On σ-convex subsets in spaces of scatteredly continuous functions

Анотація

We prove that for any topological space X of countable tightness, each convex subspace F of the space SC_p(X) of scatteredly continuous real-valued functions on X has network weight nw(F)≤nw(X). This implies that for a metrizable separable space X, each compact convex subset inthe 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(ω^ω) over the space ω^ω of irrationals.