On σ-convex subsets in spaces of scatteredly continuous functions

Abstract

We prove that for any topological space $X$ of countable tightness, each σ-convex subspace $\F$ of the space $S{C}_p(X)$ 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 $S{C}_p(X)$ is metrizable. Another corollary says that two Tychonoff spaces $X,Y$ with countable tightness and topologically isomorphic linear topological spaces $S{C}_p(X)$ and $S{C}_p(Y)$ 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 $S{C}_p({\omega }^{\omega })$ over the space ${\omega }^{\omega }$ of irrationals.

6 pages