Algebraic topology of Peano continua

Let X be a Peano continuum. Then the following hold: (1) The singular cohomology group H1(X) is isomorphic to the Čech cohomology group Hˇ1(X). (2) For each homomorphism h:π1(X)→*i∈IGi there exists a finite subset F of I such that Im(h)⊆*i∈FGi. (3) For each injective homomorphism h:π1(X)→G0*G1 there exists a finitely generated subgroup F0 of G0 or a finitely generated subgroup F1 of G1 such that Im(h)⊆F0*G1 or Im(h)⊆G0*F1.