Prime subspaces in free topological groups (Q1346486)

Let \(F(X)\) \((A(X))\) be the free (Abelian) topological group over \(X\). The authors prove: If \(P\) is one of the spaces \(\mathbb{R}\), \(\mathbb{Q}\), \(\mathbb{R} \setminus \mathbb{Q}\), \(\beta \omega\), \(\beta \omega \setminus \omega\) and \(2^ \kappa\) for an infinite \(\kappa\) and if \(F(X)\) or \(A(X)\) contains a copy of \(P\), then \(X\) contains a copy of \(P\). If \(P\) is the one-point compactification of an infinite discrete space or \(\omega_ 1 + 1\), this is not true. If \(P = \omega_ 1\), this holds for \(F(X)\) but is independent of ZFC for \(A(X)\).