On locally compact Hausdorff spaces with finite metrizability number (Q5939250)

The metrizability number \(m(X)\) of a space X is the smallest cardinal number \(\kappa\) such that \(X\) can be represented as a union of \(\kappa\) many metrizable subspaces. For example, the space \(\psi\) (also called \({\mathcal N}\cup{\mathcal R}\)) has metrizability number 2, and its one-point compactification has metrizability number 3. The authors give more interesting examples. For one, they prove that the one-point compactification of any ladder system on \(\omega_1\) has metrizability number 3. For another, they prove that for every \( 3\leq n <\omega\), there exists a compact separable Hausdorff space \(X\) such that \(X\) has weight \(\omega_1\), metrizability number \(n\) and \(X\) is the increasing union of countably many closed subsets each of metrizability number 2. The main result is the theorem: If X is a locally compact Hausdorff space with \(m(X) =n<\omega\), then for each \(1 \leq k<n \), \(X\) can be represented as \(X = G\cup F \), where \(G\) is an open dense subspace, \(m(G)=k\), \(F=X\setminus G \), and \(m(F)=n-k\). It follows that by choosing \(k=n-1\), the open dense set \(G\) satisfies \(m(G) = n-1\), and \(F= X\setminus G\) is metrizable. The authors give some results on the metrizability number of the product of two spaces. For example, if \(X, Y\) are Hausdorff spaces with \(X\) compact and \(Y\) locally compact, and both with finite metrizability number, then \(m(X\times Y)\geq (m(X) + m(Y) -1)\). Some related topics are also considered. Among them are the metrizability number at a point, and the \(m\)-spectrum of a space. The paper continues a sequence of papers by the authors [Topology Appl. 59, No. 3, 287-298 (1994; Zbl 0840.54025); ibid. 63, No. 1, 69-77 (1995; Zbl 0860.54005) and ibid. 71, No. 2, 179-191 (1996; Zbl 0864.54001)].