Research of topological and geometric structures of topological groups

我们可以在可度量空间X上研究自由拓扑群F(X)的拓扑结构。对于每个自然数n，设F_n(X)是由长度小于或等于n的所有词构成的F_(X)的子空间。则F_n(X)是由空间（X【对称】X^<-1>【对称】{e}）^n的自然映射i_n得到的连续像，其中e是F(X)的单位元素。众所周知，F(X)的拓扑结构非常复杂。事实上，紧度量空间上的自由拓扑群，例如具有极限的收敛序列，是不可第一可数的，因此是不可度量的。另一方面，已知如果空间X是紧化的可度量空间，则每个F_n(X)是可度量的。在2000年，我们得到了每个F_n(X)是可度量空间X的一个充要条件。在2001年，我们得到了一个老问题“当X是可度量的，求出空间X的每个自然映射i_n是商映射”的充分必要条件的答案。进一步，作为该结果的一个应用，我们可以刻画一个可度量空间X，使得F(X)的自由群拓扑具有一个简单的描述，如F(X)的子集U是开放的当且仅当i_n^<-1>(U∩F_n(X))对于每个自然数n在（X【对称】X^<-1>【对称】{e}）^n中是开放的。

We could make progress in studying of topological structures of the free topological group F(X) on a metrizable space X. For each natural number n, let F_n(X) be a subspace of F_(X) formed by all words whose length is less than or equal to n. Then F_n(X) is a continuous image by the natural mapping i_n from the space (X 【symmetry】 X^<-1>【symmetry】{e})^n, where e is the unit element of F(X). It is well-known that the topology of F(X) has a complicated structure. In fact, the free topological group on a compact metric space, for example a convergent sequence with its limit, is not first countable, and hence not metrizable. On the other hand, it is known that if a space X is a compact metrizable space, then each F_n(X) is metrizable. In 2000, we obtained a necessary and sufficient condition of a metrizable space X such that each F_n(X) is metrizable. In 2001, we got an answer to an old problem "Find a necessary and sufficient condition of a space X such that each natural mapping i_n is a quotient mapping", when X is metrizable. Furthermore, as an application of the result, we could characterize a metrizable space X such that the free group topology of F(X) has a simple description such as a subset U of F(X) is open if and only if i_n^<-1>(U ∩ F_n(X)) is open in (X 【symmetry】 X^<-1> 【symmetry】 {e})^n for each natural number n.