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

Abstract

If Υ 1 {\Upsilon _1} and Υ 2 {\Upsilon _2} are topologies defined on the set X X , then the intersection topology w.r.t. Υ 1 {\Upsilon _1} and Υ 2 {\Upsilon _2} is the topology Υ \Upsilon on X X such that { U 1 ∩ U 2 | U 1 ∈ Υ 1 and U 2 ∈ Υ 2 } \{ {U_1} \cap {U_2}|{U_1} \in {\Upsilon _1}\;{\text {and}}\;{U_2} \in {\Upsilon _2}\} is a basis for ( X , Υ ) (X,\Upsilon ) . In this paper, the author considers spaces in the class C \mathcal {C} , where ( X , Υ ) ∈ C (X,\Upsilon ) \in \mathcal {C} iff X = { x α | α > ω 1 } ⊆ R X = \{ {x_\alpha }|\alpha > {\omega _1}\} \subseteq {\mathbf {R}} , Υ R {\Upsilon _{\mathbf {R}}} is the inherited real line topology on X X , Υ ω 1 {\Upsilon _{{\omega _1}}} is the order topology on X X of type ω 1 {\omega _1} , and Υ \Upsilon is the intersection topology w.r.t. Υ R {\Upsilon _{\mathbf {R}}} and Υ ω 1 {\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 ∈ C X \in \mathcal {C} , then X 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) ( MA + ¬ CH ) (\operatorname {MA} + \neg \operatorname {CH} ) If X ∈ C X \in \mathcal {C} , then X X is perfect. (3) There exists in ZFC an X ∈ C X \in \mathcal {C} such that X X is not normal. (4) ( CH ) (\operatorname {CH} ) There exists X ∈ C X \in \mathcal {C} such that X X is collectionwise normal and ω 1 {\omega _1} -compact but not perfect.