Imbeddings into topological groups preserving dimensions

Abstract

We give a negative answer to the following question of Bel'nov: Can every Tychonoff space X be imbedded as a subspace of a topological group G so that dim G ⩽ dim X? We show that if n ≠ 0, 1, 3, 7, then the n-dimensional sphere S n cannot be imbedded into an n-dimensional topological group G (no matter which dimension function, ind, Ind or dim, is considered). However, in case dim X = 0 the answer to Bel'nov's question is “yes”. We prove that, for every Tychonoff space X, dim X =0 implies (in fact, equivalent to) dim F ∗(X) = 0 and dim A ∗(X) = 0 , where F ∗(X) (A ∗(X)) is the free precompact (Abelian) group of X. As a corollary we obtain that every precompact group G is a quotient group of a precompact group H such that dim H = 0 and w( H) = w( G). A complete metric space X 1 and a pseudocompact Tychonoff space X 2 are constructed such that ind X i = 0, while ind F ∗(X i) ≠ 0 and ind A ∗(X i) ≠ 0 (i = 1, 2) . The equivalence of ind G = 0 and dim G = 0 for a precompact group G is established. We prove that dim H ⩽ dim G whenever H is a precompact subgroup of a topological group G. We also show that for every Tychonoff topology T on a set X with ind(X, T) = 0 one can find a precompact Hausdorff group topology T ̃ on the free (Abelian) group G( X) of X such that w(G(X), T ̃ ) = w(X, T), T ̃ | x = T and dim(G(X), T ̃ ) = 0 .