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 $f$.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 2000: James R. Munkres: Topology (2nd ed.): $10$: Separation Theorems in the Plane: $\S 62$: Invariance of Domain