ИСТИНА

We discuss how much we know about the structure of topological spaces X which can be represented as the union of a not too large collection of subspaces with a certain nice property.In particular, we consider finite unions of subspaces with the weight less or equal \tau, finite unions of subspaces with a point-countable base, and finite unions of metrizable subspaces. As a corollary of this approach, the classical A.S. Mischenko's Theorem on metrizability of compacta with a point-countable base is extended to fininite unions.