\(s\)-point finite refinable spaces (Q1301927)

The authors call a topological space \(X\) \(s\)-point finite refinable (respectively \(ds\)-point finite refinable) if for every open cover \({\mathcal U}\) of \(X\) there exists an open refinement \({\mathcal V}\) of \({\mathcal U}\) and a (closed discrete) subset \(A({\mathcal U})\) of \(X\) such that (i) for all nonempty \(V \in{\mathcal V}\), \(V\cap A({\mathcal U})\neq\emptyset\), and (ii) for all \(a\in A({\mathcal U})\) the set \(\{V\in{\mathcal V}\mid a\in V\}\) is finite. Obviously, every \(ds\)-point finite refinable space is \(s\)-point finite refinable. Since it follows from two theorems of \textit{J. R. Boone} [Pac. J. Math. 62, 351-357 (1976; Zbl 0327.54012)] that every weak \(\overline\theta\)-refinable space is irreducible, and that every irreducible space is \(ds\)-point finite refinable, both covering properties are very weak. It is shown that if \(\lambda\) is an ordinal with \(cf( \lambda) =\lambda> \omega\) and \(X\) is a stationary subset of \(\lambda\) then \(X\) is not \(s\)-point finite refinable. Moreover, there exists a countably compact \(s\)-point finite refinable LOTS which is not \(ds\)-point finite refinable. However, it is not known whether there exist \(ds\)-point finite refinable spaces which are not irreducible. It is shown that a topological space is \(ds\)-point finite refinable if and only if it is irreducible of order \(\omega\) in the sense of \textit{J. R. Boone} [ibid. 62, 359-364 (1976; Zbl 0331.54009]. It is also shown that every strongly collectionwise Hausdorff \(ds\)-point finite refinable space without isolated points is irreducible. It is mentioned that H. R. Bennett has shown that a LOTS is paracompact if and only if it is \(ds\)-point finite refinable. Concerning irreducible spaces it is shown that every topological space can be embedded as a closed subspace into an irreducible space.