Borsuk Null-Homotopy Lemma/Corollary

Corollary
Let $a,b \in \R^2$. Let $\struct {A, \tau_A}$ be a compact topological space. Let $f : A \to \R^2 \setminus \set {a,b}$ be a continuous injective mapping.

Let $f$ be null-homotopic.

Then $a$ and $b$ lie in the same component of $\R^2 \setminus \Img f$. Here, $\Img f$ denotes the image of $f$.