×

zbMATH — the first resource for mathematics

Sums and products of ultracomplete topological spaces. (English) Zbl 1019.54015
In a previous paper [Acta Math. Hung. 92, 19-26 (2001; Zbl 0997.54037)] the authors showed that strongly complete topological spaces (introduced by V. I. Ponomarev and V. V. Tkachuk) and cofinally Čech complete topological spaces (introduced by S. Romaguera) are equivalent notions and called such spaces ultracomplete. In the present paper the authors study sums and products of such spaces. Sample result: Theorem 4.12. Let \(X\) and \(Y\) be two ultracomplete spaces. Then \(X\times Y\) is ultracomplete if and only if one of the following conditions holds: (i) both \(X\) and \(Y\) are locally compact, or (ii) either \(X\) or \(Y\) is countably compact, locally compact, or (iii) both \(X\) and \(Y\) are countably compact. Further, the authors ask whether each Čech complete, countably compact space is ultracomplete. They show that for \(GO\) spaces the answer is yes.

MSC:
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54B10 Product spaces in general topology
54D99 Fairly general properties of topological spaces
Citations:
Zbl 0997.54037
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arhangel’skiı̌, A.V., Bicompact sets and the topology of spaces, Trudy moskov. mat. obshch., Trans. Moscow math. soc., 13, 1-62, (1965) · Zbl 0162.26602
[2] D. Buhagiar, I. Yoshioka, Ultracomplete topological spaces, Acta Math. Hungar., to appear · Zbl 0997.54037
[3] Corson, H.H., The determination of paracompactness by uniformities, Amer. J. math., 80, 185-190, (1958) · Zbl 0080.15803
[4] Császár, A., Strongly complete, supercomplete and ultracomplete spaces, (), papers dedicated to Prof. L. Iliev’s 60th Anniversary, Sofia
[5] Engelking, R., General topology, (1989), Heldermann Berlin · Zbl 0684.54001
[6] Garcı́a-Máynez, A.; Romaguera, S., Perfect pre-images of cofinally complete metric spaces, Comment. math. univ. carolinae, 40, 2, 335-342, (1999) · Zbl 0976.54032
[7] Hewitt, E., Rings of real-valued continuous functions, I, Trans. amer. math. soc., 64, 45-99, (1948) · Zbl 0032.28603
[8] Howes, N.R., On completeness, Pacific J. math., 38, 431-440, (1971) · Zbl 0202.54002
[9] Howes, N.R., Paracompactifications, preparacompactness and some problems of K. Morita and H. tamano, Questions answers gen. topology, 10, 191-204, (1992) · Zbl 0803.54022
[10] Howes, N.R., Modern analysis and topology, (1995), Springer Berlin · Zbl 0853.54002
[11] Nagata, J., Modern general topology, (1985), Elsevier Science Amsterdam
[12] Ponomarev, V.I.; Tkachuk, V.V., Countable character of X in βX versus the countable character of the diagonal in X×X, Vestnik MGU, 5, 16-19, (1987), (in Russian) · Zbl 0637.54003
[13] Romaguera, S., On cofinally complete metric spaces, Questions answers gen. topology, 16, 165-169, (1998) · Zbl 0941.54030
[14] Walker, R.C., The stone-čech compactification, (1974), Springer Berlin
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.