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 $f$.

Corollary

Let $a,b \in \R^2$.

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 $\R^2 \setminus \Img f$.

Proof

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Also known as

This lemma is usually referred to as the Borsuk Lemma.

The name Borsuk Null-Homotopy Lemma is used by $\mathsf{Pr} \infty \mathsf{fWiki}$ to distinguish it from other "Borsuk Lemmas" in litterature.

Source of Name

This entry was named for Karol Borsuk.

Sources

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