Some cardinal properties of stratifiable spaces mamadaliev N. K., Nurmatova M. Ya

SOME CARDINAL PROPERTIES OF STRATIFIABLE SPACES Mamadaliev N.K., Nurmatova M.Ya.
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In the paper we study cardinal properties of stratifiable spaces and the space of linked systems. It is proved that the Lindelof number, the density, the weakly density, the Souslin number, the network weight and the -network weight of stratifiable spaces are equal. As well as we study the -weight, the character, the caliber and the pre-caliber of the space of maximal linked systems with compact elements.

A collection of nonempty subsets of a topological space is said to be a network of the space if for each point and for each neighborhood of there exists such that . If a network consists of open sets then is called a base of the space [1].
A collection of nonempty subsets of a topological space is said to be a network of the space if for each open set of there exists such that . If a network consists of open sets then is called a base of the space .
A set is called dense in if . The density of a space is defined as the smallest cardinal number of the form , where is a dense subset of ; this cardinal number is denoted by . If , then we say that the space is separable.
The smallest cardinal number such that every family of pairwise disjoint nonempty open subsets of has the cardinality , is called the Souslin number of a space and is denoted by . If , we say that the space has the Souslin property.
The Lindelof number is defined as following way:
for any open cover there exists a subcover such that . A topological space is said to be a Lindelof space if every open cover of has a countable subcover [1].
We say that the weakly density of a topological space is equal to if is the smallest cardinal number such that there exists a base in coinciding with centered systems of open sets, i.e. there exists a base , where - centered system of open sets for every , .
The weakly density of a topological space is denoted by . If , then the topological space is called weakly separable [2].