M'_n)\) so that for all \(k \ge 0\), \(M_k\) and \(M_{k+1}\) are \(\frac{1}{2^{k}}\)-interleaved. By the definition of interleaving, there exist natural transformations \(\varphi _{k} : M_k \Rightarrow M_{k+1} T_{\frac{1}{2^{k}}} \) and \(\psi _{k} : M_{k+1} \Rightarrow M_k T_{\frac{1}{2^{k}}}\) such that the triangles corresponding to (2.3) commute.

Now we define shifted versions of \(\varphi \) and \(\psi \). For \(k \ge 0\), let \(\alpha ^k = \varphi _k T_{-\frac{1}{2^{k-1}}}: M_{k-1} T_{-\frac{1}{2^{k-1}}} \Rightarrow M_k T_{-\frac{1}{2^k}}\), and \(\beta ^k = \psi _k T_{\frac{1}{2^{k}}}: M_{k} T_{\frac{1}{2^{k}}} \Rightarrow M_{k-1} T_{\frac{1}{2^{k-1}}}\). Let \(a \in {\mathbb {R}}\). For every \(k \ge 1\), \(\alpha ^k_a: M_{k-1}(a-\frac{1}{2^{k-1}}) \rightarrow M_k(a-\frac{1}{2^k})\) and \(\beta ^k_a: M_k(a+\frac{1}{2^k}) \rightarrow M_{k-1}(a+\frac{1}{2^{k-1}})\). Thus we have a direct system of vector spaces
$$\begin{aligned} M_0(a - 1) \xrightarrow {\alpha ^1_a} M_1\left( a-\frac{1}{2}\right) \xrightarrow {\alpha ^2_a} M_2\left( a-\frac{1}{4}\right) \xrightarrow {\alpha ^3_a} M_3\left( a-\frac{1}{8}\right) \xrightarrow {\alpha ^4_a} \cdots \nonumber \\ \end{aligned}$$
(4.2)
and an inverse system of vector spaces
$$\begin{aligned} \cdots \xrightarrow {\beta ^4_a} M_3\left( a+\frac{1}{8}\right) \xrightarrow {\beta ^3_a} M_2\left( a+\frac{1}{4}\right) \xrightarrow {\beta ^2_a} M_1\left( a+\frac{1}{2}\right) \xrightarrow {\beta ^1_a} M_0(a+1)\nonumber \\ \end{aligned}$$
(4.3)
given in Fig. 5. Note that it follows from the definition of interleaving that each of the trapezoids in Fig. 5 commute. Let A(a) be the colimit (i.e. direct limit) of (4.2), and let B(a) be the limit (i.e. inverse limit) of (4.3). For each \(k \ge 0\), we have maps \(\lambda ^k_a: M_k(a-\frac{1}{2^k}) \rightarrow A(a)\) and \(\mu ^k_a: B(a) \rightarrow M_k(a+\frac{1}{2^k})\). By the universal properties of the colimit and the limit, we have a map \(\theta _a: A(a) \rightarrow B(a)\), and
$$\begin{aligned} \mu ^k_a \theta _a \lambda ^k_a = M_k\left( a-\textstyle \frac{1}{2^k} \le a+\textstyle \frac{1}{2^k}\right) . \end{aligned}$$
(4.4)
Let M(a) denote the image of \(\theta _a\). Thus, \(\theta _a\) factors as follows.
Fig. 5

A direct system of vector spaces and an inverse system of vector spaces in a Cauchy sequence of persistence modules

Now observe that all of these constructions are functorial. Thus, we have persistence modules A, B and M. We also have natural transformations \(\lambda ^k: M_k T_{-\frac{1}{2^k}} \Rightarrow A\) and \(\mu ^k: B \Rightarrow M_k T_{\frac{1}{2^k}}\). In addition we have the following commutative diagram of natural transformations.
These fit into the commutative diagram in Fig. 6, where we have corresponding arrows for all \(a \in {\mathbb {R}}\).
Fig. 6

A particular subsequence of a Cauchy sequence of persistence modules and some persistence modules in the limit

Let \(a \in {\mathbb {R}}\) and \(k\ge 1\). Define \(b = a + \frac{1}{2^{k-1}}\). Then we have the following bi-infinite sequence.
$$\begin{aligned} \cdots \xrightarrow {\beta _a^{k+3}} M_{k+2}\left( a+\frac{1}{2^{k+2}}\right) \xrightarrow {\beta _a^{k+2}} M_{k+1}\left( a+\frac{1}{2^{k+1}}\right) \xrightarrow {\beta _a^{k+1}} M_{k}\left( a+\frac{1}{2^{k}}\right) \nonumber \\ \xrightarrow {\alpha _b^{k+1}} M_{k+1}\left( b-\frac{1}{2^{k+1}}\right) \xrightarrow {\alpha _b^{k+2}} M_{k+2}\left( b-\frac{1}{2^{k+2}}\right) \xrightarrow {\alpha _b^{k+3}} \cdots \nonumber \\ \end{aligned}$$
(4.6)
Notice that the left part of this sequence is an initial part of (4.3) and the right part of this sequence is a terminal part of (4.2). It follows that (4.6) has limit B(a) and colimit A(b), and there is an induced map \(\nu :B(a) \rightarrow A(b)\). We obtain the commutative diagram in Fig. 7.
Fig. 7

The bi-infinite sequence in (4.6), its limit and colimit, and three induced maps

By the universal properties of limit and colimit, we have the following commutative diagram.
By the commutativity of the bottom right part of this diagram, we have that \({{\mathrm{im}}}B(a \le b) \subset {{\mathrm{im}}}\theta _b\). So we have the following commutative diagram.
Thus
$$\begin{aligned} \rho _b \nu \iota _a = B(a \le b)|_{{{\mathrm{im}}}\theta _a} = M(a\le b). \end{aligned}$$
(4.7)
Now consider the following natural transformations.
$$\begin{aligned}&(\rho \lambda ^k) T_{\frac{1}{2^k}} : M_k \Rightarrow M T_{\frac{1}{2^k}} \end{aligned}$$
(4.8)
$$\begin{aligned}&\mu ^k \iota : M \Rightarrow M_k T_{\frac{1}{2^k}} \end{aligned}$$
(4.9)
We claim that these natural transformations provide an interleaving (Sect. 2.3). That is,
$$\begin{aligned} \left( (\mu ^k \iota ) T_{\frac{1}{2^k}}\right) \left( (\rho \lambda ^k) T_{\frac{1}{2^k}}\right) = M_k \eta _{\frac{1}{2^{k-1}}}, \text { and } \left( (\rho \lambda ^k) T_{\frac{1}{2^{k-1}}}\right) (\mu ^k \iota ) = M \eta _{\frac{1}{2^{k-1}}},\nonumber \\ \end{aligned}$$
(4.10)
where \(\eta \) is the natural transformation defined in Sect. 2.3.

A pair of natural transformations are equal if and only if their components are equal. We remark that for a natural transformations \(\alpha \) and \(\beta \), the natural transformation \(\alpha T_x\) has components \((\alpha T_x)_a = \alpha _{a+x}\), and the natural transformation \(\beta \alpha \) has components \((\beta \alpha )_a = \beta _a \alpha _a\).

Let \(a \in {\mathbb {R}}\). We will verify the identities in (4.10) using the a component. For the left hand side of the first identity, we have
$$\begin{aligned}&M(a) \xrightarrow {\lambda _{a+\frac{1}{2^k}}^k} A\left( a+\frac{1}{2^k}\right) \xrightarrow {\rho _{a+\frac{1}{2^k}}} M\left( a+\frac{1}{2^k}\right) \xrightarrow {\iota _{a+\frac{1}{2^k}}} B\left( a+\frac{1}{2^k}\right) \\&\quad \xrightarrow {\mu _{a+\frac{1}{2^k}}^k} M\left( a+\frac{1}{2^{k-1}}\right) = M(b). \end{aligned}$$
Using (4.5) the composition of the inner two maps equals \(\theta _{a+\frac{1}{2^k}}\). Then using (4.4) we see that the entire composition equals \(M_k(a \le a+\frac{1}{2^{k-1}})\), as desired.
For the left hand side of the second identity, we have
$$\begin{aligned}&M(a) \xrightarrow {\iota _a} B(a) \xrightarrow {\mu _a^k} M(a+\frac{1}{2^k}) \xrightarrow {\lambda _{a+\frac{1}{2^{k-1}}}^k} A(a+\frac{1}{2^{k-1}})\\&\quad \xrightarrow {\rho _{a+\frac{1}{2^{k-1}}}^k} M(a+\frac{1}{2^{k-1}}) = M(b). \end{aligned}$$
Using the commutativity of the induced maps in Fig. 7, the composition of the inner two maps equals \(\nu \). Then using (4.7), we see that the entire composition equals \(M(a \le b)\), as desired.

Thus (4.8) and (4.9) is a \(\frac{1}{2^k}\)-interleaving. Therefore M is a limit of the sequence \((M_k)\) and hence also a limit of the Cauchy sequence \((M'_n)\). Thus any Cauchy sequence of persistence modules has a limit.

Now assume that each of the \(M'_n\) are in \(\mathrm {(qtame)}\). We will show that \(M \in \mathrm {(qtame)}\). Let \(a<b\), and choose \(k\ge 0\) so that \(\frac{1}{2^{k-1}} < b-a\). Then the following diagram commutes.
Since \(M_k\) is q-tame, the top horizontal arrow has finite rank, and hence so does the bottom horizontal arrow. Thus \(M \in \mathrm {(qtame)}\). Therefore any Cauchy sequence of q-tame persistence modules has a limit.

Now since \(M \in \mathrm {(qtame)}\), by Theorem 3, \({{\mathrm{rad}}}M \in \mathrm {(cid)}\cap \mathrm {(qtame)}\). By Proposition 2, \(d_I(M, {{\mathrm{rad}}}M) = 0\). Therefore by the triangle inequality, \({{\mathrm{rad}}}M\) is also a limit of the Cauchy sequence. Thus \(\mathrm {(cid)}\cap \mathrm {(qtame)}\) is complete.

Finally, assume that in addition, each \(M'_n \in \mathrm {(cfid)}\cap \mathrm {(qtame)}\). Since M is \(\frac{1}{2^k}\)-interleaved with \(M_k\), which does not contain any infinite intervals in its direct sum decomposition, neither does M. Therefore \({{\mathrm{rad}}}M\) also does not contain any infinite intervals in its direct sum decomposition. That is, \({{\mathrm{rad}}}M \in \mathrm {(cfid)}\cap \mathrm {(qtame)}\). \(\square \)

Now we present a second, more concise proof of the main result in the previous proof.

Proof

We may consider the diagram in Fig. 5 to be a functor \(M: ({\mathbb {R}} \times {\mathbb {N}},\le ) \rightarrow {\underline{\mathrm{Vect}}}_{\mathbf {k}}\), where \(({\mathbb {R}} \times {\mathbb {N}},\le )\) is the poset generated by the inequalities \((a,k) \le (b,k)\), where \(a \le b \in {\mathbb {R}}\) and \(k \ge 0\), and \((a,k-1) \le (a+\frac{1}{2^k},k)\) and \((a,k) \le (a+\frac{1}{2^k},k-1)\), where \(a \in {\mathbb {R}}\) and \(k \ge 1\).

Now extend this poset to \(({\mathbb {R}} \times \overline{{\mathbb {N}}} , \le )\), by adding the generating inequalities \((a,\infty ) \le (b,\infty )\) for all \(a\le b\), and \((a-\frac{1}{2^k},k) \le (a,\infty )\) and \((a,\infty ) \le (a+\frac{1}{2^k},k)\).
We can extend the functor M to \(({\mathbb {R}} \times \overline{{\mathbb {N}}},\le )\) by taking either the left or the right Kan extension. We obtain functors corresponding to the diagram in Fig. 6, where \(A = {{\mathrm{Lan}}}_{i}M\) and \(B = {{\mathrm{Ran}}}_{i}M\). Then there is a canonical map \(\theta : A \Rightarrow B\), and the image of this map gives another extension of M. Abusing notation, let \(M = {{\mathrm{im}}}(\theta ): ({\mathbb {R}} \times \overline{{\mathbb {N}}},\le ) \rightarrow {\underline{\mathrm{Vect}}}_{\mathbf {k}}\).

For \(k \in \overline{{\mathbb {N}}}\), let \(M_k = M(-,k)\). Then by construction, \(M_{\infty }\) is \(\frac{1}{2^k}\)-interleaved with \(M_k\). Thus, \(M_{\infty }\) is a limit of the Cauchy sequence. \(\square \)

4.8 Baire spaces

Let X be a topological space. A subspace \(A \subset X\) has empty interior in X if A does not contain an open set in X. The space X is said to be a Baire space if for any countable collection of closed sets in X with empty interior in X, their union also has empty interior in X.

Theorem 8

(Baire category theorem) A complete extended pseudometric space is a Baire space.

Proof

Let X be an extended pseudometric space. Let \(\{A_n\}\) be a countable collection of closed sets in X with empty interior in X. We want to show that \(\bigcup A_n\) has empty interior in X. Let U be an open set in X. We will show that \(U \not \subset \bigcup A_n\). We need an \(x \in U\) such that for all n, \(x \not \in A_n\). By assumption, there is a \(x_1 \in U\) with \(x_1 \not \in A_1\). Since U is open and \(A_1\) is closed, there is an \(r_1 \le 1\) such that \(B_{r_1}(x_1) \subset U\) and \(B_{r_1}(x_1) \cap A_1 = \emptyset \). Let \(s_1 = \frac{r_1}{2}\). Then \(\overline{B_{s_1}(x_1)} \subset U\) and \(\overline{B_{s_1}(x_1)} \cap A_1 = \emptyset \). Given \(B_{s_n}(x_n)\) with \(\overline{B_{s_n}(x_n)} \cap A_n = \emptyset \), then by assumption, there is a \(x_{n+1} \in B_{s_n}(x)\) with \(x_{n+1} \not \in A_{n+1}\). Since \(B_{s_n}(x)\) is open and \(A_{n+1}\) is closed, there is an \(r_{n+1} \le \frac{1}{n+1}\) with \(B_{r_{n+1}}(x_{n+1}) \subset B_{s_n}(y_n)\) and \(B_{r_{n+1}}(x_{n+1}) \cap A_{n+1} = \emptyset \). Let \(s_{n+1} = \frac{r_{n+1}}{2}\). Then \(\overline{B_{s_{n+1}}(x_{n+1})} \subset \overline{B_{s_n}(y_n)}\) and \(\overline{B_{s_{n+1}}(x_{n+1})} \cap A_{n+1} = \emptyset \). Since \(\overline{B_{s_1}(x_1)} \supset \overline{B_{s_2}(x_2)} \supset \overline{B_{s_3}(x_3)} \supset \cdots \) and \((s_n) \rightarrow 0\), \((x_n)\) is a Cauchy sequence in X. Since X complete, there exists a \(x \in X\) such that \((x_n) \rightarrow x\). Since \(x_n \in \overline{B_{s_1}(x_1)}\) for all n, \(x \in \overline{B_{s_1}(x_1)} \subset U\). Also, for all n, the sequence \(x_n,x_{n+1},x_{n+2},\ldots \) in \(\overline{B_{s_n}(x_n)}\) converges to x, so \(x \in \overline{B_{s_n}(x_n)}\). Thus \(x \not \in A_n\) for all n. \(\square \)

Corollary 9

Hence \(\mathrm {(cid)}\cap \mathrm {(qtame)}\) and \(\mathrm {(cfid)}\cap \mathrm {(qtame)}\) are Baire spaces.

4.9 Topological dimension

Let X be a topological space. A collection of subsets of X has order m if there is a point in X contained in m of the subsets, but no point of X is contained in \(m+1\) of the subsets. The topological dimension of X (also called the Lebesgue covering dimension) is the smallest number m such that every open cover of X has a refinement (see Sect. 4.3) with order \(m+1\).

Theorem 9

Let \(N \ge 1\). There exists an \(\varepsilon >0\) such that there is an isometric embedding of the cube \([0,\varepsilon ]^N\) with the \(L^{\infty }\) distance into \(\mathrm {(ffid^{[c,d]})}\).

Proof

Assume \([c,d] = [0,1]\). The proof for the general case is similar. Choose \(\varepsilon < \frac{1}{100N}\). Let \(x = (x_1,\ldots ,x_N) \in [0,\varepsilon ]^N\). We will define a map \(x \mapsto M = M(x) = \bigoplus _{i=1}^N I_i\), where each interval \(I_i = I_i(x_i)\) depends only on \(x_i\). We will choose \(I_1,\ldots ,I_N\) to be far from each other and far from the zero module but so that \(I_i(x_i)\) is close to \(I_i(x'_i)\) for any \(x_i, x'_i \in [0,\varepsilon ]\).

For \(1 \le i \le N\), let \(I_i = \left[ \frac{i}{N}, \frac{i}{N} + \frac{1}{10N} + x_i \right) \). Then \(d_I(I_i(x_i),I_i(x'_i)) = |x_i-x'_i| \le \frac{1}{100N}\). Also \(d_I(I_i,0) \ge \frac{1}{20N}\). Since for \(i \ne j\), \(I_i\) and \(I_j\) are disjoint, and so we also have that \(d_I(I_i,I_j) \ge \frac{1}{20N}\). Therefore \(d_I(M(x),M(x')) = ||x-x'||_{\infty }\). \(\square \)

Corollary 10

The topological dimension of all of the topological spaces of persistence modules in Fig. 3 is infinite.

Proof

Let X be one of the spaces in Fig. 3. Then by the previous theorem, for all \(N \ge 1\), \(\dim X \ge \dim [0,\varepsilon ]^N = N\). Thus \(\dim X = \infty \). \(\square \)

5 Open questions

We end with some unresolved questions.
  • Are \(\mathrm {(cid)}\) and \(\mathrm {(cfid)}\) complete?

  • Can the results presented here be extended to multiparameter persistence modules and generalized persistence modules?

Footnotes

  1. 1.

    In particular, the multiplicity of \([a_i,a_j)\) can be calculated using the inclusion/exclusion formula \({{\mathrm{rank}}}M(a_i \le a_{j-1}) - {{\mathrm{rank}}}M(a_i \le a_j) - {{\mathrm{rank}}}M(a_{i-1} \le a_{j-1}) + {{\mathrm{rank}}}M(a_{i-1} \le a_j)\) (Cohen-Steiner et al. 2007), which is an example of Möbius inversion (Patel 2018).

Notes

Acknowledgements

The authors would like to that the anonymous referees for their helpful suggestions. In particular, we would like to thank the referee who contributed the proof that the enveloping distance from pointwise-finite dimensional persistence modules to q-tame persistence modules is zero. We also thank Alex Elchesen for proofreading an earlier draft of the paper. The first author would like to acknowledge the support of UFII SEED funds, ARO Research Award W911NF1810307, and the Southeast Center for Mathematics and Biology, an NSF-Simons Research Center for Mathematics of Complex Biological Systems, under National Science Foundation Grant No. DMS-1764406 and Simons Foundation Grant No. 594594.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Supplementary material

References

  1. Bauer, U., Lesnick, M.: Induced matchings and the algebraic stability of persistence barcodes. J. Comput. Geom. 6(2), 162–191 (2015)MathSciNetzbMATHGoogle Scholar
  2. Bauer, U., Lesnick, M.: Persistence diagrams as diagrams: a categorification of the stability theorem (2016). arXiv preprint arXiv:1610.10085
  3. Bjerkevik, H.B., Botnan, M.B.: Computational complexity of the interleaving distance. In: 34th International Symposium on Computational Geometry, vol. 12 (2017). arXiv preprint arXiv:1712.04281
  4. Blumberg, A.J., Lesnick, M.: Universality of the homotopy interleaving distance (2017). arXiv preprint arXiv:1705.01690
  5. Blumberg, A.J., Gal, I., Mandell, M.A., Pancia, M.: Robust statistics, hypothesis testing, and confidence intervals for persistent homology on metric measure spaces. Found. Comput. Math. 14(4), 745–789 (2014)MathSciNetCrossRefGoogle Scholar
  6. Botnan, M., Lesnick, M.: Algebraic stability of zigzag persistence modules. Algebra Geom. Topol. 18(6), 3133–3204 (2018)MathSciNetCrossRefGoogle Scholar
  7. Bubenik, P.: Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res. 16, 77–102 (2015)MathSciNetzbMATHGoogle Scholar
  8. Bubenik, P., Dlotko, P.: A persistence landscapes toolbox for topological statistics. J. Symb. Comput. 78, 91–114 (2017)MathSciNetCrossRefGoogle Scholar
  9. Bubenik, P., Scott, J.A.: Categorification of persistent homology. Discrete Comput. Geom. 51(3), 600–627 (2014)MathSciNetCrossRefGoogle Scholar
  10. Bubenik, P., de Silva, V., Scott, J.: Metrics for generalized persistence modules. Found. Comput. Math. 15(6), 1501–1531 (2015)MathSciNetCrossRefGoogle Scholar
  11. Bubenik, P., de Silva, V., Nanda, V.: Higher interpolation and extension for persistence modules. SIAM J. Appl. Algebra Geom. 1(1), 272–284 (2017a)MathSciNetCrossRefGoogle Scholar
  12. Bubenik, P., de Silva, V., Scott, J.: Interleaving and Gromov–Hausdorff distance and interleaving of functors (2017b). arXiv preprint arXiv:1707.06288
  13. Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46(2), 255–308 (2009)MathSciNetCrossRefGoogle Scholar
  14. Carlsson, G., Zomorodian, A.: The theory of multidimensional persistence. Discrete Comput. Geom. 42(1), 71–93 (2009)MathSciNetCrossRefGoogle Scholar
  15. Chazal, F., Michel, B.: An introduction to topological data analysis: fundamental and practical aspects for data scientists (2017). arXiv preprint arXiv:1710.04019
  16. Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L.J., Oudot, S.Y.: Proximity of persistence modules and their diagrams. In: Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry, vol. 09, ACM, New York, NY, USA, pp. 237–246 (2009)Google Scholar
  17. Chazal, F., de Silva, V., Oudot, S.: Persistence stability for geometric complexes. Geom. Dedic. 173, 193–214 (2014)MathSciNetCrossRefGoogle Scholar
  18. Chazal, F., Crawley-Boevey, W., de Silva, V.: The observable structure of persistence modules. Homol. Homotopy Appl. 18(2), 247–265 (2016a)MathSciNetCrossRefGoogle Scholar
  19. Chazal, F., de Silva, V., Glisse, M., Oudot, S.: The Structure and Stability of Persistence Modules. Springer Briefs in Mathematics. Springer, Cham (2016b)CrossRefGoogle Scholar
  20. Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37(1), 103–120 (2007)MathSciNetCrossRefGoogle Scholar
  21. Collins, A., Zomorodian, A., Carlsson, G., Guibas, L.J.: A barcode shape descriptor for curve point cloud data. Comput. Gr. 28(6), 881–894 (2004)CrossRefGoogle Scholar
  22. Crawley-Boevey, W.: Decomposition of pointwise finite-dimensional persistence modules. J. Algebra Appl. 14(5): 1550066, 8 (2015)MathSciNetzbMATHGoogle Scholar
  23. Curry, J.: Sheaves, cosheaves and applications. PhD Thesis, University of Pennsylvania (2014)Google Scholar
  24. de Silva, V., Munch, E., Patel, A.: Categorified Reeb graphs. Discrete Comput. Geom. 55(4), 854–906 (2016)MathSciNetCrossRefGoogle Scholar
  25. de Silva, V., Munch, E., Stefanou, A.: Theory of interleavings on \([0,\infty )\)-actegories (2017). arXiv preprint arXiv:1706.04095
  26. Gabriel, P.: Unzerlegbare Darstellungen. I. Manuscr. Math. 6, 71–103 (1972)MathSciNetCrossRefGoogle Scholar
  27. Ghrist, R.: Barcodes: the persistent topology of data. Bull. Am. Math. Soc. (N.S.) 45(1), 61–75 (2008)MathSciNetCrossRefGoogle Scholar
  28. Ghrist, R.: Homological algebra and data. Math. Data 25, 273 (2018)Google Scholar
  29. Lesnick, M.: The theory of the interleaving distance on multidimensional persistence modules. Found. Comput. Math. 15(3), 613–650 (2015)MathSciNetCrossRefGoogle Scholar
  30. Meehan, K., Meyer, D.: Interleaving distance as a limit (2017a). arXiv preprint arXiv:1710.11489
  31. Meehan, K., Meyer, D.: An isometry theorem for generalized persistence modules (2017b). arXiv preprint arXiv:1710.02858
  32. Mileyko, Y., Mukherjee, S., Harer, J.: Probability measures on the space of persistence diagrams. Inverse Probl. 27(12): 124007, 22 (2011)MathSciNetzbMATHGoogle Scholar
  33. Munch, E., Wang, B.: Convergence between categorical representations of Reeb space and mapper (2015). arXiv preprint arXiv:1512.04108
  34. Munkres, J.R.: Topology: A First Course. Prentice-Hall Inc., Englewood Cliffs (1975)zbMATHGoogle Scholar
  35. Patel, A.: Generalized persistence diagrams. J. Appl. Comput. Topol. 1(3), 397–419 (2018)CrossRefGoogle Scholar
  36. Puuska, V.: Erosion distance for generalized persistence modules (2017). arXiv preprint arXiv:1710.01577
  37. Turner, K., Mileyko, Y., Mukherjee, S., Harer, J.: Fréchet means for distributions of persistence diagrams. Discrete Comput. Geom. 52(1), 44–70 (2014)MathSciNetCrossRefGoogle Scholar
  38. Wasserman, L.: Topological data analysis (2016). arXiv preprint arXiv:1609.08227 [stat.ME]
  39. Webb, C.: Decomposition of graded modules. Proc. Am. Math. Soc. 94(4), 565–571 (1985)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA
  2. 2.Department of MathematicsEge UniversityIzmirTurkey

Personalised recommendations