Intersection topologies with respect to separable GO-spaces and the countable ordinals

Given two topologies, ${\displaystyle T_1}$ and ${\displaystyle T_2}$, on the same set X, the intersection topology} with respect to ${\displaystyle T_1}$ and ${\displaystyle T_2}$ is the topology with basis ${\displaystyle \{U_1\cap U_2:U_1\in T_1,U_2\in T_2\}}$. Equivalently, T is the join of ${\displaystyle T_1}$ and ${\displaystyle T_2}$ in the lattice of topologies on the set X. Following the work of Reed concerning intersection topologies with respect to the real line and the countable ordinals, Kunen made an extensive investigation of normality, perfectness and ${\displaystyle {\omega }_1}$-compactness in this class of topologies. We demonstrate that the majority of his results generalise to the intersection topology with respect to an arbitrary separable GO-space and ${\displaystyle {\omega }_1}$, employing a well-behaved second countable subtopology of the separable GO-space.