# 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 $f$.