Star-countable \(k\)-networks, and quotient images of locally separable metric spaces
This paper involves spaces that have star-countable \(k\)-networks, and establishes three kinds of theorems for such spaces. First, a reflection theorem is given showing that such spaces are \(\aleph_{0}\)-spaces whenever every small (cardinality \(\leq\omega_{1}\)) subspace is a separable \(\sigma\)-space. Also \(k\)-spaces having star-countable \(k\)-networks are shown to be quotient (compact-covering) \(s\)-images of locally separable metric spaces if and only if they contain no closed copy of \(S_{\omega_{1}}\). The final result is that the hyperspace of compact subsets of a space \(X\) has a star-countable \(k\)-network if and only if \(X\) does.
scientific article; zbMATH DE number 1121468
scientific article