Structure theorems for finite unions of subspaces of special kind

We study the internal structure of topological spaces X which can be represented as the union of a finite collection of subspaces belonging to some nice class of spaces. Several closely related structure theorems are established. In particular, they concern the finite unions of subspaces with the weight ≤τ, the finite unions of subspaces with a point-countable base, and the finite unions of metrizable subspaces. As a corollary, we extend to finite unions the classical Mischenko's Theorem on metrizability of compacta with a point-countable base [11] (see Theorem 11). A few other applications of the structure theorems are given, in particular, to homogeneous spaces ( Corollary 5 and Corollary 10).