The uniqueness and universality of a generalized ordered space

M. S. Kurilić, A. Pavlović

If $\langle L,<\rangle$ is a dense linear order without end points, $A$ and $B$ disjoint and dense subsets of $L$ and $\mathcal O_{AB}$ the topology on the set $L$ generated by closed intervals $[a,b]$, where $a\in A$ and $b\in B$, then $\langle L,\mathcal O_{AB}\rangle$ is a generalized ordered space. We show that all spaces of the form $\langle\mathbb R,\mathcal O_{AB}\rangle$, where $A,B\subset\mathbb R$ are countable sets, are homeomorphic and universal in the class of second countable zero-dimensional spaces.