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**On uniform paracompactness.**
*(English)*
Zbl 0879.54032

The definitions of uniform paracompactness of Rice (R-paracompact), Borubaev (B-paracompact) and Frolík (F-paracompact) are recalled. The authors introduce three other definitions of uniform paracompactness, referring to the second author, and make some relationships between all these notions clear.

The notion of P-paracompactness is a straightforward simplification and strengthening of B-paracompactness. It is shown that this strengthening is too restrictive in the sense that there is a uniformly locally compact space which is not P-paracompact. A natural analogue to the characterization of paracompact topological spaces in terms of maps onto metric spaces is obtained.

Introducing a notion of (\(\tau\)-) decomposability, it is shown that a uniform space is P-paracompact if and only if it is B-paracompact and finitely decomposable. Related classes of \(\tau\)-P-paracompact spaces are introduced. Finitely P-paracompact coincides with B-paracompact whereas every countably P-paracompact space is F-paracompact. Thence, the class of F-paracompact uniform spaces is shown to be strictly larger than the class of B-paracompact ones.

In the class of “countably bounded” uniform spaces, all the introduced classes of uniform paracompactness coincide with uniform spaces for which the related topological space is Lindelöf and also with one more, in general wider, class of uniform spaces referred to the second author, too.

The notion of P-paracompactness is a straightforward simplification and strengthening of B-paracompactness. It is shown that this strengthening is too restrictive in the sense that there is a uniformly locally compact space which is not P-paracompact. A natural analogue to the characterization of paracompact topological spaces in terms of maps onto metric spaces is obtained.

Introducing a notion of (\(\tau\)-) decomposability, it is shown that a uniform space is P-paracompact if and only if it is B-paracompact and finitely decomposable. Related classes of \(\tau\)-P-paracompact spaces are introduced. Finitely P-paracompact coincides with B-paracompact whereas every countably P-paracompact space is F-paracompact. Thence, the class of F-paracompact uniform spaces is shown to be strictly larger than the class of B-paracompact ones.

In the class of “countably bounded” uniform spaces, all the introduced classes of uniform paracompactness coincide with uniform spaces for which the related topological space is Lindelöf and also with one more, in general wider, class of uniform spaces referred to the second author, too.

Reviewer: Petr Holický (Praha)

### MSC:

54E15 | Uniform structures and generalizations |

54D20 | Noncompact covering properties (paracompact, Lindelöf, etc.) |

### Keywords:

uniform paracompact space
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\textit{D. Buhagiar} and \textit{B. Pasynkov}, Czech. Math. J. 46, No. 4, 577--586 (1996; Zbl 0879.54032)

### References:

[1] | A. A. Borubaev: Uniform spaces and uniformly continuous maps. Frunze, “Ilim”. · Zbl 0688.54018 |

[2] | R. Engelking: General Topology. PWN, Warsaw. · Zbl 0684.54001 |

[3] | Z. Frolík: On paracompact uniform spaces. Czechoslovak Math. J. 33, 476-484. · Zbl 0542.54023 |

[4] | M. D. Rice: A note on uniform paracompactness. Proc. Amer. Math. Soc. 62.2, 359-362. · Zbl 0353.54011 · doi:10.2307/2041044 |

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