Zero-Dimensional Spaces

Abstract

This chapter is devoted to 0-dimensional topological spaces.

Appendices

Notes

The terminology used in this chapter follows that of Bourbaki [18]. However, the terms “scattered”, “totally disconnected”, and “totally separated” have sometimes different meanings in the literature. For example, spaces that are called “scattered” in the present book are called “zero-dimensional” in [102], while a “scattered” space in [102] is a topological space in which every non-empty subset admits an isolated point.

Table 2.1 Summary Table (X non-empty)

The Cantor ternary set was described by Cantor in [21, note 11 p. 46]. It can be shown that every totally disconnected compact metrizable space that is perfect is homeomorphic to the Cantor set (see for example [48, Corollary 2–98]).

A non-empty topological space X is scattered if and only if ${{\mathrm{ind}}}(X) = 0$ (see the Notes on Chap. 1, p.19, for the definition of the small inductive dimension ${{\mathrm{ind}}}(X)$). The question of the existence of scattered metrizable spaces with positive topological dimension remained open for many years (cf. [18, note 1 p. IX.119]). An affirmative answer to this question was finally given by Roy [96, 98] who constructed a scattered metrizable space X with $\dim (X) = 1$.

The notion of a totally disconnected space and that of a totally separated space were respectively introduced by Hausdorff [47] and by Sierpinski [99]. In [99], Sierpinski described a totally disconnected subset of $\mathbb {R}^2$ that is not totally separated and a totally separated subset of $\mathbb {R}^2$ with positive topological dimension.

Exercises

• 2.1 Does the real number $1/\pi$ belong to the Cantor set?

• 2.2 Show that the Cantor set has Lebesgue measure 0.

• 2.3 Show that every countable product of Cantor spaces is a Cantor space.

• 2.4 Let H denote the Hilbert space of square-summable real sequences $(u_n)_{n \ge 1}$. Show that the subset $X \subset H$ consisting of all sequences $(u_n)_{n \ge 1}$ such that $\vert u_n \vert \le 1/n$ for all $n \ge 1$ is homeomorphic to the Hilbert cube $[0,1]^\mathbb {N}$.

• 2.5 Let G be a group. Let $\mathcal {B}$ denote the set of all left cosets of subgroups of finite index of G, i.e., the subsets of the form gH, where $g \in G$ and $H \subset G$ is a subgroup with $[G:H] < \infty$.

  • (a) Show that there is a unique topology on G admitting $\mathcal {B}$ as a base. This topology is called the profinite topology on G.

  • (b) Show that the profinite topology on G is scattered.

  • (c) Show that the profinite topology on G is discrete if and only if G is finite.

  • (d) Show that the profinite topology on the additive group $\mathbb {Q}$ of rational numbers is the trivial topology.

  • (e) Show that the profinite topology on G is Hausdorff if and only if G is residually finite. (Recall that the group G is called residually finite if the intersection of all its subgroups of finite index is reduced to the identity element.)

• 2.6 (Furstenberg’s topological proof of the infinitude of primes [38]). Let $\mathbb {Z}$ denote the group of integers equipped with its profinite topology (see Exercise 2.5).

  • (a) Show that $n\mathbb {Z}$ is a closed subset of $\mathbb {Z}$ for every $n \in \mathbb {Z}$.

  • (b) Show that every non-empty open subset of $\mathbb {Z}$ is infinite.

  • (c) Let $\mathcal {P}:= \{2,3,5,7,11,\dots \}$ denote the set of prime numbers. Use the results obtained in (a) and (b) to recover Euclid’s theorem that $\mathcal {P}$ is infinite. Hint: observe that $\bigcup _{p \in \mathcal {P}} p\mathbb {Z}= \mathbb {Z}{\setminus }\{-1,1\}$ is not closed in $\mathbb {Z}$.

• 2.7 Let $f :X \rightarrow Y$ be a continuous map from a Lindelöf space X into a topological space Y. Show that f(X) is a Lindelöf space.

• 2.8 Show that every locally compact Lindelöf space is $\sigma$-compact.

• 2.9 Let X be an uncountable set equipped with its cofinite topology. Show that X is not first-countable.

• 2.10 Show that every open subset of a separable space is separable.

• 2.11 Show that every subspace of a separable metrizable space is separable.

• 2.12 Show that the set consisting of all isolated points of a separable space is countable.

• 2.13 Show that every countable product of separable spaces is separable.

• 2.14 Let (X, d) be a separable metric space. Consider the Banach space $\ell ^\infty (\mathbb {R})$ consisting of all bounded sequences of real numbers $u = (u_n)_{n \in \mathbb {N}}$ with the supremum norm $\Vert u \Vert = \sup _{n \in \mathbb {N}} \vert u_n \vert$. Fix a point $x_0 \in X$ and a sequence $(a_n)_{n \in \mathbb {N}}$ of points of X such that the set $\{a_n \mid \; n \in \mathbb {N}\}$ is dense in X. Show that the sequence $(d(x,a_n) - d(x_0,a_n))_{n \in \mathbb {N}}$ is in $\ell ^\infty (\mathbb {R})$ for every $x \in X$ and that the map $\varphi :X \rightarrow \ell ^\infty (\mathbb {R})$ defined by $\varphi (x) = (d(x,a_n) - d(x_0,a_n))_{n \in \mathbb {N}}$ is an isometric embedding.

• 2.15 Show that the Banach space $\ell ^\infty (\mathbb {R})$ is not separable.

• 2.16 Show that every second-countable scattered accessible space is homeomorphic to a subset of the Cantor set.

• 2.17 A metric space (X, d) is called an ultrametric space if one has

  $$ d(x,y) \le \max (d(x,z),d(y,z)) $$

  for all $x,y, z \in X$. Let (X, d) be a non-empty ultrametric space.

  • (a) Let A be a closed subset of X and $\rho > 0$. Show that the set consisting of all $x \in X$ such that ${{\mathrm{dist}}}(x,A) = \rho$ is a clopen subset of X.

  • (b) Let A and B be disjoint closed subsets of X. Show that the set consisting of all $x \in X$ such that ${{\mathrm{dist}}}(x,A) \le {{\mathrm{dist}}}(x,B)$ is a clopen subset of X.

  • (c) Show that $\dim (X) = 0$.

  • (d) Show that the metric completion $(X',d')$ of (X, d) is also an ultrametric space.

• 2.18 Let p be a prime integer. Every non-zero rational number $q \in \mathbb {Q}{\setminus }\{0\}$ can be written in the form $q = p^n \dfrac{a}{b}$, where $n \in \mathbb {Z}$ and $a,b \in \mathbb {Z}{\setminus }p\mathbb {Z}$ are integers not divisible by p. The integer $v_p(q) := n \in \mathbb {Z}$ is well defined and called the p-valuation of q. Define the map $d :\mathbb {Q}\times \mathbb {Q}\rightarrow \mathbb {R}$ by

  $$ d(x,y) := {\left\{ \begin{array}{ll} p^{- v_p(x - y)} &{}\text { if } x \not = y \\ 0 &{}\text { if } x = y \end{array}\right. } $$

  for all $x,y \in \mathbb {Q}$.

  • (a) Show that $(\mathbb {Q},d)$ is an ultrametric space.

  • (b) Show that the metric completion $\mathbb {Q}_p$ of $(\mathbb {Q},d)$ satisfies $\dim (\mathbb {Q}_p) = 0$. (The set $\mathbb {Q}_p$ is the set of p-adic numbers.)

• 2.19 Show that every totally disconnected topological space that is locally connected is discrete. (Recall that a topological space X is called locally connected if every point $x \in X$ admits a neighborhood base consisting of connected subsets.)

• 2.20 Let X be a non-empty subset of $\mathbb {R}$. Show that one has $\dim (X) = 0$ if and only if X is totally disconnected.

• 2.21 Let X be the topological space described in Example 2.5.7.

  • (a) Show that X is compact.

  • (b) Show that X is not normal.

  • (c) Show that $\dim (X) = 1$.

• 2.22 A topological space X is called extremally disconnected if the closure of any open subset of X is open in X.

  • (a) Show that if a set X is equipped with its trivial (resp. discrete) topology then X is extremally disconnected.

  • (b) Show that every extremally disconnected Hausdorff space is totally separated.

  • (c) Show that every extremally disconnected metrizable space is discrete.