On the Existence of Free Topological Groups
نویسندگان
چکیده
Given 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) GE U, and (c) every continuous f:X+ H with H EV extends uniquely to a continuous homomorphism _?: 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 of pseudocompact Abelian groups and the class of compact Abelian groups. Then (a) there is a free (PA, PA)-group over X iff X =0; and (b) there is for each X a free (PA, CA)-group over X in which X is closed.
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