Tightness in products of fans and pseudo-fans. (English) Zbl 0867.54001
Summary: Give \(\kappa\) the discrete topology and \(\omega+1\) the order topology. The \(\kappa\)-fan \(F_\kappa\) is the quotient space obtained by identifying the nonisolated points of \(\kappa\times (\omega+1)\) to a point \(\infty\). We investigate the tightness of products of the form \(F_\kappa\times F_\lambda\), where \(\lambda\leq\kappa\). Some of our results are \(t(F_\kappa\times F_\lambda)\leq \lambda^\omega\), \(t^+(F_{\aleph_\omega}\times F_{\aleph_\omega})\leq \aleph_\omega\), and a new connection between the tightness of \(F_\kappa\times F_\lambda\) and the failure of \(\leq \kappa \text{-cwH}\) in first countable \(< \kappa\text{-cwH}\) spaces.
MSC:
| 54A25 | Cardinality properties (cardinal functions and inequalities, discrete subsets) |
| 54A35 | Consistency and independence results in general topology |
| 54D15 | Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) |
| 03E05 | Other combinatorial set theory |
| 54B15 | Quotient spaces, decompositions in general topology |
Citations:
Zbl 0867.54002References:
| [1] | Arhangel’skii, A., The frequency spectrum of a topological space, Soviet Math. Dokl., 13, 1185-1189 (1972) · Zbl 0275.54004 |
| [2] | J. Brendle and T. LaBerge, Forcing tightness in products of fans, Preprint.; J. Brendle and T. LaBerge, Forcing tightness in products of fans, Preprint. · Zbl 0867.54002 |
| [3] | Ben-David, S.; Magidor, M., The weak \(□\(^∗\) is really weaker than the full □, J. Symbolic Logic, 51, 1029-1033 (1986) · Zbl 0621.03035 |
| [4] | van Douwen, E. K., The integers and topology, (Kuren, K.; Vaughan, J. E., Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 111-167 · Zbl 0561.54004 |
| [5] | Dow, A.; Tall, F. D.; Weiss, W., New proofs of the consistency of the normal Moore space conjecture I, Topology Appl., 37, 33-51 (1990) · Zbl 0719.54038 |
| [6] | Dow, A.; Watson, S., A subcategory of TOP, Trans. Amer. Math. Soc., 337, 825-837 (1993) · Zbl 0823.54005 |
| [7] | K. Eda, G. Gruenhage, P. Koszmidor, K. Tamano and T. Todorčević, Sequential fans in topology, Preprint.; K. Eda, G. Gruenhage, P. Koszmidor, K. Tamano and T. Todorčević, Sequential fans in topology, Preprint. · Zbl 0868.54001 |
| [8] | Fleissner, W. G., On λ-collectionwise Hausdorff spaces, (Topology Proc., 2 (1977)), 445-456 · Zbl 0413.54026 |
| [9] | Fleissner, W. G.; Shelah, S., Collectionwise Hausdorff: Incompactness at singulars, Topology Appl., 31, 101-107 (1989) · Zbl 0659.54016 |
| [10] | Gruenhage, G., \(k\)-spaces and products of closed images of metric spaces, (Proc. Amer. Math. Soc., 80 (1980)), 478-482 · Zbl 0453.54012 |
| [11] | Gruenhage, G.; Tanaka, Y., Products of \(k\)-spaces and spaces of countable tightness, Trans. Amer. Math. Soc., 273, 299-308 (1982) · Zbl 0491.54019 |
| [12] | Hechler, S., On the existence of certain cofinal subsets of \(^ωω\), (Jech, T., Axiomatic Set Theory. Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematics, Volume XIII (1974), American Mathematical Society: American Mathematical Society Providence, RI), 155-173, Part II |
| [13] | Husek, M., Special classes of compact spaces, Categorical Topology, (Lecture Notes in Mathematics, 719 (1978), Springer: Springer Berlin), 167-175 · Zbl 0447.54029 |
| [14] | Kanamori, A.; Magidor, M., The evolution of large cardinal axioms in set theory, (Higher Set Theory. Higher Set Theory, Lecture Notes in Mathematics, 669 (1978), Springer: Springer Berlin), 99-275 · Zbl 0381.03038 |
| [15] | P. Koszmidor, Kurepa trees and topological nonreflection, Preprint.; P. Koszmidor, Kurepa trees and topological nonreflection, Preprint. |
| [16] | LaBerge, T.; Landver, A., Reflection and weakly collectionwise Hausdorff spaces, (Proc. Amer. Math. Soc., 122 (1994)), 203-291 · Zbl 0834.54012 |
| [17] | Shelah, S., Remarks on λ-collectionwise Hausdorff spaces, (Topology Proc., 2 (1977)), 583-592 · Zbl 0439.54009 |
| [18] | Stone, A. H., Metrizability of decomposition spaces, (Proc. Amer. Math. Soc., 7 (1956)), 690-700 · Zbl 0071.16001 |
| [19] | Tall, F. D., Topological applications of generic huge embeddings, Trans. Amer. Math. Soc., 341, 45-68 (1994) · Zbl 0805.54006 |
| [20] | Tanaka, Y., A characterization for the product of closed images of metric spaces to be a \(k\)-space, (Proc. Amer. Math. Soc., 74 (1979)), 166-170 · Zbl 0419.54003 |
| [21] | S. Todorčević, My new fan, Preprint.; S. Todorčević, My new fan, Preprint. |
| [22] | Watson, S., Comments on separation, (Topology Proc., 14 (1989)), 315-372 · Zbl 0742.54014 |
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