Countable Successor Ordinals as Generalized Ordered Topological Spaces

TL;DR

This paper characterizes spaces whose continuous images are generalized ordered spaces, showing they are homeomorphic to countable successor ordinals, thus linking order topology and topological properties.

Contribution

It proves that Hausdorff spaces with all continuous images as GO-spaces are precisely the countable successor ordinals, establishing a new classification.

Findings

Spaces with all continuous images as GO-spaces are countable successor ordinals.

The characterization provides a topological and order-theoretic link.

The converse, that countable successor ordinals have all images as GO-spaces, is trivial.

Abstract

A topological space $L$ is called a linear ordered topological space (LOTS) whenever there is a linear order $\leq$ on $L$ such that the topology on $L$ is generated by the open sets of the form $(a, b)$ with $a < b$ and $a, b \in L \cup \{ -\infty, +\infty \}$. A topological space $X$ is called a generalized ordered space (GO-space) whenever $X$ is topologically embeddable in a LOTS. Main Theorem: Let $X$ be a Hausdorff topological space. Assume that any continuous image of $X$ is a GO-space. Then $X$ is homeomorphic to a countable successor ordinal (with the order topology). The converse trivially holds.