A Schreier theorem for free topological groups

Peter Nickolas
1975 Bulletin of the Australian Mathematical Society  
M.I.Graevhas shown that subgroups of free topological groups need not be free. Brown and Hardy, however, have proved that any open subgroup of the free topological group on a k -space is again a free topological group: indeed, this is true for any closed subgroup for which a Schreier transversal can be chosen continuously. This note provides a proof of this result more direct than that of Brown and Hardy. An example is also given to show that the condition stated in the theorem is not a
more » ... em is not a necessary condition for freeness of a subgroup. Finally, a sharpened version of a particular case of the theorem is obtained, and is applied to the preceding example. The well-known Nielsen-Schreier theorem [4] states that every subgroup of a free group is free. Examples have been given, however, to show that the analogous statement for subgroups of a free topological group is false ([3] and [ / ] ) . Nevertheless, Brown and Hardy have proved in [2] (see also [5] ) that a closed subgroup of the free topological group on a k -space will again be free, provided that a Schreier transversal for the subgroup can be chosen continuously. (Two immediate corollaries deserve mention. Firstly, any open subgroup satisfies the condition and is therefore free.
doi:10.1017/s0004972700024308 fatcat:i7d34ccyd5gtff32vfsbj6gdwy