Algebraic topology of Peano continua

Let X be a Peano continuum. Then the following hold: (1) The singular cohomology group H 1 ( X ) is isomorphic to the Čech cohomology group H ˇ 1 ( X ) . (2) For each homomorphism h : π 1 ( X ) → * i ∈ I G i there exists a finite subset F of I such that Im ( h ) ⊆ * i ∈ F G i . (3) For each injective homomorphism h : π 1 ( X ) → G 0 * G 1 there exists a finitely generated subgroup F 0 of G 0 or a finitely generated subgroup F 1 of G 1 such that Im ( h ) ⊆ F 0 * G 1 or Im ( h ) ⊆ G 0 * F 1 .