CLP-compactness for topological spaces and groups (Q876524)

A topological space \(X\) is said to be CLP-compact if every cover of \(X\) consisting of clopen subsets has a finite subcover. This concept (under a different name) was introduced by \textit{A. Sostak} [Gen. Topol. Relat. mod. Anal. Algebra, Proc. 4th Prague Topol. Symp. 1976, Part B, 445--451 (1977; Zbl 0369.54006)] as a convenient simultaneous generalization of compactness and connectedness. The author provides a powerful insight into CLP-spaces and CLP-topological groups using methods which rely on the study of the totally disconnected space (resp. group) of quasicomponents of the given topological space (resp. topological group). We will only single out some of the many results contained in this paper. In the general setting, it is proved (Theorem 2.10) that a Tychonoff space is CLP-compact if and only if it meets every connected component of its Stone-Čech compactification, and (Theorem 3.4.) that the product of a family of topological spaces is CLP-compact if and only if each space is CLP-compact and the product is CLP-rectangular (meaning that the topology generated by all clopen sets in the product coincides with the product of the topologies generated by all clopen sets in the factor spaces). The finite version of this latter result was proved by \textit{J. Steprans} and \textit{A. Sostak} [Topology Appl. 101, No. 3, 213--229 (2000; Zbl 0962.54020)]. CLP-compactness proves to be a fruitful concept in the framework of topological groups, especially when combined with precompactness or pseudocompactness. It is shown that an abelian group without nontrivial divisible subgroups admits a totally disconnected precompact CLP-group topology if and only if it admits a compact group topology (Theorem 5.6), that divisible CLP-compact precompact groups are connected (Corollary 5.7.) and that the product of a family of pseudocompact groups is CLP-compact if and only if all groups in the family are CLP-compact (Theorem 6.6). Totally disconnected pseudocompact CLP-compact abelian groups of arbitrary finite dimension are constructed.