Borsuk Null-Homotopy Lemma

Theorem
Let $\Bbb S^2$ denote the unit sphere in $\R^3$.

Let $a,b \in \Bbb S^2$.

Let $\struct {A, \tau_A}$ be a compact topological space.

Let $f : A \to \Bbb S^2 \setminus \set {a,b}$ be a continuous injective mapping.

Let $f$ be null-homotopic.

Then $a$ and $b$ lie in the same component of $\Bbb S^2 \setminus \Img f$.

Here, $\Img f$ denotes the image of $f$.

Also known as
This lemma is usually referred to as the Borsuk Lemma.

The name Borsuk Null-Homotopy Lemma is used by to distinguish it from other "Borsuk Lemmas" in litterature.