连续映射的扩张、拼接以及连续映射在列紧集上的性质

The extension and the conjunction of continuous mapping and its propertiesz on a sequential compact set

现有文献对于连续映射扩张的讨论都基于闭集,对于一致连续映射的扩张都局限于En.本文在一般的度量空间中对非闭集上连续映射的扩张、任意集合上一致连续映射的扩张以及任意两个集合上连续映射的拼接(扩张的一种形式)进行了讨论.得出:①若连续映射在非闭集的一些边界点存在极限,则它可连续扩张到这些边界点.②一个集合上的一致连续映射在向一个紧集做连续扩张时,它必然是一致连续扩张.③可连续扩张到边界的连续映射在列紧集上具有若干与紧集上相同的性质.

In present literature, extension of continuous mapping discussions is discussed on a closed set, extension of uniformly continuous mapping are all confined to En. In a metric space, extension of continuous mapping on a non-closed set and extension of uniformly continuous mapping on any set, as well as the conjunction （a way of extension） of continuous mapping on any two sets are discussed in this paper. The main conclusions are as follows： （1） If a continuous mapping has the limit at some points of the boundary of a non-closed set, then the mapping can be continuously extended to those points. （2） When a uniformly continuous mapping on a set is extended to a compact set,the extension must be uniformly continuous. （3） A continuous mapping on a sequential compact set that can be continuously extended to the boundary has the same properties as on a compact set.