A space X is of countable type (resp. subcountable type) if every compact subspace F of X is contained in a compact subspace K that is of countable character (resp. countable pseudocharacter) in X. In this paper, we mainly show that: (1) For a functionally Hausdorff space X, the free paratopological group FP(X)and the free abelian paratopological group AP(X) are of countable type if and only if X is discrete; (2) For a functionally Hausdorff space X, if the free abelian paratopological group AP(X) is of subcountable type then X has countable pseudocharacter. Moreover, we also show that, for an arbitrary Hausdorff μ-space X, if AP2(X) or FP2(X) is locally compact, then X is a topological sum of a compact space and a discrete space.
A.V.Arhangel’skiĭClasses of topological groups, Russian Math. Surveys, 36(3)(1981), 151–174. Russian original in: Uspekhi Mat. Nauk, 36 (1981), 127–146.
A.V.Arhangel’skiĭClasses of topological groups, Russian Math. Surveys, 36(3)(1981), 151–174. Russian original in: Uspekhi Mat. Nauk, 36 (1981), 127–146.)| false
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Studia Scientiarum Mathematicarum Hungarica
2021 Volume 58
Magyar Tudományos Akadémia
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