Linear equivalence of (pseudo) compact spaces

Abstract

Given Tychonoff spaces X and Y, Uspenskij proved in [15] that if X is pseudocompact and Cp(X) is uniformly homeomorphic to C_p(Y), then Y is also pseudocompact. In particular, if C_p(X) is linearly homeomorphic to C_p(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 C_p(X) is linearly homeomorphic to C_p(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 C_p*(X) and C_p*(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.

Mathematics Subject Classification (2020):

• 54C35
• 57N17

Keywords:

• Function space
• l_p*-equivalence
• metric space
• pseudocompact space
• compact space