Compact spaces and the pseudoradial property, I. (English) Zbl 1019.54004
Author’s abstract: “We investigate two properties and their connection to the property of pseudoradiality in the context of compact spaces. The first is the WAP property introduced by P. Simon and the second is the \(\aleph_0\)-pseudoradial property introduced by B. Shapirovskiĭ. We show that \(\diamondsuit\) implies that there is a compact space which is pseudoradial but not WAP. We show that there is a model in which CH fails and in which all compact spaces of weight at most \(\aleph_2\) are \(\aleph_0\)-pseudoradial”.
Reviewer: Bernhard Behrens (Göteborg)
MSC:
| 54A20 | Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) |
| 03E10 | Ordinal and cardinal numbers |
| 54D55 | Sequential spaces |
| 03E65 | Other set-theoretic hypotheses and axioms |
Keywords:
property of weak approximation by points; Fréchet space; sequential space; pseudoradiality; compact spaces; WAPReferences:
| [1] | Bella, A.; Dow, A.; Tironi, G., Pseudoradial spaces: separable subsets and maps onto Tychonoff cubes, (Proceedings of the International School of Mathematics “G. Stampacchia” (Erice, 1998), 111 (2001)), 71-80 · Zbl 0969.54005 |
| [2] | Juhász, I.; Szentmiklóssy, Z., Sequential compactness versus pseudo-radiality in compact spaces, Topology Appl., 50, 1, 47-53 (1993) · Zbl 0794.54005 |
| [3] | Kunen, K., Set Theory—An Introduction to Independence Proofs (1980), North-Holland: North-Holland Amsterdam · Zbl 0443.03021 |
| [4] | Shapirovskiı̆, B., The equivalence of sequential compactness and pseudoradialness in the class of compact \(T_2\)-spaces, assuming CH, (Papers on General Topology and Applications (Madison, WI, 1991) (1993), New York Academy of Sciences: New York Academy of Sciences New York), 322-327 · Zbl 0855.54032 |
| [5] | Simon, P., On accumulation points, Cahiers Topologie Géom. Différentielle Catégoriques, 35, 4, 321-327 (1994) · Zbl 0858.54008 |
| [6] | Weiss, W., The equivalence of a generalized Martin’s axiom to a combinatorial principle, J. Symbolic Logic, 46, 4, 817-821 (1981) · Zbl 0493.03026 |
| [7] | Weiss, W., Versions of Martin’s axiom, (Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 827-886 · Zbl 0571.54005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
