In Hatcher's algebraic topology, I read the following:
Let $h: D^k \to S^n$ be an embedding. Then $\tilde{H}_i(S^n \setminus
h(D^k)) = 0$ for all $i$.
Here an embedding is a map that is a homeomorphism onto its image.
Would it be correct to replace the word "embedding" by continuous injection?
Because if $h$ is a continuous injection, then $h$ is a continuous bijection onto its image and since $D^k$ is compact and a subspace of the sphere is Hausdorff we get that $h$ is a homeomorphism, so $h$ is an embedding?
Best Answer
In this situation, yes you are correct -- although (I'm pointing it out but you probably already know it) in general the two notions aren't equivalent.