The intersection topology w.r.t. the real line and the countable ordinals

Abstract:

If ${\Upsilon _1}$ and ${\Upsilon _2}$ are topologies defined on the set $X$, then the intersection topology w.r.t. ${\Upsilon _1}$ and ${\Upsilon _2}$ is the topology $\Upsilon$ on $X$ such that $\{ {U_1} \cap {U_2}|{U_1} \in {\Upsilon _1}\;{\text {and}}\;{U_2} \in {\Upsilon _2}\}$ is a basis for $(X,\Upsilon )$. In this paper, the author considers spaces in the class $\mathcal {C}$, where $(X,\Upsilon ) \in \mathcal {C}$ iff $X = \{ {x_\alpha }|\alpha < {\omega _1}\} \subseteq {\mathbf {R}}$, ${\Upsilon _{\mathbf {R}}}$ is the inherited real line topology on $X$, ${\Upsilon _{{\omega _1}}}$ is the order topology on $X$ of type ${\omega _1}$, and $\Upsilon$ is the intersection topology w.r.t. ${\Upsilon _{\mathbf {R}}}$ and ${\Upsilon _{{\omega _1}}}$. This class is shown to be a surprisingly useful tool in the study of abstract spaces. In particular, it is shown that: (1) If $X \in \mathcal {C}$, then $X$ is a completely regular, submetrizable, pseudo-normal, collectionwise Hausdorff, countably metacompact, first countable, locally countable space with a base of countable order that is neither subparacompact, metalindelöf, cometrizable, nor locally compact. (2) $(\operatorname {MA} + \neg \operatorname {CH} )$ If $X \in \mathcal {C}$, then $X$ is perfect. (3) There exists in ZFC an $X \in \mathcal {C}$ such that $X$ is not normal. (4) $(\operatorname {CH} )$ There exists $X \in \mathcal {C}$ such that $X$ is collectionwise normal and ${\omega _1}$-compact but not perfect.