Corollary

A topological space homotopic to a simply connected topological space is simply connected.

Let B and $B'$ be topological spaces, let $f: B\rightarrow B'$ be a homotopy equivalence and let $g: B'\rightarrow B$ be a continuous mapping, inverse to $f$ up to homotopy. Let E be a covering of B. Since $g\circ f$ is homotopic to the identity mapping of B, the covering E is isomorphic to the covering $f^*(g^*(E))$ (prop. 1). If the space $B'$ is simply connected, the covering $g^*(E)$ is trivializable. Consequently, the covering E is trivializable. This proves that the space B is simply connected.