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On the uniform paracompactness of the product of two uniform spaces. (English) Zbl 0533.54013

The concept of uniform paracompactness was introduced by M. D. Rice [Proc. Am. Math. Soc. 62, 359-362 (1977; Zbl 0353.54011)]. The main result concerns the behavior of this property under products. The author generalizes a theorem of A. Hohti, stated for metric spaces, to uniform spaces: The product \({\mathcal U}\times {\mathcal V}\) of two uniformly paracompact uniform spaces \({\mathcal U}\), \({\mathcal V}\), which are both first- countable as topological spaces, is uniformly paracompact, if and only if either both spaces are locally compact or at least one of them is compact. Some interesting examples are given.
Reviewer: R.Domiaty

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54E15 Uniform structures and generalizations
54B10 Product spaces in general topology

Citations:

Zbl 0353.54011
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References:

[1] G. Artico - R. Moresco , \omega \mu -additive topological spaces , Rend. Sem. Mat. Univ. Padova , 67 ( 1982 ). Numdam | Zbl 0499.54024 · Zbl 0499.54024
[2] H.H. Corson , The determination of paracompactness by uniformities , Amer. J. Math. , 80 ( 1958 ), pp. 185 - 190 . MR 94780 | Zbl 0080.15803 · Zbl 0080.15803 · doi:10.2307/2372828
[3] P. Fletcher - W.F. Lindgren , C-complete quasi-uniform spaces , Arch. Math. , 30 ( 1978 ), pp. 175 - 180 . MR 474216 | Zbl 0402.54024 · Zbl 0402.54024 · doi:10.1007/BF01226037
[4] L. Gillman - M. Jerison , Rings of continuous functions , D. Van Nostrand C. ( 1960 ). MR 116199 | Zbl 0093.30001 · Zbl 0093.30001
[5] A. Hohti , On uniform paracompactness , Ann. Ac. Sc. Fennicae , Series A , Math. Diss. , 36 ( 1981 ). MR 625526 | Zbl 0502.54021 · Zbl 0502.54021
[6] J.R. Isbell , Uniform spaces , Math. Surveys , no. 12 , Amer. Math. Soc. ( 1964 ). MR 170323 | Zbl 0124.15601 · Zbl 0124.15601
[7] E. Michael , The product of a normal space and a metric space need not be normal , Bull. Amer. Math. Soc. , 69 ( 1963 ), pp. 375 - 376 . Article | MR 152985 | Zbl 0114.38904 · Zbl 0114.38904 · doi:10.1090/S0002-9904-1963-10931-3
[8] M.D. Rice , A note on uniform paracompactness , Proc. Amer. Math. Soc. , 62 , no. 2 ( 1977 ), pp. 359 - 362 . MR 436085 | Zbl 0353.54011 · Zbl 0353.54011 · doi:10.2307/2041044
[9] B.M. Scott , Review of [FL] , Math. rev. ( 1979 ), 57 # 13863.
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