## $$s$$-point finite refinable spaces.(English)Zbl 0932.54024

The authors call a topological space $$X$$ $$s$$-point finite refinable (respectively $$ds$$-point finite refinable) if for every open cover $${\mathcal U}$$ of $$X$$ there exists an open refinement $${\mathcal V}$$ of $${\mathcal U}$$ and a (closed discrete) subset $$A({\mathcal U})$$ of $$X$$ such that (i) for all nonempty $$V \in{\mathcal V}$$, $$V\cap A({\mathcal U})\neq\emptyset$$, and (ii) for all $$a\in A({\mathcal U})$$ the set $$\{V\in{\mathcal V}\mid a\in V\}$$ is finite. Obviously, every $$ds$$-point finite refinable space is $$s$$-point finite refinable. Since it follows from two theorems of J. R. Boone [Pac. J. Math. 62, 351-357 (1976; Zbl 0327.54012)] that every weak $$\overline\theta$$-refinable space is irreducible, and that every irreducible space is $$ds$$-point finite refinable, both covering properties are very weak. It is shown that if $$\lambda$$ is an ordinal with $$cf( \lambda) =\lambda> \omega$$ and $$X$$ is a stationary subset of $$\lambda$$ then $$X$$ is not $$s$$-point finite refinable. Moreover, there exists a countably compact $$s$$-point finite refinable LOTS which is not $$ds$$-point finite refinable. However, it is not known whether there exist $$ds$$-point finite refinable spaces which are not irreducible. It is shown that a topological space is $$ds$$-point finite refinable if and only if it is irreducible of order $$\omega$$ in the sense of J. R. Boone [ibid. 62, 359-364 (1976; Zbl 0331.54009]. It is also shown that every strongly collectionwise Hausdorff $$ds$$-point finite refinable space without isolated points is irreducible. It is mentioned that H. R. Bennett has shown that a LOTS is paracompact if and only if it is $$ds$$-point finite refinable. Concerning irreducible spaces it is shown that every topological space can be embedded as a closed subspace into an irreducible space.

### MSC:

 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)

### Citations:

Zbl 0327.54012; Zbl 0331.54009
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