Sequentially compact, FranklinRajagopalan spaces
Authors:
P. J. Nyikos and J. E. Vaughan
Journal:
Proc. Amer. Math. Soc. 101 (1987), 149155
MSC:
Primary 54D30; Secondary 03E35
MathSciNet review:
897087
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Abstract: A locally compact space is called a FranklinRajagopalan space (or FRspace) 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 FRspace can be identified with a space constructed from a tower on , and (2) for an ultrafilter on , a sequentially compact FRspace is not compact if and only if there exists an ultrafilter on such that , and is below in the RudinKeisler order on . As one application of these results we show that in certain models of set theory there exists a family of towers such that , and is a product of sequentially compact FRspaces which is not countably compact (a new solution to the ScarboroughStone problem). As further applications of these results, we give consistent answers to questions of van Douwen, Stephenson, and Vaughan concerning initially chain compact and totally initially compact spaces.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198708970872
PII:
S 00029939(1987)08970872
Keywords:
FranklinRajagopalan space,
sequentially compact,
countably compact,
initially compact,
initially chain compact,
totally initially compact,
strongly compact,
product spaces,
points,
points,
RudinKeisler order,
towers,
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© Copyright 1987
American Mathematical Society
