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Keywords:
countably compact; initially $\kappa $-compact; weakly $\delta \theta $-refinable; $\kappa $-refinable; sequential
countably compact; initially $\kappa $-compact; weakly $\delta \theta $-refinable; $\kappa $-refinable; sequential
Summary:
We present short and elementary proofs of the following two known theorems in General Topology: (i) [H. Wicke and J. Worrell] A $T_1$ weakly $\delta \theta $-refinable countably compact space is compact. (ii) [A. Ostaszewski] A compact Hausdorff space which is a countable union of metrizable spaces is sequential.
References:
[A] Arhangel'skiĭ A.V.: The star method, new classes of spaces and countable compactness. Soviet Math. Dokl. 21 (1980), 550-554. MR 0569369
[B] Burke D.K.: Covering properties. Handbook of Set Theoretic Topology, North Holland, 1984, pp. 347-422. MR 0776628 | Zbl 0569.54022
[O] Ostaszewski A.J.: Compact $\sigma $-metric Hausdorff spaces are sequential. Proc. Amer. Math. Soc. 68 (1978), 339-343. MR 0467677 | Zbl 0392.54014
[S] Stephenson R.M., Jr.: Initially $\kappa $-compact and related spaces. Handbook of Set Theoretic Topology, North Holland (1984), pp. 603-632. MR 0776632 | Zbl 0588.54025
[WW] Wicke H.H., Worrell J.M., Jr.: Point countability and compactness. Proc. Amer. Math. Soc. 55 (1976), 427-431. MR 0400166 | Zbl 0323.54013
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