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

Abstract

We search for conditions on a countably compact (pseudocompact) topological semigroup under which: (i) each maximal subgroup H(e) in S is a (closed) topological subgroup in S; (ii) the Cliord part H(S) (i.e. the union of all maximal subgroups) of the semigroup S is a closed subset in S; (iii) the inversion inv : H(S) → H(S) is continuous; and (iv) the projection π : H(S) → E(S), π : x −→ xx −1 , onto the subset of idempotents E(S) of S, is continuous.In this paper all topological spaces will be assumed to be Hausdor. We shall follow the terminology of [7,8,11]. We shall denote the cardinality of continuum by c and the topological closure of subset A in a topological space by A. We shall call a T 3 -space a regular topological space.A topological space X is said to be countably compact if any countable open cover of X contains a nite subcover [11]. A topological space X is called pseudocompact if each continuous real-valued function on X is bounded [11].

Pseudobounded or ω-pseudobounded paratopological groups

“…It is well known that paratopological groups is a good generalization of topological groups. The topic of paratopological groups is quite popular nowadays and one can see a lot of activities going on in what concerns of the study of these objects, see [1,3,7,10,11].…”

Metrizability of Clifford topological semigroups

“…The following "countably compact" version of Theorem 1.1 was proved by Gutik, Pagon, and Repovš in [7].…”

“…The following "countably compact" version of Theorem 1.1 was proved by Gutik, Pagon, and Repovš in [7]. There are also some other conditions guaranteeing that a countably compact Clifford topological semigroup is a topological Clifford semigroup.…”