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Dimensions of Spaces

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Geometric Aspects of General Topology

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Abstract

For an open cover \(\mathcal{U}\) of a space X, \(\mathrm{ord}\,\mathcal{U} =\sup \{ \mathrm{card}\,\mathcal{U}[x]\mid x \in X\}\) is called the order of \(\mathcal{U}\). Note that \(\mathrm{ord}\,\mathcal{U} =\dim N(\mathcal{U}) + 1\), where \(N(\mathcal{U})\) is the nerve of \(\mathcal{U}\). The (covering) dimension of X is defined as follows: dimX ≤ n if each finite open cover of X has a finite open refinement \(\mathcal{U}\) with \(\mathrm{ord}\,\mathcal{U}\leq n + 1\). and then, dimX = n if dimX ≤ n and dimX ≮ n. By \(\dim X = -1\), we mean that X = ∅. We say that X is \(\boldsymbol{n}\)-dimensional if dimX = n and that X is finite-dimensional (f.d.) (dimX < ∞) if dimX ≤ n for some n ∈ ω. Otherwise, X is said to be infinite-dimensional (i.d.) (dimX = ∞). The dimension is a topological invariant (i.e., dimX = dimY if X ≈ Y ).

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Notes

  1. 1.

    Such a map r is called a retraction, which will be discussed in Chap. 6.

  2. 2.

    In this chapter, spaces are assumed normal, but the small inductive dimension also makes sense for regular spaces.

  3. 3.

    In many articles, the infinite dimensionality is not assumed, i.e., w.i.d. = not s.i.d., so f.d. implies w.i.d. However, here we assume the infinite dimensionality because we discuss the difference among infinite-dimensional spaces.

  4. 4.

    It is known that \(\ell_{1} \approx \ell_{2} \approx {\mathbb{R}}^{\mathbb{N}}\), where the latter homeomorphy was proved by R.D. Anderson.

  5. 5.

    Since \(\mathrm{rint}\,\boldsymbol{Q}\) and \({\mathbf{I}}^{\mathbb{N}} \setminus {(0, 1)}^{\mathbb{N}}\) are not completely metrizable, they are not homeomorphic to \({\mathbb{R}}^{\mathbb{N}}\), but it is known that \(\mathrm{rint}\,\boldsymbol{Q} \approx {\mathbf{I}}^{\mathbb{N}} \setminus {(0, 1)}^{\mathbb{N}}\).

  6. 6.

    Refer to Engelking’s book “Theory of Dimensions, Finite and Infinite,” Problem 6.1.E.

  7. 7.

    When Y is paracompact, \(\{\mathcal{V}(f)\mid \mathcal{V}\in \mathrm{cov\,(Y )}\}\) is a neighborhood basis of each f ∈ C(X, Y ) and the topology is Hausdorff.

  8. 8.

    In this case, a proper map coincides with a perfect map (Proposition 2.1.5).

  9. 9.

    Usually, the phrase “the class of” is omitted.

  10. 10.

    Recall that \(\nu^0\) denotes the space \(\mathbb{R} \setminus \mathbb{Q}\). Then, \(\nu_0 \subsetneqq \nu^0\) but \(\nu _{0} \approx ((-1, 1) \setminus \mathbb{Q})^{\mathbb{N}} \approx \nu ^{0}\).

  11. 11.

    This is different from the usual notation. In the literature for Dimension Theory, this space is represented by \(K_{\omega}(\aleph_0)\) and \(K_{\omega }\) stands for \(\mathbf{I}_{f}^{\mathbb{N}}\).

  12. 12.

    Recall \({U}^{-} =\{ A \subset Y \mid A \cap U\not =\emptyset \}\) and \({U}^{+} =\{ A \subset Y \mid A \subset U\}\).

  13. 13.

    In a metric space X = (X, d), A ⊂ X is said to be \(\boldsymbol{\varepsilon }\)-dense if \(d(x,A) <\varepsilon\) for each x ∈ X.

  14. 14.

    Refer to the last Remark of Sect. 2.2.

  15. 15.

    Recall that a continuum is a compact connected metrizable space.

  16. 16.

    In general, each link U i is not assumed to be connected and open.

  17. 17.

    This condition is stronger than usual, and is adopted to simplify our argument. Usually, it is said that \((U_{1},\ldots ,U_{n})\) is a simple chain if \(U_{i} \cap U_{j}\not =\emptyset \Leftrightarrow \vert i - j\vert \leq 1\). However, in our definition, \(U_{i} \cap U_{j}\not =\emptyset \Leftrightarrow \mathrm{cl}\,U_{i} \cap \mathrm{cl}\,U_{j}\not =\emptyset \Leftrightarrow \vert i - j\vert \leq 1\).

  18. 18.

    Recall that an arc is an injective path, i.e., an embedding of I.

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  1. Department of Mathematics Faculty of Engineering, Kanagawa University, Yokohama, Kanagawa, Japan

    Katsuro Sakai

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Sakai, K. (2013). Dimensions of Spaces. In: Geometric Aspects of General Topology. Springer Monographs in Mathematics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54397-8_5

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