A note on the Brouwer dimension of chainable spaces
For a normal space \(X\), Brouwer's dimensionsgrad is defined as follows. \(\roman{Dg} X = -1\) if and only if \(X = \emptyset\). For \(X \neq \emptyset\), \(\roman{Dg} X\) is the smallest non-negative integer \(n\) for which between any pair of disjoint closed subsets \(A\) and \(B\) of \(X\), there exists a \textit{cut} \(C\) with \(\roman{Dg} C \leq n-1\), if such an integer exists. If no such integer exists, \(\roman{Dg} X = \infty\). In general, the inequality \(\roman{Dg} X \leq \roman{Ind} X\) clearly holds. Moreover, dimensionsgrad, \(\roman{Dg} X\), is Brouwer's original definition of dimension and for long years it was believed that \(\roman{Dg} X\) and \(\roman{Ind} X\) coincide for any locally connected separable metric space \(X\). However, \textit{V. V. Fedorčuk} and \textit{Jan van Mill} [Fundam. Math. 163, 77--82 (2000; Zbl 0944.54025)] constructed for any given \(n > 1\) an \(n\)-dimensional locally connected Polish space \(X\) with \(\roman{Dg} X = 1\). On the other hand, \textit{V. V. Fedorčuk}, \textit{M. Levin} and \textit{E. V. Shchepin} [Russ. Math. Surv. 54, 432--433 (1999; Zbl 0995.54032 ); translation from Usp. Mat. Nauk 54, 193--194 (1999)] proved that for compact metric spaces \(\dim = \roman{Dg}\), and \textit{V. V. Fedorčuk} [Math. Notes 73, 271--279 (2003; Zbl 1027.54052 ); translation from Mat. Zametki 73, 295--304 (2003)] showed that \(\dim \leq \roman{Dg}\) for compact Hausdorff spaces. He also pointed out that the equality \(\dim X= \text{Dg }X\) does not hold even for a first countable compact space \(X\). \textit{V. V. Fedorčuk} and \textit{V. A. Chatyrko} [ Vestn. Mosk. Univ., Ser. I 2005, No. 2, 22--27 (2005; Zbl 1102.54036); translation in Mosc. Univ. Math. Bull. 60, No. 2, 22--27 (2005)] showed that Chatyrko's examples of first countable, separable, chainable compact Hausdorff spaces with \(\dim = 1 < \roman{ind}\) satisfy \(\roman{Dg} = 1\). Then they naturally posed the problem whether every chainable compact Hausdorff space had \(\roman{Dg} = 1\). In this paper the author shows that Charalambous' first countable, separable compact Hausdorff spaces \(X_n\), \(n \geq 1\), with \(\dim X_n = 1\) and \(\roman{ind} X_n = \roman{Ind} X_n = 1\) are chainable. Therefore for \(n > 1\), \(\roman{Dg} X_n < \roman{Ind} X_n = n\).
scientific article; zbMATH DE number 5013539
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