Weak selections and suborderable metrizable spaces

Abstract

Each continuous weak selection for a space X defines a coarser topology on X, called a selection topology. Spaces whose topology is determined by a collection of such selection topologies are called continuous weak selection spaces. For such spaces, García-Ferreira, Miyazaki, Nogura and Tomita considered the minimal number cws(X) of selection topologies which generate the original topology of X, and called it the cws-number of X. In this paper, we show that cws(X)≤2 for every semi-orderable space X, and that cws(X)=2 precisely when such a space X has two components and is not orderable. Complementary to this result, we also show that cws(X)=1 for each suborderable metrizable space X which has at least 3 components.