AB-compacta. (English) Zbl 1212.54016

Gorelic, Isaac; Juhász, István

Commentat. Math. Univ. Carol. 49, No. 1, 141-146 (2008).

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

Keywords:

compact space; first countable space; character of a point

Citations:

Zbl 0937.54022