Sequentially compact, Franklin-Rajagopalan spaces

Abstract

ABSTRACT. 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 uonu such that v D T, and v is below u in the RudinKeisler order on lj* . As one application of these results we show that in certain models of set theory there exists a family T of towers such that \T\ < 2W, and JT{X(T) : T € 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.

Ultrafilter Convergence in Ordered Topological Spaces

“…Also, it is well-known that the product of two countably compact topological spaces is not necessarily countably compact, and many similar counterexamples are known for initial κ-compactness, when κ is regular. See, e. g., Stephenson [26], Vaughan [28], Nyikos and Vaughan [21] and more references there. Hence, for κ regular, preservation of initial κ-compactness under products is really a special property of GO spaces.…”