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Dimension of the square of a compactum and local connectedness. (English) Zbl 1039.54018

A compactum \(X\) has cohomological dimension at most \(n\) with respect to an abelian group \(G\), denoted by \(\dim_G X \leq n\) provided that \(\check{H}^{n+1}(X,A;G) = 0\) for any closed subset \(A\) of \(X\). K. Borsuk [Math. Ann. 109, 376–380 (1934; Zbl 0008.18301)] showed that for an \(n\)-dimensional ANR compactum \(X\) there exists a prime number \(p\) such that \(\dim_{\mathbb {Z}_p} X = n\). The author employs modern language from J. Dydak and A. Koyama [Topology Appl. 113, No. 1–3, 39–50 (2001; Zbl 1094.55002)] and shows that if a locally (\(n-1\))connected compactum \(X\) has \(\dim_{\mathbb Z} X = n\), there exists a prime number \(p\) such that \(\dim_{\mathbb{Z}_p} X = n\). As an application he obtains that for an \(n\)-dimensional locally (\(n-1\))-connected compactum \(X\), \(\dim (X \times X) = 2n\). This is also a generalization of an interesting property of compact ANR pointed at K. Borsuk’s book [Theory of retracts. (Warsaw: PWN - Polish Scientific Publishers) (1967; Zbl 0153.52905)].

MSC:

54F45 Dimension theory in general topology
54D05 Connected and locally connected spaces (general aspects)
55M10 Dimension theory in algebraic topology
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References:

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