Countable metacompactness in Ψ-spaces

Szeptycki, Paul

doi:10.1090/S0002-9939-1994-1169890-9

Countable metacompactness in $\Psi$-spaces
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Proc. Amer. Math. Soc. 120 (1994), 1241-1246 Request permission

Abstract:

We prove under a variety of assumptions including $\mathfrak {c} = {\aleph _2}$ that, for every maximal almost disjoint family $\mathcal {A}$ of countable subsets of ${\omega _1},\;\Psi (\mathcal {A})$ is not countably metacompact. In addition, a first countable, countably metacompact, regular space with a closed discrete set which is not a ${G_\delta }$ is constructed from the mutually consistent assumptions that $\mathfrak {b} = {\omega _1}$ and there can exist a Q-set.

References

Zoltán T. Balogh and Dennis K. Burke, A total ladder system space by ccc forcing, Proceedings of the Symposium on General Topology and Applications (Oxford, 1989), 1992, pp. 37–44. MR 1173241, DOI 10.1016/0166-8641(92)90077-D
Dennis K. Burke, Closed discrete subsets in first countable, countably metacompact spaces, Proceedings of the Symposium on General Topology and Applications (Oxford, 1989), 1992, pp. 63–67. MR 1173244, DOI 10.1016/0166-8641(92)90080-J
Dennis K. Burke, PMEA and first countable, countably paracompact spaces, Proc. Amer. Math. Soc. 92 (1984), no. 3, 455–460. MR 759673, DOI 10.1090/S0002-9939-1984-0759673-7
Kenneth Kunen and Jerry E. Vaughan (eds.), Handbook of set-theoretic topology, North-Holland Publishing Co., Amsterdam, 1984. MR 776619
William G. Fleissner and Arnold W. Miller, On $Q$ sets, Proc. Amer. Math. Soc. 78 (1980), no. 2, 280–284. MR 550513, DOI 10.1090/S0002-9939-1980-0550513-4
Stephen H. Hechler, On the existence of certain cofinal subsets of $^{\omega }\omega$, 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. MR 0360266
Fritz Rothberger, Sur les familles indénombrables de suites de nombres naturels et les problèmes concernant la propriété $C$, Proc. Cambridge Philos. Soc. 37 (1941), 109–126 (French). MR 4281, DOI 10.1017/S0305004100021617
Petr Simon, A compact Fréchet space whose square is not Fréchet, Comment. Math. Univ. Carolin. 21 (1980), no. 4, 749–753. MR 597764
S. Shelah, A consistent counterexample in the theory of collectionwise Hausdorff spaces, Israel J. Math. 65 (1989), no. 2, 219–224. MR 998672, DOI 10.1007/BF02764862
Stevo Todorčević, Partition problems in topology, Contemporary Mathematics, vol. 84, American Mathematical Society, Providence, RI, 1989. MR 980949, DOI 10.1090/conm/084