Resumen de Free Topological Groups over om-Metrizable Spaces

Manuel Sanchís López , M. Sakai, Salvador García Ferreira

• Let 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,UX) is an -metrizable uniform space, where is the universal uniformity on X.

  2. A(X) is topologically orderable and ?(A(X)) =?µ .

  3. The derived set is ?µ-compact and X is ?µ-metrizable.

  Theorem. Let X be a non-discrete space. Then, the following are equivalent.

  1. X is ?µ-compact and ?µ-metrizable.

  2. (X,UX) is ?µ-metrizable and X is ?µ-compact.

  3. F(X) is topologically orderable and ?(F(X)) =?µ .

  We also prove that an ?µ-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).