Type: Article

Free topological groups over omega_mu-metrizable spaces

S. García-Ferreira; M. Sakai; M. Sanchis;

Abstract:

Let omega(mu) be an uncountable regular cardinal. For a Tychonoff space X, we let A(X) and F(X) be the free Abelian topological group and the free topological group over X, respectively. In this paper, we establish the next equivalences. Theorem. Let X be a space. The following are equivalent. 1. (X, U-X) is an omega(mu)-metrizable uniform space, where U-X is the universal uniformity on X. 2. A(X) is topologically orderable and chi(A(X)) = omega(mu). 3. The derived set X-d is omega(mu)-compact and X is omega(mu)-metrizable. Theorem. Let X be a non-discrete space. Then, the following are equivalent. 1. X is omega(mu)-compact and omega(mu)-metrizable. 2. (X, U-X) is omega(mu)-metrizable and X is omega(mu)-compact. 3. F(X) is topologically orderable and chi(F(X)) = omega(mu). We also prove that an omega(mu)-metrizable uniform space (X, U) is a retract of its uniform free Abelian group A( X, U) and of its uniform free group F( X, U).