Corollary 2

Let X be a normal topological space, let A be a closed subspace of X and let U be a neighbourhood of A in X. Let $f$ and $f'$ be continuous homotopic mappings of U into a topological space Z. If $f$ possesses a continuous extension to X, the same is true of the mapping $f'|A$.

Let F be a continuous mapping of X into Z such that $F|U =f$. Let us choose a homotopy $\sigma : A\times \mathbf{I}\rightarrow Z$ joining $f'$ to $f$. By Cor. 1, there exists a homotopy $\widetilde{\sigma}: X\times \mathbf{I}\rightarrow Z$ with term F which extends $\sigma$. The origin of $\widetilde{\sigma}$ is then a continuous mapping of X into Z which coincides with $f'$ on A.