|
ИСТИНА |
Войти в систему Регистрация |
ФНКЦ РР |
||
Generalizing dyadicity and Esenin-Vol'pin's Theoremстатья
- Автор: Arhangel'skii A.V.
- Журнал: Topology and its Applications
- Год издания: 2020
- Издательство: Elsevier BV
- Местоположение издательства: Netherlands
- DOI: 10.1016/j.topol.2020.107234
- Аннотация: For a pair (G, X), where G is a topological group and X is a subspace of G, we consider continuous mappings f of G onto a topological space Y such that f(G)=Y=f(X). If, in addition, X is compact (σ-compact), then Y is called a gc-image of G (respectively, a gσc-image of G). For example, all dyadic compacta are gc-images of compact topological groups. Below Esenin-Vol'pin's Theorem on metrizability of first-countable dyadic compacta is extended, in several ways, to gc-images and gσc-images, and to certain generalizations of them. Essential new features of these generalizations are: the compactness assumption and the topological group assumption are completely separated, compactness is weakened to σ-compactness, and first-countability is replaced by the G_\delta-points requirement. In particular, in Section 2 the following theorem is proved. Suppose that G is a topological group, X is a Lindelöf Σ subspace of G, and f is a continuous mapping of G onto a space Y such that f(G)=f(X)=Y. Suppose also that Z is a closed subset of Y such that every point of Z is a G_\delta-point in Y. Then Z has a countable network. Hence, if in addition Z is compact, or a p-space, then Z is separable and metrizable. The last statement is generalizied to the case of paratopological groups.
- Добавил в систему: Архангельский Александр Владимирович