Proposition

Let $B \subset \mathbb{R}^n$ be any convex set, $X$ be any topological space, and $A$ be any subset of $X$. Show that any two continuous maps $f, g: X \rightarrow B$ that agree on $A$ are homotopic relative to $A$.

Solution

We claim that the straight-line homotopy $F(x, t) = f(x) + t(g(x) - f(x))$ is actually a homotopy relative to $A$.

• It makes sense to consider the straight-line homotopy since $B$ is a convex set.
• $\forall x \in A, \forall t \in I, F(x, t) = f(x) + t(g(x) - f(x)) = f(x) + 0 = f(x)$ since $f, g$ agree on $A$. Therefore, $f\mid_A = g\mid_A$.

Thus $f$ and $g$ are homotopic relative to $A$.