Spaces determined by selections

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. García-Ferreira and A.H. Tomita [S. García-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].