On the existence of free topological groups

AbstractGiven a Tychonoff space X and classes U and V of topological groups, we say that a topological group G = G(X, U, V) is a free (U,V)-group over X if (a) X is a subspace of G, (b) G ϵ U, and (c) every continuous f: X → H with H ϵ V extends uniquely to a continuous homomorphism f̄: G→H. For certain classes U and V, we consider the question of the existence of free (U,V)- groups. Our principal results are the following. Let PA and CA denote, respectively, the class ofpseudocompact Abelian groups and the class of compact Abelian groups. Then 1.(a) there is a free (PA,PA)-group over X iff; X=Ø and2.(b) there is for each X a free (PA,CA)-group over X in which X is closed