An example of a space which is countably compact whose square is countably paracompact but not countably compact
HTML articles powered by AMS MathViewer
- by Lee Parsons PDF
- Proc. Amer. Math. Soc. 65 (1977), 351-354 Request permission
Abstract:
A subspace P of $\beta N - N$ is obtained whose square is disjoint from the graph, G, of a pre-selected homeomorphism $f:\beta N \to \beta N$ that has no fixed points. The construction is performed in such a way that, for $X = P \cup N$, all countable subsets of ${X^2} - G$ will have a limit point in ${X^2}$. We use the following lemma: If $K \subset {(\beta N)^2} - G$ is countably infinite, then $|{\text {cl}_{{{(\beta N)}^2}}}K - G| = {2^c}$.References
- W. W. Comfort, A nonpseudocompact product space whose finite subproducts are pseudocompact, Math. Ann. 170 (1967), 41–44. MR 210070, DOI 10.1007/BF01362285
- R. Engelking, Outline of general topology, North-Holland Publishing Co., Amsterdam; PWN—Polish Scientific Publishers, Warsaw; Interscience Publishers Division John Wiley & Sons, Inc., New York, 1968. Translated from the Polish by K. Sieklucki. MR 0230273
- Zdeněk Frolík, Generalisations of compact and Lindelöf spaces, Czechoslovak Math. J. 9(84) (1959), 172–217 (Russian, with English summary). MR 105075
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
- John Ginsburg and Victor Saks, Some applications of ultrafilters in topology, Pacific J. Math. 57 (1975), no. 2, 403–418. MR 380736
- James Keesling, Normality and compactness are equivalent in hyperspaces, Bull. Amer. Math. Soc. 76 (1970), 618–619. MR 254812, DOI 10.1090/S0002-9904-1970-12459-4
- Kiiti Morita, A survey of the theory of $M$-spaces, General Topology and Appl. 1 (1971), no. 1, 49–55. MR 286060
- J. Novák, On the Cartesian product of two compact spaces, Fund. Math. 40 (1953), 106–112. MR 60212, DOI 10.4064/fm-40-1-106-112
- A. K. Steiner, On the topological completion of $M$-space products, Proc. Amer. Math. Soc. 29 (1971), 617–620. MR 282339, DOI 10.1090/S0002-9939-1971-0282339-8
- Lynn A. Steen and J. Arthur Seebach Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1970. MR 0266131
- Hidetaka Terasaka, On Cartesian product of compact spaces, Osaka Math. J. 4 (1952), 11–15. MR 51500
- Russell C. Walker, The Stone-Čech compactification, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 83, Springer-Verlag, New York-Berlin, 1974. MR 0380698
- R. Grant Woods, The structure of small normal $F$-spaces, Topology Proceedings, Vol. I (Conf., Auburn Univ., Auburn, Ala., 1976) Math. Dept., Auburn Univ., Auburn, Ala., 1977, pp. 173–179. MR 0454921
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 65 (1977), 351-354
- MSC: Primary 54G20; Secondary 54D20
- DOI: https://doi.org/10.1090/S0002-9939-1977-0451218-0
- MathSciNet review: 0451218