拓扑空间上一类函数的极限及连续性

Limit and Continuity of a Class of Functions on Topological Spaces

摘要: 极限和连续是实数集理论最重要的概念，而实数集可以看作特殊的拓扑空间，因此如何定义拓扑空间到实数域上函数的极限及连续是具有重要研究意义的。本文借助于拓扑空间中邻域的概念给出了极限和连续的定义，证明了它们的充要条件并讨论了相关性质。

Abstract: Limit and continuity are the most important concepts in the theory of the real number set, and the real number set can be regarded as a special topological space, so how to define the limit and con-tinuity of the function from the topological space to the real number field is of great research sig-nificance. In this paper, we give the definition of limit and continuity by means of the concept of neighborhood in topological space, prove their sufficient and necessary conditions, and discuss their related properties.