TITLE: Good Pairs in Algebraic Topology
QUESTION [8 upvotes]: Hatcher’s book says that $ \left( \mathbb{D}^{n},\mathbb{S}^{n - 1} \right) $ is a good pair; that is, there exists an open neighborhood $ V $ of $ \mathbb{D}^{n} $ containing $ \mathbb{S}^{n - 1} $ that deformation retracts onto $ \mathbb{S}^{n - 1} $.
What is the open neighborhood $ V $ in this case? Is it an annulus (without its boundary, of course)?
Thanks!

REPLY [12 votes]: Take $V=\{x\in D^n: \|x\|\not= 0\}$. Then $V$ is an open neighborhood of $S^{n-1}$ in $D^n$ that deformation retracts into $S^{n-1}$ (can you see why?).