On $\infty$-convex sets in spaces of scatteredly continuous functions

TL;DR

This paper investigates the structure of $ abla$-convex subsets within the space of scatteredly continuous functions on a topological space, revealing conditions under which these sets exhibit weak discontinuity and bounded network weight.

Contribution

It establishes that in spaces with countable strong fan tightness, $ abla$-convex subsets of scatteredly continuous functions are weakly discontinuous and have controlled network weight.

Findings

Potentially bounded $ abla$-convex sets are weakly discontinuous.

Such sets have network weight at most that of the underlying space.

The results connect topological properties of $X$ with the structure of function spaces.

Abstract

Given a topological space $X$, we study the structure of $\infty$-convex subsets in the space $SC_p(X)$ of scatteredly continuous functions on $X$. Our main result says that for a topological space $X$ with countable strong fan tightness, each potentially bounded $\infty$-convex subset $F\subset SC_p(X)$ is weakly discontinuous in the sense that each non-empty subset $A\subset X$ contains an open dense subset $U\subset A$ such that each function $f|U$, $f\in F$, is continuous. This implies that $F$ has network weight $nw(F)\le nw(X)$.