Universal metric spaces

The thesis covers the properties of isometric embeddings of metric spaces into the Urysohn universal space U (P.S. Urysohn, 1927) and its generalizations (M. Katětov, 1988). The examination of various metric properties of the space U leads to the question of extendability of the embedding ϕ: M → U from a subspace M of a space P onto an embedding Φ: P → U. We approach to this question in situation P = M ∪ {p} in finer form. If ϕ denotes an embedding M → U, let Rϕ denotes the set of images of the point p in U under all possible isometric extensions of the embedding ϕ (we call Rϕ the space of realizations). The main objective of this thesis is answering the following question: Which forms do the spaces Rϕ assume, if ϕ passes all embeddings of the space M into the space U? Corollary 1 and theorem 3 in the II. part of the thesis metrically characterize the family {Rϕ|ϕ: M → U}. We use previous results in part III in order to determine the number of classes of metrically equivalent embeddings of the space M into the space U. As a consequence, we obtain the result of J. Melleray about the homogeneity of the space U