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 -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 can be identified with a space constructed from a tower on , and (2) for an ultrafilter on , a sequentially compact FR-space is not -compact if and only if there exists an ultrafilter on such that , and is below in the Rudin-Keisler order on . As one application of these results we show that in certain models of set theory there exists a family of towers such that , and 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 -chain compact and totally initially -compact spaces.

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DOI:
https://doi.org/10.1090/S0002-9939-1987-0897087-2

Keywords:
Franklin-Rajagopalan space,
sequentially compact,
countably compact,
initially -compact,
initially -chain compact,
totally initially -compact,
strongly -compact,
product spaces,
-points,
-points,
Rudin-Keisler order,
towers,

Article copyright:
© Copyright 1987
American Mathematical Society