Abstract. We answer a question of van Mill and Wattel by showing that there is a separable locally compact space which admits a continuous weak selection but is not weakly orderable. Furthermore, we show that a separable space which admits a continuous weak selection can be covered by two weakly orderable spaces. Finally, we give a partial answer to a question of Gutev and Nogura by showing that a separable space which admits a continuous weak selection admits a continuous selection for all finite sets.

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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].

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