Maps generating the same primal space

Abstract

Alexandroff topologies play an enigmatic role in topology. An important family of Alexandroff topologies are the functional Alexandroff spaces introduced by Shirazi and Golestani, and called primal topologies by O. Echi. The primal topology P(ƒ) on χ determined by a function ƒ : χ → χ is the topology whose closed sets are the f-invariant subsets of χ. If (χ;P(ƒ)) is a primal space, we investigate the collection F = {g : χ → χ : P(g) = P(ƒ)} of functions on χ which determine the primal topology. We give a necessary and sufficient condition for F to be nite, and when it is nite, we give an enumeration of F.

Mathematics Subject Classication (2010): 18A40, 54B30, 54D10, 54F65.

Key words: Alexandroff space, functional Alexandroff space, primal space, T0-space.