On locally compact Hausdorff spaces with finite metrizability number

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. 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 < omega, then for each k, 1 less than or equal to k < n, X can be represented as X = G U F, where G is an open dense subspace, F = X \ G, m(G) = k, and m(F) = n - k. (C) 2001 Elsevier Science B.V. All rights reserved.