The Frechet-Urysohn Properties And Metrizable Problems In Topological Groups

It is obvious that the strong connections exists between the metrization of topology group and its sequential properties. We can answer the problem that wether weak first countable topological groups are metrizable in two ways.If the T2condition is assumed, then the answer will be affirmative. We will prove this conclusion using a Freehet-Urysohn property of topological groups in the second chapter of this article. Furthermore, the problem that under what kind of condition topological groups are Freehet-Urysohn has been dicussed. The author provided a improved proof for the condition given by P.J.Nyikos in [5].On the other hand, if the T2condition removed, then the answer will be negative. We will first build an example of topological vector space E in the third chapter. The counterexample is built upon E, that is the topological group G which will be introduced in chapter four. G is weak first countable but not a pseudometrizable space. The author discovered some futher properties about topological vector space E including the sequential order of E and the necessary and sufficient condition for a subset of E to be closed. For the example G, the author pointed out a wrong conclusion in [5] by constructed an counterexample in chapter four.