Selectors for dense subsets of function spaces

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 X R.Φ ↑ ,Ψ ↑ 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 that that USC ⋆ p (X) satisfies the selection principles S 1 (Φ ↑ ,Ψ ↑ ), 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 f in (Φ ↑ ,Ψ ↑ ).

Spaces of Real Functions, Covers and Dense Subsets

“…We tried to extend the known results also for covers respecting a bornology on X and spaces of bounded upper semicontinuous functions with the corresponding topology defined by the bornology. This paper is a natural generalization the research done in papers [BL1,BL2,BH,BO,Osi,Osi3,Osi4,Osi6,Osi7] for spaces of continuous real-valued functions with the topology of pointwise convergence and with the compact-open topology.…”