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Certain sequences with compact closure. (English) Zbl 1169.54003

The authors deal with the question when a \(\beta\)-sequence in a \(T_2\)-space \(X\) has a compact closure. They show that every \(\beta\)-sequence has a feebly compact closure and, if \(X\) is monotonically normal, a compact closure. In regular spaces and in Tychonoff spaces, they give several equivalent conditions for the compactness of the closure of a \(\beta\)-sequence. Further, they express \(\beta\)-sequences with countable closure by fairly concrete forms and give several answers to the question when a \(\beta\)-space is a strong \(\beta\)-space [Y. Yajima, Houston J. Math. 33, No. 2, 531–540 (2007; Zbl 1243.54046)]. Examples show the sharpness of the results.

MSC:

54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54D30 Compactness
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54E99 Topological spaces with richer structures
54G20 Counterexamples in general topology

Citations:

Zbl 1243.54046
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References:

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