On some properties of the space of upper semicontinuous functions

TL;DR

This paper characterizes Tychonoff spaces where the space of upper semicontinuous functions is sequentially separable, revealing that this class matches those with a stronger form of separability for Baire class 1 functions.

Contribution

It identifies the exact class of Tychonoff spaces for which USC_p(X) is sequentially separable, linking it to the separability properties of Baire class 1 functions.

Findings

The class of spaces with sequentially separable USC_p(X) coincides with those with stronger Baire class 1 separability.

The paper provides a characterization of Tychonoff spaces based on function space properties.

It reveals an unexpected equivalence between two classes of spaces related to different function spaces.

Abstract

For a Tychonoff space $X$, we will denote by $USC_{p}(X)$ ($B_1(X)$) a set of all real-valued upper semicontinuous functions (a set of all Baire functions of class 1) defined on $X$ endowed with the pointwise convergence topology. In this paper we describe a class of Tychonoff spaces $X$ for which the space $USC_{p}(X)$ is sequentially separable. Unexpectedly, it turns out that this class coincides with the class of spaces for which a stronger form of the sequential separability for the space $B_1(X)$ holds.