On resolvable primal spaces

Abstract

A topological space is called resolvable if it is a union of two disjoint dense subsets, and is n-resolvable if it is a union of n mutually disjoint dense subsets. Clearly a resolvable space has no isolated points. If ƒ is a selfmap on X, the sets A ⊆ X with ƒ(A) ⊆ A are the closed sets of an Alexandroff topology called the primal topology Ρ(ƒ) associated with ƒ. We investigate resolvability for primal spaces (X;Ρ(ƒ)). Our main result is that an Alexandroff space is resolvable if and only if it has no isolated points. Moreover, n-resolvability and other related concepts are investigated for primal spaces.

Mathematics Subject Classication (2010): 54B25, 18B30.

Key words: Primal spaces, categories, resolvable spaces.