Ultracomplete topological spaces
If \(X\) is a set and \(\Sigma\) is a set of coverings of \(X\), then we can consider \((X,\Sigma)\) to be a generalized uniform space. This point of view originates from the work of \textit{J. W. Tukey} [Convergence and uniformity in topology, Princeton (1940)] who used \((X,\Sigma)\), where \(\Sigma\) fulfills some appropriate conditions, as another approach to define uniform spaces. Is now \(\phi\) a filter on \(X\), one can define \(\phi\) to be Cauchy, iff \(\forall \alpha\in \Sigma: \phi\cap\alpha\neq\emptyset\) holds. For a Tukey-uniform space \((X,\Sigma)\), \textit{H. H. Corson} [Am. J. Math. 80, 185-190 (1958; Zbl 0080.15803)] defined a filter \(\phi\) to be weakly Cauchy, iff for all \(\alpha\in \Sigma\) there exists a refining filter \(\psi\supseteq\phi\), such that \(\psi\cap\alpha\neq\emptyset\). In 1937, in his famous paper [Ann. Math. (2) 38, 823-844 (1937; Zbl 0017.42803)], \textit{E. Čech} also defined the now called Čech-complete spaces: A Tikhonov space \(X\) is said to be Čech complete, iff it is a \(G_{\delta}\)-set in one (equivalently in all) of its Hausdorff compactifications. An internal characterization of Čech complete spaces was established independently by \textit{Z. Frolík} [Czech. Math. J. 10, 359-379 (1960; Zbl 0100.18701)] and by \textit{A. V. Arkhangel'skij} [Vestn. Mosk. Univ., Ser. I 16, 37-39 (1961; Zbl 0106.15702)]: A Tikhonov space \(X\) is Čech complete, iff there exists a countable generalized uniform structure \(\Sigma=(\alpha_n)_{n\in \mathbb N}\) of open covers of \(X\) such that every \(\Sigma\)-Cauchy filter clusters in \(X\). There are some interesting topological notions which are related to Čech completeness. For Tukey-uniform spaces, \textit{N. R. Howes} [Pac. J. Math. 38, 431-440 (1971; Zbl 0202.54002)] introduced the notion of a confinally complete uniform space and proved that a uniform space is confinally complete, iff every weak Cauchy filter has a cluster point. This notion independently was also defined by \textit{Á. Császár} [Strongly complete, supercomplete and ultracomplete spaces, Math. Struct., Comput. Math., Math. Modelling, Sofia (1975; Zbl 0269.54018)]. Császár called these spaces ultracomplete. In 1998, \textit{S. Romaguera} [Quest. Answers Gen. Topology 16, No. 2, 165-170 (1998; Zbl 0941.54030)] defined the notion of a confinally Čech complete space: A Tikhonov space \(X\) is called confinally Čech complete, iff there is a countable family \(\Sigma=(\alpha_n)_{n\in \mathbb N}\) of open covers of \(X\) such that, whenever \(\psi\) is a filter on \(X\) such that for each \(n\) there is some \(\alpha_n\in \Sigma\) which meets all the members of \(\psi\), then \(\psi\) clusters in \(X\). On the other hand, in 1987 \textit{V. I. Ponomarev} and \textit{V. V. Tkachuk} [Mosc. Univ. Math. Bull. 42, No.5, 14-17 (1987); translation from Vestn. Mosk. Univ., Ser. I 1987, No. 5, 16-19 (1987; Zbl 0637.54003)] characterized strongly complete topological spaces as those spaces, which have countable character in their Stone-Čech compactification. As the main result of the present paper, the authors prove that the completeness notions of Romaguera and Ponomarev/Tkachuk are equivalent, where now strongly complete spaces are called ultracomplete.
scientific article; zbMATH DE number 1821677