×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bohuslav Balcar, Ryszard Frankiewicz, and Charles Mills, More on nowhere dense closed \?-sets, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), no. 5-6, 295 – 299 (1981) (English, with Russian summary). · Zbl 0467.54024
[2] Allen R. Bernstein, A new kind of compactness for topological spaces, Fund. Math. 66 (1969/1970), 185 – 193. · Zbl 0198.55401
[3] W. W. Comfort and S. Negrepontis, The theory of ultrafilters, Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 211. · Zbl 0298.02004
[4] Eric K. van Douwen, Hausdorff gaps and a nice countably paracompact non-normal space, Topology Proc. 1 (1976), 239-242. · Zbl 0406.54018
[5] Kenneth Kunen and Jerry E. Vaughan , Handbook of set-theoretic topology, North-Holland Publishing Co., Amsterdam, 1984. · Zbl 0546.00022
[6] -, The product of two normal initially \( \kappa \)-compact spaces (to appear). · Zbl 0776.54010
[7] S. P. Franklin and M. Rajagopalan, Some examples in topology, Trans. Amer. Math. Soc. 155 (1971), 305 – 314. · Zbl 0217.48104
[8] Stephen H. Hechler, On the existence of certain cofinal subsets of ^\?\?, Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1974, pp. 155 – 173.
[9] I. Juhász, Zs. Nagy, and W. Weiss, On countably compact, locally countable spaces, Period. Math. Hungar. 10 (1979), no. 2-3, 193 – 206. · Zbl 0418.54009 · doi:10.1007/BF02025892 · doi.org
[10] Ronnie Levy, Pseudocompactness and extension of functions in Franklin-Rajagopalan spaces, Topology Appl. 11 (1980), no. 3, 297 – 303. · Zbl 0456.54017 · doi:10.1016/0166-8641(80)90029-2 · doi.org
[11] M. Rajagopalan, Some outstanding problems in topology and the \?-process, Categorical topology (Proc. Conf., Mannheim, 1975) Springer, Berlin, 1976, pp. 501 – 517. Lecture Notes in Math., Vol. 540.
[12] M. Rajagopalan and R. Grant Woods, Products of sequentially compact spaces and the \?-process, Trans. Amer. Math. Soc. 232 (1977), 245 – 253. · Zbl 0383.54015
[13] Mary Ellen Rudin, A technique for constructing examples, Proc. Amer. Math. Soc. 16 (1965), 1320 – 1323. · Zbl 0141.20403
[14] R. M. Stephenson Jr., Initially \?-compact and related spaces, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 603 – 632.
[15] Victor Saks and R. M. Stephenson Jr., Products of \?-compact spaces, Proc. Amer. Math. Soc. 28 (1971), 279 – 288. · Zbl 0207.52802
[16] C. T. Scarborough and A. H. Stone, Products of nearly compact spaces, Trans. Amer. Math. Soc. 124 (1966), 131 – 147. · Zbl 0151.30001
[17] J. E. Vaughan, Products of perfectly normal, sequentially compact spaces, J. London Math. Soc. (2) 14 (1976), no. 3, 517 – 520. · Zbl 0349.54022 · doi:10.1112/jlms/s2-14.3.517 · doi.org
[18] J. E. Vaughan, Products of [\?,\?]-chain compact spaces, Houston J. Math. 3 (1977), no. 4, 569 – 578. · Zbl 0383.54013
[19] Jerry E. Vaughan, Countably compact and sequentially compact spaces, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 569 – 602. · Zbl 0562.54031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.