×

Uniform powers of compacta and the proximal game. (English) Zbl 1375.54007

The authors answer a question of P. Nyikos proving that if \(X\) is a proximal compactum, then the countable uniform power \(^\omega X\) of \(X\) is also proximal. It then follows from known results that the countable uniform power of a Corson compactum is collectionwise normal, countably paracompact and Fréchet-Urysohn.
They also present interesting results about first countability and realcompactness in countable uniform powers of compacta. Furthermore they contribute to the question of which properties of proximal spaces also hold in semi-proximal spaces.
The countable uniform power (or uniform box product) of a uniform space \(X\) is a specific topology on \(^\omega X\) that lies between the Tychonoff topology and the box topology.

MSC:

54B10 Product spaces in general topology
54C45 \(C\)- and \(C^*\)-embedding
54D30 Compactness
54D60 Realcompactness and realcompactification
54E15 Uniform structures and generalizations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Balogh, Zoltán; Eisworth, Todd; Gruenhage, Gary; Pavlov, Oleg; Szeptycki, Paul J., Uniform and anti-uniformizable properties of ladder systems, Fundam. Math., 181, 3, 189-213 (2004) · Zbl 1051.03034
[2] Bell, Jocelyn R., The uniform box product, Proc. Am. Math. Soc., 142, 6, 2161-2171 (2014) · Zbl 1294.54011
[3] Bell, Jocelyn R., An infinite game with topological consequences, Topol. Appl., 175, 1-14 (2014) · Zbl 1301.54041
[4] Clontz, Steven; Gruenhage, Gary, Proximal spaces are Corson compact, Topol. Appl., 173, 1-8 (2014) · Zbl 1319.54005
[5] Engelking, Ryszard, General Topology, Sigma Series in Pure Mathematics, vol. 6 (1989), Heldermann Verlag: Heldermann Verlag Berlin, viii+529 pp · Zbl 0684.54001
[6] Gillman, Leonard; Jerison, Meyer, Rings of Continuous Functions, The University Series in Higher Mathematics (1960), D. Van Nostrand Co., Inc.: D. Van Nostrand Co., Inc. Princeton, NJ, Toronto, London, New York, ix+300 pp · Zbl 0093.30001
[7] Kemoto, Nobuyuki; Ohta, Haruto; Tamano, Ken-ichi, Products of spaces of ordinal numbers, Proceedings of the Tsukuba Topology Symposium. Proceedings of the Tsukuba Topology Symposium, Tsukuba, 1990. Proceedings of the Tsukuba Topology Symposium. Proceedings of the Tsukuba Topology Symposium, Tsukuba, 1990, Topol. Appl., 45, 3, 245-260 (1992) · Zbl 0789.54006
[8] Nyikos, Peter, Proximal and semi-proximal spaces, Quest. Answ. Gen. Topol., 32, 2, 79-91 (2014) · Zbl 1314.54015
[9] Todorčević, Stevo, Trees and linearly ordered sets, (Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 235-293 · Zbl 0557.54021
[10] Walker, Russel C., The Stone-Čech Compactification, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 83 (1974), Springer-Verlag: Springer-Verlag New York, Berlin, x+332 pp · Zbl 0292.54001
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.