The Dieudonne τ-complete spaces and free topological groups of uniform spaces

摘要

For a Tychonoff space X, the Dieudonne tau-completion of X, denoted by mu TX, is investigated. The space mu TX is defined as the completion of X with respect to the uniformity uTX, where uTX is generated by all continuous mappings of X to metric spaces of weight = tau. It is proved that a dense subspace Y of X is PT -(equivalently, PT-)embedded in X iff uTX |y = uY iff Y subset of X subset of mu TX. In this case, the free topological group F(uYY) of the uniform space uYY is the topological subgroup of the group F(uTXX) generated by Y iff Y is PT-embedded in X. This result implies the Nummela-Pestov's Theorem: A dense subspace Y is P-embedded in a Tychonoff space X iff the free topological group F(Y) is topologically isomorphic to the subgroup of the group F(X) generated by Y. It is proved that the Weil completion of the free topological group coincides with its Rakov completion. It is also shown that the free topological groups in the sense of Nakayama and Nummela coincide. (C) 2020 Elsevier B.V. All rights reserved.