Linear homeomorphisms of function spaces and properties close to countable compactness (Q2850615)

Let \(X\) be a Tychonoff space and \(C_p(X)\) denote the space of all continuous functions on \(X\) endowed with the pointwise topology. The symbols \(F(X)\) and \(A(X)\) stand for the free topological group and the free Abelian topological group of \(X\) in the sense of Markov (see [\textit{A. Markov}, Izv. Akad. Nauk SSSR, Ser. Mat. 9, 3--64 (1945; Zbl 0061.04210)]). Two topological spaces \(X\) and \(Y\) are called \(M\)-equivalent (or \(A\)-equivalent) if their free topological groups (or the free Abelian topological groups) are topologically isomorphic. The space \(X\) and \(Y\) are \(l\)-equivalent if \(C_p(X)\) and \(C_p(Y)\) are linearly homeomorphic. The paper studies several topological properties of countable compactness type that are preserved by the relations of \(l\)-equivalence, \(M\)-equivalence or \(A\)-equivalence.NEWLINENEWLINEThe first result states that a topological property invariant with respect to to continuous images, closed subspaces, finite products and finite unions is preserved by \(A\)-equivalence within the class of pseudocompact spaces. As a corollary the authors obtain that total countable compactness, \(\omega\)-boundedness, \(p\)-compactness, sequential compactness and initial \(\kappa\)-compactness with \(\kappa\) a singular strong limit cardinal are preserved by \(A\)-equivalence.NEWLINENEWLINEThe main result of the paper is then a theorem stating that \(\omega\)-boundedness is preserved by \(l\)-equivalence (recall that a space is \(\omega\)-bounded if its every countable subset has a compact closure).