Countable Successor Ordinals as Generalized Ordered Topological Spaces

A topological space $L$ is called a linear ordered topological space (LOTS) whenever there is a linear order $\le$ on $L$ such that the topology on $L$ is generated by the open sets of the form $(a,b)$ with $a<b$ and $a, b \in L \cup { -\infty, +\infty }$\mathrm{.}Atopologicalspace$X$ 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.