Extensions of semicontinuous and quasi-continuous functions from dense subspaces

Abstract

In this paper we give a sufficient condition for existence of an extension of a lower (upper) semicontinuous function ?₀ defined on a given dense subset of a topological space X to a lower (upper) semicontinuous function ? : X → ℝ. Moreover, we present an equivalent condition for existence of an extension of a quasi-continuous (lower and upper semicontinuous quasi-continuous) function ?₀ defined on a given dense subset of a topological space X to a quasi-continuous (lower and upper semi-continuous quasi-continuous, respectively) function ? : X → R.

Mathematics Subject Classification (2010): Primary: 54C20, 26A15; Secondary: 54C08.