Selections generating new topologies

Abstract

Every (continuous) selection for the non-empty 2-point subsets of a space X naturally defines an interval-like topology on X. In the present paper, we demonstrate that, for a second-countable zerodimensional space X, this topology may fail to be first-countable at some (or, even any) point of X. This settles some problems stated in [7].

Coarser Compact Topologies

“…Each selection topology T σ is Tychonoff [16,Theorem 2.7]. On the other hand, T σ may lack several of the other strong properties of the open interval topology, see [4,13].…”

“…Any weak selection σ for X generates a natural topology T σ on X [9], called a selection topology and defined following the pattern of the open interval topology, see Section 2. If X is a space and σ is continuous, then T σ is a coarser topology on X, but σ is not necessarily continuous with respect to T σ [9] (see also [11,13]). A weak selection σ for a space X is called properly continuous if T σ is a coarser topology on X and σ is continuous with respect to T σ [12,Definition 4.4].…”