On the metrizability number and related invariants of spaces

The metrizability number of a space X, m(X), is the smallest cardinal κ such that X can be represented as a union of κ metrizable subspaces; the first countability number, fc(X), is defined analogously. In this paper we study the relationships between m(X), fc(X), z.sfnc X z.sfnc and other invariants of spaces. We show that for a large class of compact Hausdorff spaces those three invariants are equal.