Sequentially compact, Franklin-Rajagopalan spaces

Abstract:

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 $\omega \left ( {X = X\left ( T \right )} \right )$, and (2) for an ultrafilter $u$ on $\omega$, a sequentially compact FR-space $X\left ( T \right )$ 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 $\left | \mathcal {T} \right | < {2^\omega }$, and $\prod \left \{ {X\left ( T \right ):T \in \mathcal {T}} \right \}$ 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.