Saturated classes of bases


Abstract


In this paper we consider classes consisting of pairs (B,X), where B is a base of cardinality \leq\tau for the open subsets of a space X. Such classes are called classes of bases.  For such a class I\!\! P we define the notion of a universal element: an element (B^T,T) of I\!\! P is said to be universal in I\!\! P if for every (B^X,X)\in I\!\! P there exists an embedding i^X_T of X into T such that B^X=\{(i^X_T)^{-1}(U):U \in B^T\}. 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 \tau 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 (Q,B,X), where Q is a subset of a space X and B is a set of open subsets of X such that the set \{ Q\cap U:U\in B\} is a base for the open subsets of the subspace Q. 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.


DOI Code: 10.1285/i15900932v22n2p141

Keywords: Universal space; Containing space; Saturated class of spaces; Saturated class of subsets; Saturated class of bases; Saturated class of p-bases

Classification: 54C25; 54E99

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