Александровские чтения-2016 | 22-26 мая 2016
Участник: Mamadaliev Nodirbek Kamoldinovich
Имя: Mamadaliev Nodirbek Kamoldinovich
Город, страна: Tashkent, Uzbekistan
Организация: Institute of Mathematics of National University of Uzbekistan
The local density of topological spaces is investigated. It is proved that for stratifiable spaces the local density and the local weak density coincide, these cardinal numbers are preserved under open mappings, are inverse invariant of a class of closed irreducible mappings. Moreover, it is showed that the functor of probability measures of finite supports preserves the local density of compacts. The local properties play an important role in general topology. For instance, compactness in $R^{n}$ is equivalent to total boundedness and closedness. In the case of local compactness boundedness is not necessary. In a case of locality some properties can be lost or some new properties may appear. For example, an open subspace of a compact space may often be non-compact. But any open subspace of a locally compact space is locally compact. In the paper we consider the local density and the local weak density of topological spaces. We investigate what properties are preserved in a local cases or what properties may appear. It is known that the density is preserved under continuous onto mappings, is hereditary with respect to $F_{\sigma}$-sets and closed domains. The continuum product of separable spaces is separable, any compact extension of a weakly separable space is weakly separable. It turns out that the local density is not preserved under a continuous mapping, a compact extension of a locally separable space can be not locally separable, the product of infinitely many locally separable spaces is not always locally separable.

© 2016 Механико-математический факультет МГУ им. М.В. Ломоносова