Can the Fundamental (Homotopy) Group of a Space be the Rationals?

Saharon Shelah
1988 Proceedings of the American Mathematical Society  
We prove that for any topological space which is metric, compact (hence separable) path connected and locally path connected, its homotopy group is not the additive group of the rational, moreover if it is not finitely generated then it has the cardinality of the continuum.
doi:10.2307/2047190 fatcat:pyc57wa5drghtctnekgwigpaia