## 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

Zbl 0518.54030