### Saturated classes of bases

#### Abstract

In this paper we consider classes consisting of pairs , where is a base of cardinality for the open subsets of a space . Such classes are called classes of bases. For such a class we define the notion of a universal element: an element of is said to be universal in if for every there exists an embedding of into such that . We define also the notion of a (weakly) saturated class of bases similar to that of a saturated class of spaces in [2] and a saturated class of subsets in [3]. For the (weakly) saturated classes of bases we prove the universality property (that is, in any such class there exist universal elements) and the intersection property (that is, the intersection of not more than many saturated classes of bases is also saturated). We give some relations between these classes and the classes of spaces and classes of subsets. Furthermore, we give a method of construction of saturated classes of bases by saturated classes of subsets.

Also, we consider classes consisting of triads , where is a subset of a space and is a set of open subsets of such that the set is a base for the open subsets of the subspace . Such classes are called classes of p-bases (positional bases). For such classes we also define the notion of a universal element and the notion of a saturated class of p-bases and prove the universality and the intersection properties. Some examples are given.

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