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Short proofs of two theorems in topology. (English) Zbl 0780.54023
Summary: We present short and elementary proofs of the following two known theorems in General Topology: (i) [H. H. Wicke and J. M. Worrell jun., Proc. Am. Math. Soc. 55, 427-431 (1976; Zbl 0323.54013)] A \(T_ 1\) weakly \(\delta\theta\)-refinable countably compact space is compact. (ii) [A. Ostaszewski, ibid 68, 339-343 (1978; Zbl 0392.54014)]. A compact Hausdorff space which is a countable union of metrizable spaces is sequential.

MSC:
54D30 Compactness
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D55 Sequential spaces
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