On σ-convex subsets in spaces of scatteredly continuous functions

Taras Banakh, Bohdan Bokalo, Nadiya Kolos


We prove that for any topological space X of countable tightness, each convex subspace F of the space SCp(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 SCp(X) is metrizable. Another corollary says that two Tychonoff spaces X,Y with countable tightness and topologically isomorphic linear topological spaces SCp(X) and SCp(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 SCpω) over the space ωω of irrationals.

Ключові слова

scatteredly continuous map; weakly discontinuous map; σ-convex subset; network weight

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