Sequentially Compact Cancellative Topological Semigroups: Some Progress on the Wallace Problema

Abstract

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.

Countably Compact Spaces which Cannot Be Cancellative Topological Semigroups

“…Therefore, Yo is a sequentially compact subsemigroup of Y. So by Grant's result [9], YO is a topological group. Let e be the identity element of YO.…”

“…D.L. Grant [9] in 1993 extended this result for the completely regular case by showing that every sequentially compact, cancellative topological semigroup is a topological group.…”

“…Besides the consistency of Wallace's problem, related questions arise. Watson [9] asked whether a pseudocompact, first countable, cancellative topological semigroup is a topological group. Using convergence properties of cancellative topological semigroups, we show that the obvious types of candidates for counterexamples to Watson's question do not admit a cancellative topological semigroup structure.…”

Feebly compact paratopological groups and real-valued functions

“…In particular, compact Hausdorff paratopological groups are topological groups. The latter fact has been considerably generalized by Ellis, Grant, Brand, Bouziad, Bokalo and Guran, Romaguera and Sanchis, KenderovKortezov-Moors, and some others (see [11,15,9,8,7,30,18]). …”

The continuity of the inversion and the structure of maximal subgroups in countably compact topological semigroups

“…Also Grant [13] and Yur'eva [29] showed that a Hausdor cancellative sequential countably compact topological semigroup is a topological group. Bokalo and Guran [5] established that an analogous theorem is true for cancellative sequentially compact semigroups.…”