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Coset spaces, algebraically structured mappings, and problems on homogeneity, 1статья
- Автор: Arhangel'skii A.V.
- Журнал: Questions and Answers in General Topology
- Том: 35
- Номер: 2
- Год издания: 2017
- Первая страница: 89
- Последняя страница: 101
- Аннотация: Suppose that G is a topological group and H is a closed subgroup of G. G/H stands for the quotient space of G which consists of left cosets xH, where x ∈ G. We call the space G/H a coset space. It is well known that a coset space is always homogeneous and Tikhonov. In this paper, the author introduces P-algebraically structured maps, a(a1)-maps and c1-maps to investigate coset spaces. The author obtains some interesting results: (1) Suppose that X is a Lindel¨of Σ-space. Then the following are equivalent: (a) there exists an a-mapping of X onto a space with a countable network; (b) there exists a topological group G which contains X as a closed Gδ-subspace; (c) there exists a topological group G which contains X as a closed subspace and satisfies the following condition: M = {x ∈ X : there exists a Gδ-subset P of G such that x ∈ P ⊂ X} is dense in X. (2) Suppose that a nonempty Lindel¨of Σ-subspace Y of a space X is a Gδ-subset of the free topological group F(X) of X. Then X and F(X) are submetrizable, and Y has a countable network. Many questions on coset spaces are posed in the paper. These questions will give some directions for the future study of coset spaces. There are also many old unsolved questions on homogeneous spaces that are asked again in the paper.
- Добавил в систему: Архангельский Александр Владимирович