Free topological groups

Abstract Main Theorem. The free abelian topological group over a Tychonoff1 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 P is a closed hereditary property of Tychonoff spaces, and if the free abelian topological group over a Tychonoff spaces has P , 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 group: normal, k- sequential, Frechet, Lindelof, paracompact, pseudocompact, countably compact, sequentially compact, etc.