Left-ray Right-ray Hybrid Topologies on the Real Line

Abstract

Given a nonempty set $A \subseteq \mathbb{R}$, we consider the smallest topology on $\mathbb{R}$ which contains the open left rays containing points $a \in A$ and the open right rays containing points $b \in \mathbb{R} - A$. We present a natural model for this hybrid topology and show that it is quasi-metrizable. We investigate other variations of this topology.