ABSTRATT. Some results of Mukherjea and Tserpes are generalized by showing that any sequentially compact, cancellative topological semigroup is a topological group. Hence, any countably compact, cancellative topological semigroup with any additional condition that would imply sequential compactness is also a topological group. Finally, it is shown that any w-bounded, cancellative topological semigroup is also a topological group. In 1952, Numakura [12] showed that every compact, cancellative topological semigroup is a topological group. While the example of the half-open interval (0, 11 under multiplication shows that the same is not true if "locally compact" replaces "compact," Ellis showed [6,7] that a locally compact group with continuous multiplication (called a paratopological group by Bourbaki [ 3 ] ) must have continuous inversion and thus be a topological grqup. Brand [2] and Pfister [14] showed that Ellis's result remains true if either "Cech-complete" or "locally countably compact" replaces "locally compact." (When consulting Pfister's paper [14], the reader should be sure to rcad the proof of proposition (A), in which a distinctly more general result is obtained, and explicitly acknowledged, than is given in the formal statement of thc proposition.) The author [8] produced some conditions undcr which a pscudocompact paratopological group is a topological group.In 1955, Wallace [16] asked whether every countably compact, canccllativc topological scmigroup is a topological group, and this question remains unsolved to date. Mukherjea and Tserpcs, in 1972, showed [Ill that every cancellative topological semigroup that is countably compact and first countable (and hence sequentially compact) is in fact a topological group. The present author then notcd [8] that "first countable" can be replaced by the concept called weakfirst countability by Nyikos [ 131 and "the gf-axiom of countability" by Arkhangelskii [ 11. We here further generalize the results of Mukherjea and Tserpes by producing weaker additional conditions related to first countability under which their theorem rcmains true. We note further that any o-bounded, cancellativc topological semigroup is also a topological group.In what follows, all spaces are assumcd to be completely regular and Hausdorff.Mathematics Subjecf Classification. Primary 22A15,54D20,54D55.
It is a question of Arhangel'skiĭ [1] (Problem 2) whether the identity ψ(G) = X(G) holds for every minimal Hausdorff topological group G = 〈G,u〉). (Here, as usual, ψ(G), the pseudocharacter of G, is the least cardinal number K for which there is such that and and x(G), the character of G,is the least cardinality of a local base at e for (〈G,u〉.) That 〈G, u〉 is minimal means that, if v is a Hausdorff topological group topology for G and v ⊂ u, then v = u.In this paper, we give some conditions on G sufficient to ensure a positive response to Arhangel'skiï's question, and we offer an example which responds negatively to a question on minimal groups posed some years ago (cf. [6] (p. 107) and [4] (p. 259)).
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