Applications of cluster sets in minimal topological spaces

Abstract

Given a function f f from a topological space X X into a topological space Y Y and a point x ϵ X x\epsilon X , the cluster set of f f at x x is C ( f ; x ) = ∩ { Cl ⁡ ( f ( U ) ) : U is a neighborhood of  x } \mathcal {C}(f;x) = \cap \{ \operatorname {Cl} (f(U)):\;U\;{\text {is a neighborhood of }}x\} , where Cl ⁡ ( U ) \operatorname {Cl} (U) denotes the closure of U U . In this paper, Y Y is taken to be a minimal topological space and C ( f ; x ) \mathcal {C}(f;x) is used as a tool to obtain information about the continuity of f f .