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Sequentially compact, Franklin-Rajagopalan spaces. (English) Zbl 0626.54004
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 \(\omega\) \((X=X(T))\), and (2) for an ultrafilter u on \(\omega\), a sequentially compact FR-space X(T) 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 \(| {\mathcal T}| <2^{\omega}\), and \(\prod \{X(T):\) \(T\in {\mathcal T}\}\) 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 spaces and totally initially m-compact spaces.

54A35 Consistency and independence results in general topology
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D30 Compactness
03E35 Consistency and independence results
54B10 Product spaces in general topology
Full Text: DOI
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