Spaces determined by selections

摘要

A function ψ: |X|~2 → X is a called a weak selection if ψ({x, y}) ∈ {x, y) for every x, y ∈ X. To each weak selection ψ, one associates a topology τ_ψ, generated by the sets (←,x) = {y ≠ x: ψ(x, y) =y) and (x, →) = [y ≠ x: ψ(x,y) =x). Answering a question of S. Garcia-Ferreira and A.H. Tomita [S. Garcia-Ferreira, A.H. Tomita, A non-normal topology generated by a two-point selection, Topology Appl. 155 (10) (2008) 1105-1110], we show that (X, τ_ψ) is completely regular for every weak selection ψ. We further investigate to what extent the existence of a continuous weak selection on a topological space determines the topology of X. In particular, we answer two questions of V. Gutev and T. Nogura [V. Gutev, T. Nogura, Selection problems for hyperspaces, in: E. Pearl (Ed.), Open Problems in Topology 2, Elsevier B.V., 2007, pp. 161-170].