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, Garc\'ıa-Ferreira, Miyazaki, Nogura and Tomita considered the minimal number $\text{cws}(X)$ of selection topologies which generate the original topology of $X$, and called it the cws-number of $X$. In this paper, we show that $\text{cws}(X)\le 2$ for every semi-orderable space $X$, and that $\text{cws}(X)=2$ precisely when such a space $X$ has two components and is not orderable. Complementary to this result, we also show that $\text{cws}(X)=1$ for each suborderable metrizable space $X$ which has at least 3 components.