Free topological groups

Abstract

The free abelian topological group over aTychonoff space contains as a closed subspace a homeomorphic copy of each finite power of the space.A major and immediate corollary of this theorem is: if ~ is a closed hereditary property of Tychonoff spaces and if the free abelian topological group over a Tychonoff space has ~ then so does every finite power of the space.In particular, the corollary shows that the following properties are not preserved by passage to the free abelian topological group: normal, k-, sequential~ Frechet, Lindel~f~ paracompact~ pseudocompact, countably compact~ sequentially compact~ etc. In fact, the Main Theorem arose from studying the problem of preservation of topological properties by the free abelian topological group.The notions of free topological groups and free abelian topological groups were introduced by A. A. Markov in 1941 [ii] . He showed that every Tychonoff space has a free topological group which is Hausdorff~ and which contains as a closed subspace a homeomorphic copy of the original space. Markov's proof involves a complicated construction of multinorms on the free group over the underlying set of the space.Kakutani [8] and Samuel [16] have provided proofs of the existence of free topological groups~ and free abelian topological groups

Free topological groups

“…The first application extends Proposition 3.9 of Thomas (1974) to all free groups. The second extends his main theorem to the non-abelian free groups, with a much simpler proof which does not apply to the abelian case (an elementary proof of both the abelian and non-abelian cases appears in Hardy, Morris, Thompson (to appear)).…”

Free Boolean Topological Groups

“…. x n : x i ∈ X for i = 1 ≤ n} of F (X) [43] and to the closed subset [44]. (Arkhangel'skii announced the result for F (X) in [43] and proved it in [31] by considering the Stone-Čech compactification of X and its free topological group; details can be found in Theorem 7.1.13 of [17].…”

Free topological groups and dimension

“…The proof uses the Fundamental Lemma and is very much like the standard proof used to show the topological product of X with itself is not normal. Additional relevant information can be found in [20].…”