Linear equivalence of (pseudo) compact spaces. Issue 3 (4th March 2023)

• Record Type: Journal Article
• Authors: Baars, Jan; van Mill, Jan; Tkachuk, Vladimir V.
• Is Part Of: Quaestiones mathematicae. Volume 46:Issue 3(2023)
• Page Start: 513
• Page End: 518
• Publication Date: 2023-03-04
• Subjects: 54C35 -- 57N17; Function space -- lp*-equivalence -- metric space -- pseudocompact space -- compact space; Mathematics -- Periodicals; 510.5
• DOI: 10.2989/16073606.2022.2034066
• Languages: English
• ISSNs: 1607-3606

Abstract

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.