## On the product of a compact space with an hereditarily absolutely countably compact space.(English)Zbl 0937.54013

M. V. Matveev defined absolutely countably compact (shortly, acc) spaces $$X$$ in 1994 [Topology Appl. 58, No. 1, 81-92 (1994; Zbl 0801.54021)]: for every open cover $$\mathcal U$$ and a dense set $$D$$ there is a finite set $$F$$ such that the star of $$F$$ with respect to $$\mathcal U$$ covers $$X$$. A space $$X$$ is hereditarily acc (shortly, hacc) if every of its closed subspaces is acc. The author proves analogous results on products of hacc spaces to those proved by Vaughan for acc spaces (all spaces are Hausdorff): (1) Product of a compact sequential space with a regular hacc is hacc. (2) Product of a compact space of countable tightness with $$Y$$ is hacc provided $$Y$$ is either regular $$\omega$$-bounded hacc or regular acc with countable tightness. (Unlike Vaughan’s results for acc spaces, the last occurrence of countable tightness in (2) cannot be replaced by countable density-tightness.).
Reviewer: M.Hušek (Praha)

### MSC:

 54D30 Compactness

### Keywords:

absolutely countably compact space; cartesian product

Zbl 0801.54021
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