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Some Classical Counterexamples

Authors: Michel Coornaert,

Publish Date: 2015
Volume: , Issue:, Pages: 87-104
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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.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 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.


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