Sequentially Compact, Franklin-Rajagopalan Spaces
نویسندگان
چکیده
A locally compact T2-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 w (X = X(T)), and (2) for an ultrafilter u on w, a sequentially compact FR-space X(T) is not u-compact if and only if there exists an ultrafilter v on w such that v D T, and v is below u in the RudinKeisler order on w*. As one application of these results we show that in certain models of set theory there exists a family T of towers such that I T I < 2W, and nH{X(T): T E 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 and totally initially m-compact spaces.
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