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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.

##### MSC:
 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
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##### References:
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