## 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

### Keywords:

compact space; first countable space; character of a point

Zbl 0937.54022
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