Topological spaces of persistence modules and their properties
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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 distanceMathematics Subject Classification
55N99 54D99 54E99 18A251 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 (CohenSteiner 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 (CohenSteiner 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 zigzag 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 finitedimensional persistence modules (CrawleyBoevey 2015), qtame persistence modules (Chazal et al. 2014), intervaldecomposable 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 intervaldecomposable persistence modules and qtame persistence modules are not sets (Corollary 5), though the classes of pointwise finitedimensional persistence modules and persistence modules decomposable into countablemany 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 finitedimensional persistence modules is not complete (Theorem 6), but that the space of persistence modules that are both qtame and that decompose into countablymany 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 qtame 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 (CohenSteiner 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 finitedimensional (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 (CohenSteiner 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 Ncube 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 straightforward extensions of the standard results for metric spaces (Lemmas 12, 15, 16, and 17 and Theorem 8). However, in order to keep the material accessible to applied mathematicians without a background in pointset 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 pWasserstein 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
 (i)
for all a, \(v_a^a : M(a) \rightarrow M(a)\) is the identity map, and
 (ii)
if \(a \le b \le c\) then \(v_a^c=v_b^c\circ v_a^b\).
Example 1
Example 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.
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 [a, b) to the interval module [c, d) 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 CrawleyBoevey].
A special case of the following theorem follows from work of Gabriel (1972), but the general case was proved by CrawleyBoevey (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
Remark 1
Two persistence modules are 0interleaved 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
Example 6
The interval modules [0, 2] and (0, 2) are not 0interleaved. 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
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
 M1)
\(d(x,x)=0\),
 M2)
\(d(x,y)=d(y,x)\), and
 M3)
\(d(x,y)\le d(x,z)+d(z,y)\)
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.
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

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

\(\mathrm {(id)}\) is the class of intervaldecomposable 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 intervaldecomposable persistence modules, is the subclass of \(\mathrm {(id)}\) where the index set A is countable.

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

\(\mathrm {(fid)}\), the finitely intervaldecomposable 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 finiteinterval 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 qtame 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

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 \(i\in \{1,\ldots ,n1\}\) , and

for \(a_n\le s\le t\), \(M(s\le t)\) is an isomorphism.
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 [r, r] 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
 1.
\(\mathrm {(eph)}\not \subset \mathrm {(pfd)}\): \(\bigoplus _{k=1}^{\infty } [0,0]\) is in \(\mathrm {(eph)}\) but not in \(\mathrm {(pfd)}\).
 2.
\(\mathrm {(eph)}\not \subset \mathrm {(cid)}\): \(\bigoplus _{r \in {\mathbb {R}}} [0,0]\) is in \(\mathrm {(eph)}\) but not in \(\mathrm {(cid)}\).
 3.
\(\mathrm {(ffid^{[c,d]})}\not \subset \mathrm {(eph)}\): [c, d] is in \(\mathrm {(ffid^{[c,d]})}\) but is not in \(\mathrm {(eph)}\).
 4.
\(\mathrm {(fid)}\not \subset \mathrm {(cfid)}\): \([0,\infty )\) is in \(\mathrm {(fid)}\) but is not in \(\mathrm {(cfid)}\).
 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.
\(\mathrm {(cfid)}\not \subset \mathrm {(qtame)}\): \(\bigoplus _{k=1}^{\infty }[0,1)\) is in \(\mathrm {(cfid)}\) but is not in \(\mathrm {(qtame)}\).
 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).
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.
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 onepoint 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 nonsymmetric distance between classes of persistence modules and calculate its value for all of the pairs in Fig. 1.
Definition 5
For example, as we will demonstrate later in this section, \(E(\mathrm {(0)},\mathrm {(ffid^{[c,d]})}) = \frac{dc}{2}\) 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 sinterleaved and persistence modules B and C are tinterleaved, 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 sinterleaved 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 A, B are sinterleaved. 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 sinterleaved.
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 sinterleaved. 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.
We have the following enveloping distances: \(E(\mathrm {(0)},\mathrm {(ffid^{[c,d]})}) = \frac{dc}{2}\) and \(E(\mathrm {(ffid^{[c,d]})},\mathrm {(ffid)}) = \infty ^\). Also, \(E(\mathrm {(0)},\mathrm {(ffid)}) = \infty ^\), \(E(\mathrm {(eph)},\mathrm {(ffid)}) = \infty ^\) and \(E(\mathrm {(eph)},\mathrm {(ffid^{[c,d]})}) = \frac{dc}{2}\).
 2.
In addition, \(E(\mathrm {(cfid)},\mathrm {(fid)})=\infty \) and \(E(\mathrm {(qtame)},\mathrm {(cfid)})=\infty \).
 3.
Furthermore, \(E(\mathrm {(pfd)},\mathrm {(qtame)})=0\).
 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.

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

\(\mathrm {(ffid^{[c,d]})}\subset \mathrm {(ffid)}\): 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.

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

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

 3.
This case is more interesting and its proof will occupy the remainder of this subsection below.
 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
Proposition 4
Let M be a persistence module and let \(p\ge 0\). Then M and \(M^{(p)}\) are pinterleaved.
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 qtame persistence module. Let \(p>0\). Then by definition, \(M^{(p)}\) is a pointwise finitedimensional persistence module. By Proposition 4, M and \(M^{(p)}\) are pinterleaved. Thus, by definition, \(E(\mathrm {(pfd)},\mathrm {(qtame)})=0\). \(\square \)
3.5 Sets of persistence modules
Proposition 5
The class \(\mathrm {(cid)}\) is a set.
Proof
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 directsum intervalmodule decomposition of a pointwise finitedimensional 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
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 Intervaldecomposable 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 intervaldecomposable 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
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

\(\mathrm {(ffid^{[c,d]})}\subset \mathrm {(ffid)}\). 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 (c, d), 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 \(\varepsilon < \frac{dc}{2}\). For \(n\ge 0\) consider \(M_n = \bigoplus _{k=1}^n [c,d)\). Then for \(m\ne n\), \(d_I(M_m,M_n) = \frac{dc}{2}\). 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 \(\delta < \varepsilon \), \(\delta < dc\) and \(\delta < \frac{1}{4} \min _{j} {{\mathrm{diam}}}I_j\). Choose an interval I of diameter \(\delta \) contained in [c, d]. 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 (X, d) 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 \(M^{(t)} = [c + t\frac{dc}{2}, d  t\frac{dc}{2})\). Then for \(0 \le s \le t \le 1\), \(d_I(M^{(s)},M^{(t)}) = (ts)\frac{dc}{2}\). 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 \(M^{(t)} = \bigoplus _{k=1}^N [c_k + t \frac{d_kc_k}{2}, d  t \frac{d_kc_k}{2})\). Then for \(0 \le s \le t \le 1\), \(d_I(M^{(s)},M^{(t)}) \le (ts)\max _{1 \le k \le N} \frac{d_kc_k}{2}\). 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 \(h_k = \frac{d_kc_k}{2}\). Let \(M^{(0)} = M\), \(M^{(1)} = 0\) and for \(0<t<1\), \(M^{(t)} = \bigoplus _{k \in A} [c_k+th_k,dth_k)\).
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
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 [c, d]. \(\square \)
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
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
 1.
second countable;
 2.
separable; and
 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 [c, d], 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 qtame 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.
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 \((\beta \alpha )_a = \beta _a \alpha _a\).
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 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 \((a,k1) \le (a+\frac{1}{2^k},k)\) and \((a,k) \le (a+\frac{1}{2^k},k1)\), where \(a \in {\mathbb {R}}\) and \(k \ge 1\).
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 \(d_I(I_i(x_i),I_i(x'_i)) = x_ix'_i \le \frac{1}{100N}\). 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 \(d_I(M(x),M(x')) = xx'_{\infty }\). \(\square \)
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

Are \(\mathrm {(cid)}\) and \(\mathrm {(cfid)}\) complete?

Can the results presented here be extended to multiparameter persistence modules and generalized persistence modules?
Footnotes
 1.
In particular, the multiplicity of \([a_i,a_j)\) can be calculated using the inclusion/exclusion formula \({{\mathrm{rank}}}M(a_i \le a_{j1})  {{\mathrm{rank}}}M(a_i \le a_j)  {{\mathrm{rank}}}M(a_{i1} \le a_{j1}) + {{\mathrm{rank}}}M(a_{i1} \le a_j)\) (CohenSteiner 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 pointwisefinite dimensional persistence modules to qtame 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 NSFSimons Research Center for Mathematics of Complex Biological Systems, under National Science Foundation Grant No. DMS1764406 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
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