Some Classical Counterexamples

Abstract

The topological spaces presented in this chapter are spaces with amazing properties. Their analysis reveals the validity limits of certain statements in dimension theory and they may be used as counterexamples to various plausible-sounding conjectures. Despite their pathological nature, each of them has its strange intrinsic beauty.

Appendices

Notes

The counterexamples gathered in this chapter played an important role in the history of dimension theory (see [33, 50, 93]). They are named after the mathematicians who discovered them.

The space X of Sect. 5.1 was introduced by Erdös in [34]. It has topological dimension $\dim (X) = {{\mathrm{ind}}}(X) = {{\mathrm{Ind}}}(X) = 1$ (see [34]) and Roberts [95] proved that it can be embedded in the Euclidean plane $\mathbb {R}^2$. Note that X is a subgroup of the additive group H and hence inherits a structure of a topological group. This gives an interesting example of a totally disconnected abelian group. It turns out that X is isomorphic, as a topological group, to certain homeomorphism groups of manifolds (see [29] and the references therein).

The Knaster-Kuratowski fan was described in [60] (see [102, Example 129 p. 145] and [33, p. 29]). It is also sometimes called the Cantor teepee . The point $y_0$ is a dispersion point of the Knaster-Kuratowski fan Y (a point p in a connected space C is called a dispersion point if the space $C{\setminus }\{p\}$ is totally disconnected). Propositions 5.2.3, 2.6.6, and Corollary 2.3.3 imply that the punctured Knaster-Kuratowski fan $X = Y{\setminus }\{y_0\}$ satisfies $\dim (X) \ge 1$. As $X \subset Y \subset \mathbb {R}^2$, we deduce that $1 \le \dim (X) \le \dim (Y) \le 2$ by applying Theorem 1.8.3 and Corollary 3.5.7. Actually, it can be shown that $\dim (X) = \dim (Y) = 1$ by using the fact that every subset of $\mathbb {R}^n$ whose topological dimension is n has non-empty interior (see for example [50, Th. IV.3 p. 44]).

The counterexample of Sect. 5.3 was described by Bing in a one-page paper [14, Example 1] (see [102, p. 93] and [17, I p. 108 exerc. 21 and I p. 115 exerc. 1]). The Bing space has covering dimension $\infty$ and small inductive dimension 1 (cf. [14]). The first examples of countably infinite connected Hausdorff spaces were given by Urysohn in his posthumous article [109]. Subsequently, many other interesting examples of such spaces were discovered (see the paper by Miller [75] and the references therein). A topological space X is called a Urysohn space if any two distinct points of X admit disjoint closed neighborhoods. Of course, every Urysohn space is Hausdorff. It immediately follows from Lemma 5.3.3 that the Bing space is not a Urysohn space. An example of a countably-infinite connected Urysohn space admitting a dispersion point was constructed by Roy in [97].

The Tychonoff plank was introduced in [108]. The smallest uncountable ordinal is sometimes denoted $\omega _1$ instead of $\Omega$. It can be shown that the punctured Tychonoff plank X has covering dimension $\dim (X) = 1$.

The Sorgenfrey topology was used by Sorgenfrey in [100] to show that a product of paracompact spaces is not necessarily paracompact, thus settling in the negative a question previously raised by Dieudonné [28]. According to Cameron [20], it seems that the copaternity of the Sorgenndroff and Urysohn [10] for priority reasons.

Exercises

 

• 5.1 Let X denote the Erdös space (cf. Sect. 5.1).

  • (a) Show that X and $\mathbb {Q}^\mathbb {N}$ are isomorphic as vector spaces over $\mathbb {Q}$ and hence as additive groups.

  • (b) Describe an injective continuous map $f :X \rightarrow \mathbb {Q}^\mathbb {N}$.

  • (c) Show that the space $\mathbb {Q}^\mathbb {N}$ is scattered.

  • (d) Show that there is no subspace of $\mathbb {Q}^\mathbb {N}$ that is homeomorphic to X.

• 5.2 Show that $\dim (\mathbb {Q}^\mathbb {N}) = 0$.

• 5.3 Show that the Erdös space X described in Sect. 5.1 cannot be embedded into $\mathbb {R}$.

• 5.4 Show that the space $\mathbb {Q}^\mathbb {N}$ can be embedded into the Cantor set (and hence into $\mathbb {R}$).

• 5.5 Let X denote the Erdös space (cf. Sect. 5.1). Show that $X \times X$ is homeomorphic to X.

• 5.6 In this exercise, we use the notation of Sect. 5.2. Let x be a point in X and let $c \in K$ such that $x \in Y_c$. Show that the quasi-component of x in X is $Y_c{\setminus }\{y_0\}$.

• 5.7 Show that the Bing space X described in Sect. 5.3 is not regular. (Recall that a topological space X is called regular if for every closed subset A of X and every point $x \in X{\setminus }A$, there exist disjoint open subsets U and V of X such that $A \subset U$ and $x \in V$.)

• 5.8 Let X be the Bing space described in Sect. 5.3. Consider the subsets Y and Z of X defined by $Y := \mathbb {Q}\times \{0\} \cong \mathbb {Q}$ and

  $$ Z := X{\setminus }Y = \{(x,y) \in X \mid \; y >0 \}. $$

  • (a) Show that the topology induced by X on Z is the discrete one.

  • (b) Show that the topology induced by X on Y is the usual topology on $\mathbb {Q}$.

  • (c) Show that Y is an open dense subset of X.

  • (d) Give a direct proof of the fact that X is not compact by showing that the cover of X consisting of Y and all the subsets of the form $Y \cup \{z\}$, where $z \in Z$, is an open cover admitting no finite subcover.

• 5.9 Let X denote the Bing space described in Sect. 5.3. Show that X admits a base consisting of open subsets whose topological boundary is clopen.

• 5.10 Show that a countable connected Hausdorff space having more than one point cannot be compact.

• 5.11 Show that a countable accessible topological space having more than one point cannot be path-connected. Hint: use Baire’s theorem to prove that the unit segment [0, 1] cannot be expressed as the union of a countably-infinite family of pairwise disjoint non-empty closed subsets.

• 5.12 Let X be a countable connected space. Show that every continuous map $f :X \rightarrow \mathbb {R}$ is constant.

• 5.13 (The relatively prime topology on the positive integers [39], [102, Example 60 p. 82]). Let $\mathbb {Z}_+ := \{1,2,\ldots \}$ denote the set of positive integers. Given coprime integers $x,r \in \mathbb {Z}_+$, define the subset $V_r(x) \subset \mathbb {Z}_+$ by

  $$ V_r(x) := \{x + rn \mid \; n \in \mathbb {Z}\} \cap \mathbb {Z}_+. $$

  • (a) Show that there is a unique topology on $\mathbb {Z}_+$ admitting as a base the set consisting of all $V_r(x)$, where $x,r \in \mathbb {Z}_+$ are coprime. In the sequel, the set $\mathbb {Z}_+$ is equipped with this topology.

  • (b) Show that the space $\mathbb {Z}_+$ is Hausdorff.

  • (c) Show that if $x,r \in \mathbb {Z}_+$ are coprime, then every $y \in \mathbb {Z}_+$ that is a multiple of r belongs to the closure of $V_r(x)$.

  • (d) Deduce from (c) that if U and $U'$ are non-empty open subsets of $\mathbb {Z}_+$ then one has $\overline{U} \cap \overline{U'} \not = \varnothing$.

  • (e) Show that the space $\mathbb {Z}_+$ is connected.

• 5.14 The set $[0,\Omega )$, which consists of all the ordinals that are smaller than the first uncountable ordinal $\Omega$, is equipped with its order topology.

  • (a) Show that $[0,\Omega )$ is a locally compact first-countable Hausdorff space.

  • (b) Show that $[0,\Omega )$ is not Lindelöf.

  • (c) Show that $[0,\Omega )$ is not second-countable.

  • (d) Show that $[0,\Omega )$ is sequentially-compact but not compact. (Recall that a topological space X is called sequentially-compact if every sequence of points of X admits a convergent subsequence.)

  • (e) Show that $[0,\Omega )$ is not metrizable.

  • (f) Recover from (e) the fact that neither the punctured Tychonoff plank P nor the Tychonoff plank X is metrizable (cf. Corollary 5.4.12).

• 5.15 Let $\Omega$ denote the first uncountable ordinal. Let $[0,\Omega ]$ be the set consisting of all ordinals $\xi \le \Omega$, equipped with its order topology.

  • (a) Show that $[0,\Omega ]$ is not first-countable.

  • (b) Deduce from (a) that neither the Tychonoff plank P nor the Tychonoff plank X is first-countable.

• 5.16 Show that the punctured Tychonoff plank X is not $\sigma$-compact.

• 5.17 Show that neither the Tychonoff plank P nor the punctured Tychonoff plank X are separable.

• 5.18 Show that every non-empty subspace X of the Sorgenfrey line S has topological dimension $\dim (X) = 0$.

• 5.19 Show that every subspace of the Sorgenfrey line S is normal.

• 5.20 Let A and B be the subsets of the Sorgenfrey plane $S \times S$ defined by

  $$ A := \{(x,-x) \in S \times S \mid \; x \in \mathbb {Q}\} \quad \text {and} \quad B := \{(x,-x) \in S \times S \mid \; x \notin \mathbb {Q}\}. $$

  Give a direct proof that $S \times S$ is not normal (cf. Proposition 5.5.9) by showing that there do not exist disjoint open subsets U and V of $S \times S$ with $A \subset U$ and $B \subset V$.