On ∞-convex sets in spaces of scatteredly continuous functions

الأصل · EN

Given a topological space X, we study the structure of ∞-convex subsets in the space SCₚ(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 ∞-convex subset F⊂ SCₚ(X) is weakly discontinuous in the sense that each non-empty subset A⊂ X contains an open dense subset U⊂ A such that each function f|U, f∈ F, is continuous. This implies that F has network weight nw(F)≤ nw(X).