Closed maps on spaces with point-countable bases

AbstractWe prove a version of Lašnev's theorem for spaces with point-countable bases.Theorem. Let X have a point-countable base, f:X→Y be a closed map. Then there is an ω1-relatively-discrete subspace Z of Y such that |Z|≤d(Y) and f−1(y) is compact for every y∈Y\Z.Then we study the subspaces of closed images of regular spaces with point-countable bases and show that every such subspace has countable π-character and a point-countable π-base. The latter result is extended to a wider class of spaces which is invariant under closed maps and products with metrizable compacta. The proofs use a structure which controls the convergence properties of the space and those of its closed image