If \(X\) is \(\sigma\)-compact Polish, then \(C_{k}\)(\(X\)) has a \(\sigma\)-closure-preserving base
As the title itself indicates, the main result of this paper is to show that if \(X\) is a \(\sigma\)-compact Polish space, then the function space \(C_k(X)\) of all continuous functions on \(X\) with the compact open topology is an M\(_1\)-space. First, the authors show that if \(X\) is such a space, then \(C_k(X)\) is a \(\mu\)-space. This gives a positive answer to a question by Gartside. Combining the result of Gartside and Reznichenko that \(C_k(X)\) is stratifiable if \(X\) is a Polish space with this new result, \(C_k(X)\) is automatically an M\(_1\)-space. However, the authors also construct directly a \(\sigma\)-closure-preserving base for \(C_k(X)\).
scientific article; zbMATH DE number 2174907