On locally compact Hausdorff spaces with finite metrizability number

The metrizability number m(X) of a space X is the smallest cardinal number κ such that X can be represented as a union of κ many metrizable subspaces. In this paper, we study compact Hausdorff spaces with finite metrizability number. Our main result is the following representation theorem: If X is a locally compact Hausdorff space with m(X)=n<ω, then for each k, 1⩽k<n, X can be represented as X=G∪F, where G is an open dense subspace, F=X⧹G, m(G)=k, and m(F)=n−k.