Linear homeomorphisms of topological almost modules of continuous functions and coincidence of dimension

In this paper, the space of continuous functions C (X, G), where G is a topological space, is considered. If the set G is endowed with an almost ring structure, the set C (X, G) is a topological almost module. It is proved that the dimension dim of the topological space X is an isomorphic invariant of its topological almost module C (X, I), where I = [0, 1) is a naturally defined almost ring. This statement is based on ideas of G.G. Pestov's work «The coincidence of dimension dim of /-equivalent topological spaces», where the following theorem was formulated: if C (X, R) and C (Y, R) are linearly homeomorphic spaces, then dim X = dim Y. Here, X and Y are arbitrary totally regular spaces, and C (X, R) is the space of all continuous real functions on X with the pointwise convergence topology. Note that Pestov's theorem was generalized to the case of uniform homeomorphisms by S. P. Gul'ko.

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