2001 Fiscal Year Final Research Report Summary
Research of topological and geometric structures of topological groups
| Project/Area Number |
12640065
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | Shizuoka University |
|---|
Principal Investigator |
YAMADA Kohzo Shizuoka University, Faculty of Education, Department of Mathematics, Assistant Professor, 教育学部, 助教授 (00200717)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
SHAKHMATOV Dmitri Ehime University, Faculty of Science, Department of Mathematics, Professor, 理学部, 教授 (90253294)
MIYATA Yoshimasa Shizuoka University, Faculty of Education, Department of Mathematics, Professor, 教育学部, 教授 (50022207)
OHTA Haruto Shizuoka University, Faculty of Education, Department of Mathematics, Professor, 教育学部, 教授 (40126769)
|
|---|
| Project Period (FY) |
2000 – 2001
|
| Keywords | Topological group / Free topological group / k-space / Quotient mapping / Metric space |
|---|
| Research Abstract |
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.
|
