Topologies generated by closed intervals

Miloš S. Kurilić, Aleksandar Pavlović

If $\langle L,<\rangle$ is a dense linear ordering without end points and $A$ and $B$ disjoint dense subsets of $L$, then the topology $\mathcal O_{AB}$on the set $L$ generated by closed intervals $[a,b]$, where $a\in A$ and $b\in B$, is finer than the standard topology, $\mathcal O_<$, generated by all open intervals and $\langle L,\mathcal O_{AB}\rangle$ is a $\mathrm{GO}$-space. The basic properties of the topology $\mathcal O_{AB}$ (separation axioms, cardinal functions, metrizability) are investigated and compared with the corresponding results concerning the standard topology.