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- W1991624127 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 Sn 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 X1 and a pseudocompact Tychonoff space X2 are constructed such that ind Xi = 0, while ind F∗(Xi) ≠ 0 and ind A∗(Xi) ≠ 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 T̃ on the free (Abelian) group G(X) of X such that w(G(X), T̃) = w(X, T), T̃ |x = T and dim(G(X), T̃) = 0." @default.
- W1991624127 created "2016-06-24" @default.
- W1991624127 creator A5048615463 @default.
- W1991624127 date "1990-08-01" @default.
- W1991624127 modified "2023-09-23" @default.
- W1991624127 title "Imbeddings into topological groups preserving dimensions" @default.
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- W1991624127 doi "https://doi.org/10.1016/0166-8641(90)90008-p" @default.
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