Local connectedness and extension of uniformly continuous functions
Local connectedness and extension of uniformly continuous functions
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一致连续函数的局部连通性和可拓性
DOI：
10.1016/j.topol.2005.04.016
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发表时间：
2006
期刊：
影响因子：
--
通讯作者：
J. Pelant
中科院分区：
文献类型：
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作者：
A. Berarducci;D. Dikranjan;J. Pelant
度量空间X是直的，如果对X的每个有限闭集覆盖，对X上的每个真实的值函数f，如果f在每个覆盖集上一致连续，则f在整个X上一致连续。在[A. Berarducci，D. Dikranjan，J. Pelant，An additivity theorem for uniformly continuous functions，Topology and its Applications 146-147（2005）339-352]，其中包含局部连通空间类（它们是一致局部连通空间）和完全不连通空间类（它们与完全不连通Atsuji空间一致）内的直空间的特征。我们证明了直空间的完备化是直的，并刻画了直空间的稠密直子空间。为了进一步阐明直线性和空间的局部连通性之间的关系，我们引入了直线性和一致局部连通性之间的两个中间性质，并给出了各种例子来区分它们。这些性质之一与完备空间的直性相吻合，并以这种方式提供了完备直空间在空间的拟分量的行为方面的有用表征。
A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. The straight spaces have been studied in [A. Berarducci, D. Dikranjan, J. Pelant, An additivity theorem for uniformly continuous functions, Topology and its Applications 146–147 (2005) 339–352], which contains characterization of the straight spaces within the class of the locally connected spaces (they are the uniformly locally connected ones) and the class of the totally disconnected spaces (they coincide with the totally disconnected Atsuji spaces). We show that the completion of a straight space is straight and we characterize the dense straight subspaces of a straight space. In order to clarify further the relation between straightness and the level of local connectedness of the space we introduce two more intermediate properties between straightness and uniform local connectedness and we give various examples to distinguish them. One of these properties coincides with straightness for complete spaces and provides in this way a useful characterization of complete straight spaces in terms of the behaviour of the quasi-components of the space.