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Sequentially compact, Franklin-Rajagopalan spaces

Authors: P. J. Nyikos and J. E. Vaughan
Journal: Proc. Amer. Math. Soc. 101 (1987), 149-155
MSC: Primary 54D30; Secondary 03E35
MathSciNet review: 897087
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Abstract: A locally compact $ {T_2}$-space is called a Franklin-Rajagopalan space (or FR-space) provided it has a countable discrete dense subset whose complement is homeomorphic to an ordinal with the order topology. We show that (1) every sequentially compact FR-space $ X$ can be identified with a space constructed from a tower $ T$ on $ \omega \left( {X = X\left( T \right)} \right)$, and (2) for an ultrafilter $ u$ on $ \omega $, a sequentially compact FR-space $ X\left( T \right)$ is not $ u$-compact if and only if there exists an ultrafilter $ v$ on $ \omega $ such that $ v \supset T$, and $ v$ is below $ u$ in the Rudin-Keisler order on $ {\omega ^ * }$. As one application of these results we show that in certain models of set theory there exists a family $ \mathcal{T}$ of towers such that $ \left\vert \mathcal{T} \right\vert < {2^\omega }$, and $ \prod \left\{ {X\left( T \right):T \in \mathcal{T}} \right\}$ is a product of sequentially compact FR-spaces which is not countably compact (a new solution to the Scarborough-Stone problem). As further applications of these results, we give consistent answers to questions of van Douwen, Stephenson, and Vaughan concerning initially $ m$-chain compact and totally initially $ m$-compact spaces.

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Keywords: Franklin-Rajagopalan space, sequentially compact, countably compact, initially $ m$-compact, initially $ m$-chain compact, totally initially $ m$-compact, strongly $ m$-compact, product spaces, $ P$-points, $ T$-points, Rudin-Keisler order, towers, $ {\text{MA + not - CH + }}\diamondsuit \left( {c,{\omega _1} - {\text{limits}}} \right)$
Article copyright: © Copyright 1987 American Mathematical Society

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