Isometrically containing spaces
by
Stavros D. Iliadis
University of Patras, Patras, Greece

In [1] and [2] for an indexed collection S of spaces of weight less than or equal to a fixed infinite cardinal t containing spaces of weigth t for S are constructed. For each element X of S and each containing space T a natural embedding i(X, T) of X into T is defined. For the case, where S consists of separable metric spaces, in these containing spaces we can define metrics in such a manner that the corresponding natural embeddings to be isometries.

These isometrically containing spaces can be applied for the construction of isometrically universal spaces for many classes of separable metric spaces.

Definition. A class M of metric spaces is said to be uniform if the following condition are satisfied: (a) there exists an integer d such that Diam(X) is less than or equal to d for every X in M, and (b) for every e>0 there exists an integer n(e) such that every element X of M has a finite e-net the number of elements of which is less than or equal to n(e).

We study uniform classes of metric spaces from the universality point of view. In particular, we prove the following: the uniformity of a class P of compact metric spaces is a necessary and sufficient condition for the existence of an isometrically containing compact space for P.

[1] Stavros Iliadis, A construction of containing spaces, Topology and its Application 107 (2000) 97-116.

[2] S.D. Iliadis, A generalization of the construction of containing spaces, Topology Proceedings.

Date received: August 31, 2002