Almost Perfectly Continuous Functions

Abstract

The aim of the present paper is to continue the study of almost perfectly continuous (≡ regular set connected) functions initiated by Dontchev, Ganster and Reilly. It turns out that in general the notion of almost perfectly continuity is independent of continuity but represents a significantly strong form of continuity called ‘cl-supercontinuity’, if Y is a semiregular space. Basic properties of almost perfectly continuous functions are studied and their place in the hierarchy of variants of continuity that already exist in the literature is discussed. Moreover, suffcient conditions on X and Y are outlined for the function space of all almost perfectly continuous functions from X to Y to be closed in Y^X with the topology of pointwise convergence.