拓扑群上的一致与等度连续及其非标准特征

The Uniformly Continuous and Equicontinuity on Topological Group and Their Nonstandard Characteristics

文中涉及的拓扑空间约定都包含在标准全域 U 的个体集 S 中,非标准全域~*U 是扩大.m(α)表示α点的单子,x≈α表示x∈m(α).定义1 设(G,I_1,·,e_1)及(H,I_2,·,e_2)是拓扑群,f:G→H,若对每一 V∈I_2(e_2)存在 W∈I_1(e_1)使对任意的 g_1,g_2∈G,若 g_1·g_2^(-1)∈W,即有 f(g)·f^(-1)(g_2)∈V,则称 f 在 G 上一致连续.容易验证,若 f 在 G 上一致连续,则其在 G 上连续.

In this paper,the concepts of uniform continuity and equicontinuity from ma- tric spaces to topological groups is generalized.Secondly,their nonstandard charac- teristics are given.Finally,we use the nonstandard methods to study some of their proporties.