Oscillation and ω-Primitives

Abstract

We extend the results of [2], [6] in the case of topological spaces. It is shown that given an upper semicontinuous (USC) function $f:X\to [0,\infty)$ where $X$ is a massive first countable $T_1$-space satisfying some "neighborhood conditions", there exists $F:X\to [0, \infty)$ whose oscillation equals $f$ everywhere on $X$ (Theorem 2.1). The analogous result holds for USC functions $f:X\to [0, \infty]$ if, in addition, $X$ is a normal space (Theorem 2.4). A special metrizability criterion is established (Theorem 1.1). This is to show, by exhibiting corresponding examples, that the neighborhood conditions and massiveness do not imply that $X$ is Baire or metrizable. Among other related topics, sequences of $\omega$-primitives are discussed.