P0-spaces

A regular topological space X is defined to be a P0-space if it has countable Pytkeev network. A network N for X is called a Pytkeev network if for any point x∈X, neighborhood Ox⊂X of x and subset A⊂X accumulating at a x there is a set N∈N such that N⊂Ox and N∩A is infinite. The class of P0-spaces contains all metrizable separable spaces and is (properly) contained in the Michael's class of ℵ0-spaces. It is closed under many topological operations: taking subspaces, countable Tychonoff products, small countable box-products, countable direct limits, hyperspaces of compact subsets. For an ℵ0-space X and a P0-space Y the function space Ck(X,Y) endowed with the compact-open topology is a P0-space. For any sequential ℵ0-space X the free abelian topological group A(X) and the free locally convex linear topological space L(X) both are P0-spaces. A sequential space is a P0-space if and only if it is an ℵ0-space. A topological space is metrizable and separable if and only if it is a P0-space with countable fan tightness.