Sequentially compact, Franklin-Rajagopalan spaces
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- by P. J. Nyikos and J. E. Vaughan PDF
- Proc. Amer. Math. Soc. 101 (1987), 149-155 Request permission
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 $\left | \mathcal {T} \right | < {2^\omega }$, 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.References
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Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 101 (1987), 149-155
- MSC: Primary 54D30; Secondary 03E35
- DOI: https://doi.org/10.1090/S0002-9939-1987-0897087-2
- MathSciNet review: 897087