Sphere is not homeomorphic with its proper subset

Question

Can we use Borsuk-Ulam Theorem to show that $S^{n-1}\subset \mathbb{R}^n$ is not homeomorphic with proper subset $A \subset S^{n-1}$ ?

Answer 1

Yes, we can:

Suppose $h: S^{n-1} \rightarrow S^{n-1}\setminus A, A \neq \emptyset$, is a homeomorphism between the sphere and its proper subset.

By first rotating $A$ to contain the north pole, followed by the standard stereographic projection shows that there is an embedding $e:S^{n-1}\setminus A \rightarrow \mathbb{R}^{n-1}$.

Apply Borsuk-Ulam to $e \circ h: S^{n-1} \rightarrow \mathbb{R}^{n-1}$, which is continuous as the composition of continuous functions.

We immediately get a contradiction to the fact that $e \circ h$ is 1-1 (as the composition of 1-1 maps).