Selectors for dense subsets of function spaces

2019/05/24 by Lev Bukovský, Bukovský, Lev, Alexander V. Osipov +1
Mathematics · #FOS: Mathematics #General Topology (math.GN) #math.GN

paper · pdf · doi:10.48550/arxiv.1905.10287

19 pages

arxiv created 2019/07/01 · arxiv updated 2019/07/02

Abstract

Let USC^*_p(X) be the topological space of real upper semicontinuous bounded functions defined on X with the subspace topology of the product topology on ^Xℝ. Φ^\uparrow,Ψ^\uparrow are the sets of all upper sequentially dense, upper dense or pointwise dense subsets of USC^*_p(X), respectively. We prove several equivalent assertions to the assertion USC^*_p(X) satisfies the selection principles S₁(Φ^\uparrow,Ψ^\uparrow), including a condition on the topological space X. We prove similar results for the topological space C^*_p(X) of continuous bounded functions. Similar results hold true for the selection principles S_fin(Φ^\uparrow,Ψ^\uparrow).

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