Corollary 4

Let X be a paracompact topological space, let A be a closed subspace of X. Let $n$ be an integer $\geqslant 0$ and let V be an open subset of $\mathbf{R}^n$. Let $f: X\rightarrow V$ be a continuous mapping and let $\sigma : A\times \mathbf{I}\rightarrow V$ be a homotopy with terminal map $f|A$. There exists a homotopy $\widetilde{\sigma}: X\times \mathbf{I}\rightarrow V$ with terminal map $f$ which extends $\sigma$.

Let us denote by $i_A$ the canonical injection of A into X, by $\alpha_A: A\times \mathbf{I}\rightarrow$ Cyl($i_A$) and $\beta_A: X\rightarrow$ Cyl($i_A$) the canonical mappings, and also by $j_A:$ Cyl($i_A$)$\rightarrow$ $X\times \mathbf{I}$ the canonical injection. Since A is closed in X, $j_A$ defines a homeomorphism of Cyl($i_A$) onto the closed subspace $(A\times \mathbf{I})\cup (X\times \{1\})$ of $X\times \mathbf{I}$. Let $\widetilde{\sigma}_0$ be the unique mapping of Cyl($i_A$) into V such that $\widetilde{\sigma}_0\circ \alpha_A=\sigma$ and $\widetilde{\sigma}_0\circ \beta_A=f$; it is continuous (III, p. 238, prop. 4).

Since X is paracompact and $\mathbf{I}$ is compact, the space $X\times \mathbf{I}$ is paracompact (TG, I, p. 70, prop. 17), hence normal (TG, IX, p. 49, prop. 4). There therefore exists a continuous mapping $k: X\times \mathbf{I}\rightarrow \mathbf{R}^n$ such that $k\circ j_A=\widetilde{\sigma}_0$ (TG, IX, p. 45, corollary).

The set U of points $x\in$ X such that $k(x, t)\in$ V for all $t\in \mathbf{I}$ is open in X (IV, p. 439, lemma 1). Hence it is a neighbourhood of A. By corollary 1, applied to the mapping $f$ and to the homotopy $k|U\times \mathbf{I}$, there exists a homotopy $\widetilde{\sigma}: X\times \mathbf{I}\rightarrow V$ with initial map $f$ which coincides with $k$, hence with $\sigma$, on $A\times \mathbf{I}$.