Spaces determined by selections

A function psi vertical bar X vertical bar(2) -> X is a called a weak selection if psi ({x y}) is an element of {x y} for every x y is an element of X To each weak selection psi, one associates a topology tau(psi), generated by the sets (<-, x) = {y not equal x psi (x y) = y} and (x ->) = {y not equal x psi (x y) = x} Answering a question of S Garcia-Ferreira and A H Tolima [S Garcia-Ferreira, A Tomita, A non-normal topology generated by a two-point selection, Topology Appl 155 (10) (2008) 1105-1110]. we show that (X, tau(psi)) is completely regular for every weak selection psi 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 Gamey 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] (C) 2009 Elsevier B V All rights reserved