Contributions to the theory of boundedness in uniform spaces and topological groups

Abstract

First, we discuss the behavior of boundedness in uniform spaces with respect to subspaces, projective limits, and suprema in relation to precompactness. A special uniformly isomorphic embedding of un arbitrary uniform space in a bounded uniform space is presented and examined in 2.6. Hejcman’s characterization (by B-conservativity) of uniform spaces in which boundedness can be tested by a single pseudometric is proved in a new way, see 3.13, using a version 3.1 of the metrization lemma. We comment briefly on boundedness in topological vector spaces. In topological groups we investigate a hierarchy of partly new notions of boundedness, strongly interrelated among themselves, and exhibit various situations in which certain of these notions coincide. "Boundedness respecting subspaces" of a uniform space prove useful. Many examples illustrate and complement the general theory, see, e.g., Example 6.4.