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 plausiblesounding conjectures Despite their pathological nature each of them has its strange intrinsic beautyThe space X of Sect 51 was introduced by Erdös in 34 It has topological dimension dim X = mathrmindX = mathrmIndX = 1 see 34 and Roberts 95 proved that it can be embedded in the Euclidean plane mathbb R2 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 thereinThe KnasterKuratowski 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 KnasterKuratowski fan Y a point p in a connected space C is called a dispersion point if the space Csetminus p is totally disconnected Propositions 523  266 and Corollary  233 imply that the punctured KnasterKuratowski fan X = Ysetminus y 0 satisfies dim X ge 1 As X subset Y subset mathbb R2 we deduce that 1 le dim X le dim Y le 2 by applying Theorem  183 and Corollary  357 Actually it can be shown that dim X = dim Y = 1 by using the fact that every subset of mathbb Rn whose topological dimension is n has nonempty interior see for example 50 Th IV3 p 44The counterexample of Sect 53 was described by Bing in a onepage 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 533 that the Bing space is not a Urysohn space An example of a countablyinfinite connected Urysohn space admitting a dispersion point was constructed by Roy in 97The 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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