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On uniform homeomorphisms of spaces of continuous functions. (English. Russian original) Zbl 0804.54018

Proc. Steklov Inst. Math. 193, 87-93 (1993); translation from Tr. Mat. Inst. Steklova 193, 82-88 (1992).
Suppose that \(X\) and \(Y\) are completely regular spaces, \(\dim X\) denotes the Lebesgue dimension of \(X\), and \(C_ p(X)\) denotes the space of all continuous real valued functions on \(X\) with the topology of pointwise convergence. According to a result of V. G. Pestov [Sov. Math., Dokl. 26, 380-383 (1982); translation from Dokl. Akad. Nauk SSSR 266, 553-556 (1982; Zbl 0518.54030)], if \(C_ p(X)\) and \(C_ p(Y)\) are linearly homeomorphic, then \(\dim X = \dim Y\). In the present paper, the author extends this result by showing that if \(C_ p (X)\) and \(C_ p (Y)\) are uniformly homeomorphic, then \(\dim X = \dim Y\). He first proves his result for second countable spaces and then uses inverse limits to obtain the general case.
For the entire collection see [Zbl 0785.00035].

MSC:

54C35 Function spaces in general topology
54F45 Dimension theory in general topology

Citations:

Zbl 0518.54030
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