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Convergent sequences in \(\beta\) X. (English) Zbl 0545.54019
Examples of completely regular spaces are constructed in which no nontrivial sequences converge but in their Čech-Stone compactifications the following types of converge can occur: (in-out) the sequences is in X and the limit point in the remainder, (out-in) the sequence is in the remainder and the limit in X, (out-out) both are in the remainder. A point p of \(\omega^*\) is called a \(\sigma\)-\(\kappa\)- point if there is a family F of disjoint open sets in \(\omega^*\) such that \(| F| \geq \kappa\) and each \(G_{\delta}\) set containing p intersects each member of F. A question is posed whether each point of \(\omega^*\) is \(\sigma\)-2\({}^{\omega}\)-point.
Reviewer: A.Szymanski
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)