Selections and order-like relations

Abstract

<p>Every selection f for the family F<sub>2</sub>(X) of at most two-point subsets of a set X naturally defines an order-like relation on X by if and only . In the present paper we study the relationship between and the possible topologies T on X which realize the continuity of f with respect to the Vietoris topology on F<sub>2</sub>(X) generated by T.</p>

Selections and countable compactness

“…is a subbase for a natural " s -open" interval topology T s on X, called a selection topology [12]. In fact, T s is the usual open interval topology provided s is a linear ordering on X.…”

Weak orderability of second countable spaces

“…It is well-known that f ∈ W cs (X) if and only if f ⊥ ∈ W cs (X) (see, for instance, [4,Theorem 3.5]). This implies the following simple observation, which will be found useful.…”

Selections generating new topologies

“…For convenience, we write that x ≺ f y if x f y and x = y. We note that the relation " f " may fail to be transitive (see, for instance, [4,Proposition 2.2]). Nevertheless, to every continuous weak selection f for X we may associate a topology T f on X generated by all "open f -intervals" {y ∈ X : y ≺ f x} and {y ∈ X : x ≺ f y}, x ∈ X.…”

“…Nevertheless, to every continuous weak selection f for X we may associate a topology T f on X generated by all "open f -intervals" {y ∈ X : y ≺ f x} and {y ∈ X : x ≺ f y}, x ∈ X. According to [10,Lemma 7.2] (see, also, [4,Lemma 3.3]), these "f -intervals" are always open in the original topology of X. Hence, T f is a coarser topology on X, and, consequently, it is the original topology on X provided X is compact.…”

“…Some further properties of this topology were studied in [4], [7]. For instance, by [7,Corollary 2.3], T f is always a regular topology on X.…”