Linear equivalence of (pseudo) compact spaces. Issue 3 (4th March 2023) - Baars, Jan; van Mill, Jan; Tkachuk, Vladimir V.
Abstract: Given Tychonoﬀ spaces X and Y, Uspenskij proved in [15] that if X is pseudocompact and Cp ( X ) is uniformly homeomorphic to Cp ( Y ), then Y is also pseudocompact. In particular, if Cp ( X ) is linearly homeomorphic to Cp ( Y ), then X is pseudocompact if and only if so is Y . This easily implies Arhangel'skii's theorem [1] which states that, in the case when Cp ( X ) is linearly homeomorphic to Cp ( Y the space X is compact if and only if Y is compact. We will establish that existence of a linear homeomorphism between the spaces Cp *( X ) and Cp *( Y ) implies that X is (pseudo)compact if and only if so is Y . We will also show that the methods of proof used by Arhangel'skii and Uspenskij do not work in our case.