Tukey order and diversity of free Abelian topological groups

Abstract

For a Tychonoff space X the free Abelian topological group over X , denoted A ( X ) , is the free Abelian group on the set X with the coarsest topology so that for any continuous map of X into an Abelian topological group its canonical extension to a homomorphism on A ( X ) is continuous. We show there is a family A of maximal size, 2 c , consisting of separable metrizable spaces , such that if M and N are distinct members of A then A ( M ) and A ( N ) are not topologically isomorphic (moreover, A ( M ) neither embeds topologically in A ( N ) nor is an open image of A ( N ) ). We show there is a chain C = { M α : α < c + } , of maximal size, of separable metrizable spaces such that if β < α then A ( M β ) embeds as a closed subgroup of A ( M α ) but no subspace of A ( M β ) is homeomorphic to A ( M α ) . We show that the character (minimal size of a local base at 0) of A ( M ) is d (minimal size of a cofinal set in N N ) for every non-discrete, analytic M , but consistently there is a co-analytic M such that the character of A ( M ) is strictly above d . The main tool used for these results is the Tukey order on the neighborhood filter at 0 in an A ( X ) , and a connection with the family of compact subsets of an auxiliary space.