Research of dimension and topological structures of topological groups

设F(X)和A(X)分别是拓扑空间X上的自由拓扑群和自由交换拓扑群。对于每个自然数n，F_n(X)表示F(X)的一个子空间，它由所有长度小于或等于n的字组成。用A_n(X)表示A(X)的相同子空间。然后，每个F_n(X)是由空间(X[对称性]X^&lt；-1&gt；[对称性]{e})^n，其中e是F(X)的单位元。1989年，Arangel‘skii，Okunev和Pestov提出，对于度量空间X，A(X)的紧性是否等于X的所有非孤立点的空间的权重。在这项研究中，我们证明了在集合论公理V=L下，答案是肯定的。此外，在非阿贝尔的情况下，我们证明了对于度量空间X，在集合论公理V=L下，F(X)的紧性等于X的权重。当X是可度量化的时候，我们得到了一个老问题的答案“找到一个空间X使得每个映射I_n都是商映射的充要条件”。也就是说，我们刻画了一个可度量化空间X，使得映射I_n是F(X)和A(X)的每个自然数n的商映射。

Let F(X) and A(X) be the free topological group and the free abelian topological group on a topological space X, respectively. For each natural number n, F_n(X) denotes a subspace of F(X) consisting of all words whose length are less than or equal to n. In the abelian case, we. use A_n(X) to denote the same subspace of A(X). Then, each 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).In 1989, Arhangel'skii, Okunev and Pestov asked whether for a metric space X, the tightness of A(X) is equal to the weight of the space of all non-isolated points of X. In this research, we proved that the answer is affirmative under the set-theoretic axiom V=L. Furthermore, in the non-abelian case, we proved that for a metric space X, the tightness of F(X) is equal to the weight of X under the set-theoretic axiom V=L.For last 3 years, we got an answer to an old problem "Find a necessary and sufficient condition of a space X such that each mapping i_n is a quotient mapping", when X is metrizable. That is, we characterized a metrizable space X such that the mapping i_n is a quotient mapping for each natural number n for both F(X) and A(X), respectively.