On \sigma-convex subsets in spaces of scatteredly continuous functions

TL;DR

This paper investigates the properties of ta-convex subsets in spaces of scatteredly continuous functions, establishing bounds on network weight, metrizability conditions, and homeomorphism results for certain compact spaces.

Contribution

It introduces new bounds on network weight for ta-convex subsets in scatteredly continuous function spaces and characterizes their topological properties.

Findings

ta-convex subspaces have network weight at most that of the base space

Compact convex subsets in $SC_p(X)$ are metrizable for separable metrizable $X$

Zero-dimensional separable Rosenthal compact spaces embed into $SC_p(\u03c9^\u03c9)$

Abstract

We prove that for any topological space $X$ of countable tightness, each \sigma-convex subspace $\F$ of the space $SC_p(X)$ of scatteredly continuous real-valued functions on $X$ has network weight $nw(\F)\le nw(X)$. This implies that for a metrizable separable space $X$, each compact convex subset in the function space $SC_p(X)$ is metrizable. Another corollary says that two Tychonoff spaces $X,Y$ with countable tightness and topologically isomorphic linear topological spaces $SC_p(X)$ and $SC_p(Y)$ have the same network weight $nw(X)=nw(Y)$. Also we prove that each zero-dimensional separable Rosenthal compact space is homeomorphic to a compact subset of the function space $SC_p(\omega^\omega)$ over the space $\omega^\omega$ of irrationals.