Abstract
Let X X be a compact group, F G X FGX the (Graev) free topological group generated by X X , and K K the kernel of the canonical quotient morphism from F G X FGX to X X . Then K K is a (Graev) free topological group. A corollary to the abelian analogue of this theorem is that the projective dimension of a compact abelian group, relative to the class of all continuous epimorphisms admitting sections, is exactly one.
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