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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Author: G. M. Reed
Journal: Trans. Amer. Math. Soc. 297 (1986), 509-520
MSC: Primary 54A10; Secondary 03E50, 54A35, 54D18
DOI: https://doi.org/10.1090/S0002-9947-1986-0854081-9
MathSciNet review: 854081
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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}\vert{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 }\vert\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.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0854081-9
Keywords: Intersection topology, collectionwise normal, collectionwise Hausdorff, subparacompact, submetrizable, separable metric spaces
Article copyright: © Copyright 1986 American Mathematical Society

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