A countably compact topological group with the non-countably pracompact square. (English) Zbl 1508.22001
Summary: Under Martin’s Axiom we construct a Boolean countably compact topological group whose square is not countably pracompact.
MSC:
| 22A05 | Structure of general topological groups |
| 54H11 | Topological groups (topological aspects) |
| 54B10 | Product spaces in general topology |
| 54G20 | Counterexamples in general topology |
| 54A35 | Consistency and independence results in general topology |
References:
| [1] | Banakh, T.; Ravsky, A., Feebly compact paratopological groups, preprint · Zbl 1412.22005 |
| [2] | Bardyla, S.; Ravsky, A., Closed subsets of compact-like topological spaces, preprint · Zbl 1452.54012 |
| [3] | Comfort, W. W.; Ross, Kennet A., Pseudocompactness an uniform continuity in topological groups, Pac. J. Math., 16, 3, 483-496 (1966) · Zbl 0214.28502 |
| [4] | van Douwen, E. K., The product of two countably compact topological groups, Trans. Am. Math. Soc., 262, 2, 417-427 (1980) · Zbl 0453.54006 |
| [5] | Dow, A.; Porter, J. R.; Stephenson, R. M.; Woods, R. G., Spaces whose pseudocompact subspaces are closed subsets, Appl. Gen. Topol., 5, 243-264 (2004) · Zbl 1066.54024 |
| [6] | García-Ferreira, S.; Tomita, A. H., Finite powers of selectively pseudocompact groups, Topol. Appl., 248, 3, 50-58 (2018) · Zbl 1404.54024 |
| [7] | García-Ferreira, S.; Tomita, A. H.; Watson, S., Countably compact groups from a selective ultrafilter, Proc. Am. Math. Soc., 133, 3, 937-943 (2005) · Zbl 1062.54037 |
| [8] | Gutik, O.; Ravsky, A., On old and new classes of feebly compact spaces, Visnyk of the Lviv Univ. Series Mech. Math., 85, 48-59 (2018) · Zbl 07877855 |
| [9] | Hart, K. P.; van Mill, J., A countably compact topological group H such that \(H \times H\) is not countably compact, Trans. Am. Math. Soc., 323, 2, 811-821 (1991) · Zbl 0770.54037 |
| [10] | Hrušák, M.; van Mill, J.; Ramos-García, U. A.; Shelah, S., Countably compact groups without non-trivial convergent sequences, (presentable draft) · Zbl 1482.22002 |
| [11] | Madariaga-Garcia, R.; Tomita, A., Countably compact topological group topologies on free Abelian groups from selective ultrafilters, Topol. Appl., 154, 1470-1480 (2007) · Zbl 1116.54004 |
| [12] | Novák, J., On the Cartesian product of two compact spaces, Fundam. Math., 40, 106-112 (1953) · Zbl 0053.12404 |
| [13] | Scarborough, C. T.; Stone, A. H., Products of nearly compact spaces, Trans. Am. Math. Soc., 124, 1, 131-147 (1966) · Zbl 0151.30001 |
| [14] | Stephenson, R. M., Initially κ-compact and related compact spaces, (Kunen, K.; Vaughan, J. E., Handbook of Set-Theoretic Topology (1984), Elsevier), 603-632 · Zbl 0588.54025 |
| [15] | Szeptycki, P.; Tomita, A., HFD groups in the Solovay model, Topol. Appl., 156, 1807-1810 (2009) · Zbl 1173.54016 |
| [16] | Teresaka, H., On Cartesian product of compact spaces, Osaka J. Math., 4, 11-15 (1952) · Zbl 0047.41801 |
| [17] | Tkachenko, Mikhail, Generalization of the theorem by Comfort and Ross, Ukr. Math. J., 41, 3, 377-382 (1989), (in Russian) · Zbl 0698.22004 |
| [18] | Tkachenko, Mikhail, Generalization of the theorem by Comfort and Ross II, Ukr. Math. J., 41, 7, 939-944 (1989), (in Russian) · Zbl 0698.22005 |
| [19] | Tkachenko, Mikhail, Productive properties in topological groups, preprint, (version April 18, 2013) · Zbl 1285.22006 |
| [20] | Vaughan, J. E., Countably compact and sequentially compact spaces, (Kunen, K.; Vaughan, J. E., Handbook of Set-Theoretic Topology (1984), Elsevier), 569-602 · Zbl 0562.54031 |
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.
