×

AB-compacta. (English) Zbl 1212.54016

Summary: We define a compactum \(X\) to be AB-compact if the cofinality of the character \(\chi (x,Y)\) is countable whenever \(x\in Y\) and \(Y\subset X\). It is a natural open question whether every AB-compactum is necessarily first countable.
We strengthen several results from A. V. Arhangel’skii and R. Z. Buzyakova [Commentat. Math. Univ. Carol. 39, No. 1, 159–166 (1998; Zbl 0937.54022)] by proving the following results.
(1) Every AB-compactum is countably tight.
(2) If \(\mathfrak p = \mathfrak c\), then every AB-compactum is Frèchet-Urysohn.
(3) If \(\mathfrak c < \aleph _\omega \), then every AB-compactum is first countable.
(4) The cardinality of any AB-compactum is at most \(2^{< \mathfrak c}\).

MSC:

54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54A35 Consistency and independence results in general topology
54D30 Compactness

Citations:

Zbl 0937.54022
PDF BibTeX XML Cite
Full Text: EuDML EMIS