Star-countable \(k\)-networks, compact-countable \(k\)-networks, and related results (Q2705819)

Let \(X\) be a space and \({\mathcal P}\) a cover of \(X\). \({\mathcal P}\) is said to be a \(k\)-network for \(X\) if \(K\subset U\) with \(K\) compact and \(U\) open, then \(K\subset\bigcup{\mathcal P}'\subset U\) for some finite subset \({\mathcal P}'\) of \({\mathcal P}\). \({\mathcal P}\) is star-countable if the set \(\{Q\in{\mathcal P}:Q\cap P\neq\emptyset\}\) is countable for each \(P\in{\mathcal P}\) (compact subset \(P\) of \(X)\). In the theory of generalized metric spaces, the notion of \(k\)-networks has played an important role. Every locally separable metric space or CW-complex has a star-countable \(k\)-network. Every space with a star-countable \(k\)-network has a compact-countable \(k\)-network. In this paper, the authors investigate around spaces with a star-countable \(k\)-network, or a compact-countable \(k\)-network. Some interesting results are obtained. For example, let \(X\) be a \(k\)-space with a star-countable \(k\)-network. Then \(X\) is locally separable if and only if \(X\) is the topological sum of \(\aleph_0\)-subspaces, if and only if \(\chi(X) \leq\omega_1\) under CH.