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arXiv · 1207.0827

On $\infty$-convex sets in spaces of scatteredly continuous functions

Taras Banakh·Bogdan Bokalo·Nadiya Kolos
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Abstract

Given a topological space $X$, we study the structure of $\infty$-convex subsets in the space $SC_p(X)$ of scatteredly continuous functions on $X$. Our main result says that for a topological space $X$ with countable strong fan tightness, each potentially bounded $\infty$-convex subset $F\subset SC_p(X)$ is weakly discontinuous in the sense that each non-empty subset $A\subset X$ contains an open dense subset $U\subset A$ such that each function $f|U$, $f\in F$, is continuous. This implies that $F$ has network weight $nw(F)\le nw(X)$.

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Taras Banakh, Bogdan Bokalo, Nadiya Kolos. 2012-07-03. On $\infty$-convex sets in spaces of scatteredly continuous functions. https://doi.org/10.1016/j.topol.2014.02.030

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