On countably compact quasi-pseudometrizable spaces. (English) Zbl 0706.54027

Salbany, Sergio; Romaguera, Salvador

J. Aust. Math. Soc., Ser. A 49, No. 2, 231-240 (1990).

Summary: We prove the following results: (1) A quasi-metrizable space is compact if and only if every compatible quasi-metric has a quasi-metric left d- sequential completion. (2) A quasi-pseudometrizable space is countably compact if and only if every compatible quasi-pseudometric is pointwise bounded. (3) A quasi-pseudometrizable space is compact if and only if every compatible quasi-pseudometric is precompact.

MSC:

54E45	Compact (locally compact) metric spaces
54E52	Baire category, Baire spaces
54D30	Compactness
54E35	Metric spaces, metrizability
54E50	Complete metric spaces

Keywords:

left d-Cauchy sequence; left d-(weakly, sequentially) complete quasi- pseudometric space; quasi-metric completion; (countably) compact space; quasi-metrizable space; quasi-metric left d-sequential completion; quasi- pseudometrizable space