On a supertopological cardinal invariant
Tikhonov spaces X and Y are said to be t-equivalent if \(C_ p(X)\) is homeomorphic to \(C_ p(Y)\). In 1981 A. V. Arkhangel'skij proved that the Souslin number c(X) of a compact space X is equal to the supremum of weights of compact subspaces lying in \(C_ p(X)\). Thus the Souslin numbers of t-equivalent compact spaces coincide. However, it was an open problem for Tikhonov spaces. The new cardinal invariant \(p(X)=\sup \{| \gamma |:\) \(\gamma\) is a point-finite family of open non- empty subsets of \(X\}\) is introduced. It is known that \(p(X)=c(X)\) for any Čech-complete space X. The author proved that \(p(X)=a(C_ p(X))\) for an arbitrary Tikhonov space X, where \(a(Z)=sum\{| A|:\) \(A\subseteq Z\), A is homeomorphic to Alexandroff compactification of some infinite discrete set\(\}\). In particular, \(p(X)=p(Y)\) for any t- equivalent spaces X, Y and \(c(X)=c(Y)\) for any t-equivalent Čech- complete spaces X, Y. Recently the author constructed two spaces X and Y such that the free topological groups F(X) and F(Y) are isomorphic, but \(c(X)<c(Y)\). This answers Arkhangelskij's question in the negative.
supremum of weights of compact subspaces
scientific article; zbMATH DE number 3895870