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

Definition 1[3] We say that -topological space is stratifiable provided that to each open set one can assign a sequence of open subsets of such that
a) ,
b) ,
c) whenever .
The construction in the definition is called the stratification.
It is obvious that every metrizable space is stratifiable.
J.Ceder [3] proved the following statement.
Theorem 1[3]. For a stratifiable space following conditions are equivalent:
1) is finally compact;
2) is separable;
3) has the Souslin property;
4) has countable network.
Proposition 1[5]. For any -space we have

In the work R.A.Stoltenberg proved the following theorem:
Theorem 2[6]. Let be an infinite cardinal. Then for any stratifiable space following conditions are equivalent:
1) The space has dense subset of cardinality ;
2) Every open cover of the space has a subcover of cardinality
3) Every family of pairwise disjoint non-empty open subsets of the space has cardinality ;
4) The space has a network of cardinality .
Beshimov and Mamadaliyev generalize that theorem 2 as following:
Theorem 3. Let be an infinite cardinal. Then for any stratifiable space following conditions are equivalent:
1) The space has dense subset of cardinality ;
2) The weakly density of the space is ;
3) Every open cover of the space has a subcover of cardinality
4) Every family of pairwise disjoint non-empty open subsets of the space has cardinality ;
5) The space has a network of cardinality ;
6) The space has a -network of cardinality .
Proof. The implication 5) 6) is obvious. We show the implication 1) 6). Let be a dense subset of cardinality in . We must prove that the family is a -network of the space . Let be any non-empty open subset of . There exists a point such that from density of the subset Then . Hence, the family is a -network of the space .
Now we show the implication 6) 1). Let be a -network of the space . We choose a point from each set . It is obvious that the set has cardinality . We show that is dense in . Let be an arbitrary non-empty open subset of . Since is a -network of there exists such that . Thence, we have , so that is dense in . Theorem 1.5 is proved.