Research of topological and geometric structures of topological groups

在可度量化空间X上的自由拓扑群F（X）的拓扑结构的研究中，我们可以取得一些进展。对于任意自然数n，设F_n（X）是由长度小于或等于n的所有字构成的子空间。则F_n（X）是空间（X [symmetry] X^<-1>[symmetry]{e}）^n的自然映射i_n的连续象，其中e是F（X）的单位元。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，例如一个有极限的收敛序列，不是第一可数的，因此不是可度量化的。另一方面，如果空间X是紧可度量化空间，则每个F_n（X）都是可度量化的。在2000年，我们得到了度量空间X的一个充要条件，使得每个F_n（X）都是度量空间。2001年，我们对一个老问题“求空间X的一个充要条件使得每个自然映射i_n是商映射”，当X是可度量化的时，得到了一个解答。进一步地，作为结果的应用，我们可以刻画可度量化空间X，使得F（X）的自由群拓扑有一个简单的描述，如F（X）的子集U是开的当且仅当对每个自然数n，i_n^<-1>（U &lt;$F_n（X））在（X [对称] X^<-1>[对称] {e}）^n中是开的.

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.