Spaces determined by selections

Abstract

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].

Selections and countable compactness

“…We let, µ g (X) = min γ : µ g (p, X) ≤ γ for every p ∈ X . [19], (X, T g ) is a Tychonoff space. Hence, [p, →) g is itself a Tychonoff space at p. Exactly in the same way, using the reverse relation of g , the interval (←, p] g is also a Tychonoff space at p. Consequently, so is X.…”

Weak Selections and Suborderable Metrizable Spaces

“…It was called a selection topology, and was defined following exactly the pattern of the usual open interval topology utilising the collection of ≤ σ -open intervals S σ = (←, x) ≤σ , (x, →) ≤σ : x ∈ X as a subbase. It was shown in [12] that T σ is regular, and in [16] that T σ is also Tychonoff. Some pathological examples of continuous weak selections that are not continuous with respect to the selection topology they generate were given in [1,10] (see also [12,14]).…”

“…Some pathological examples of continuous weak selections that are not continuous with respect to the selection topology they generate were given in [1,10] (see also [12,14]). Subsequently, answering a question of [13], it was shown in [16] that if there is a coarsest topology on a given set so that a weak selection defined on it is continuous, then this topology must be precisely the selection topology determined by the weak selection itself.…”

“…Regarding the distinction between the original topology and a selection topology, the following spaces (X, T ) were studied in [16]: weakly determined by selections if X admits a weak selection σ with T = T σ ; determined by selections if X admits a continuous weak selection σ with T = T σ ; and strongly determined by selections if X admits a continuous weak selection and T = T σ for every continuous weak selection σ for X. Every orderable space is determined by selections, and it was shown in [16,Example 3.4] that so also is the Sorgenfrey line, which is suborderable but not orderable. However, there are suborderable spaces which are not determined by selections, for instance such a space is the subspace (1.1) X = (0, 1) ∪ {2} ⊂ R .…”

𝜎-ideals and outer measures on the real line

“…The weak selections have been studied by several mathematicians in the areas of Topology and Analysis (see for instance [2], [3], [4], [5], [6], [7], [9], [8] and [10]). One important property of the weak selections is that they give the possibility to generates topologies which have interesting topological properties (see [3], [6] and [9]). In the article [1], the authors introduced the notion of f -outer measure by using a weak selection f on the real line as follows: Certainly, the Lebesgue outer measure λ coincides with the outer measure λ f E (briefly denoted by λ) where f E is the weak selection induced by the Euclidean order of R. Given a weak selection f on R, N f will denote the σ-ideal consisting of all λ f -null sets and the family of λ f -measurable subsets will be denoted by M f .…”