
Имя: Mamadaliev Nodirbek Kamoldinovich Название: SOME PROPERTIES OF TOPOLOGICAL SPACES RELATED TO THE LOCAL DENSITY
Город, страна: Tashkent, Uzbekistan
Организация: Institute of Mathematics of National University of Uzbekistan
Абстракт: 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
noncompact. 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.

