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