Uniform Continuity of Continuous Functions on Uniform Spaces
Uniform Continuity of Continuous Functions on Uniform Spaces
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均匀空间上连续函数的一致连续性
DOI：
10.4153/cjm-1961-055-9
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发表时间：
1961
期刊：
影响因子：
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通讯作者：
M. Atsuji
中科院分区：
文献类型：
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作者：
M. Atsuji
最近几个拓扑学家呼吁注意的一致结构（在大多数情况下，粗函数），在此条件下，每个连续的真实的函数都是一致连续的（让我们称这种结构为[粗] uc-structures），并且已经发现了一些重要的结果，这些结果与uc-structures密切相关，显式或隐式，如休伊特的vS（3）和Shirota的e-完全空间（7）。在这种情况下，很自然地会像Hitotzu（4）那样提出这样的问题：哪些是具有uc-structure的一致空间？在（1 ; 2）中，我们刻画了具有这种结构的度量空间，本文将给出一致空间中该问题的一个解（§ 1），以及它在正规一致空间和度量空间的乘积中的一些应用（§ 2）。显然，一致空间上的每个连续真实的函数是一致连续的，当且仅当该空间的一致结构比由该空间上的所有连续真实的函数定义的一致结构更精细。
Recently several topologists have called attention to the uniform structures (in most cases, the coarsest ones) under which every continuous real function is uniformly continuous (let us call the structures the [coarsest] uc-structures), and some important results have been found which closely relate, explicitly or implicitly, to the uc-structures, such as in the vS of Hewitt (3) and in the e-complete space of Shirota (7). Under these circumstances it will be natural to pose, as Hitotumatu did (4), the problem: which are the uniform spaces with the uc-structures? In (1 ; 2), we characterized the metric spaces with such structures, and in this paper we shall give a solution to the problem in uniform spaces (§ 1), together with some of its applications to normal uniform spaces and to the products of metric spaces (§ 2). It is evident that every continuous real function on a uniform space is uniformly continuous if and only if the uniform structure of the space is finer than the uniform structure defined by all continuous real functions on the space.