Inverse cluster sets

Abstract

For a function f : X → Y f:X \to Y , the cluster set of f f at x ϵ X x\epsilon X is the set of all y ϵ Y y\epsilon Y such that there exists a filter F \mathcal {F} on X X converging to x x and the filter generated by f ( F ) f(\mathcal {F}) converges to y y . The inverse cluster set of f f at y ϵ Y y\epsilon Y is the set of all x ϵ X x\epsilon X such that y y belongs to the cluster set of f f at x x . General properties of inverse cluster sets are proved, including a necessary and sufficient condition for continuity. Necessary and sufficient conditions for functions to have a closed graph in terms of inverse cluster sets are also given. Finally, a known theorem giving a condition as to when a connected function is also a connectivity function is generalized and further investigated in terms of inverse cluster sets.