Sphere as a subspace of $\mathbb R^n$

Question

I was reading John Lee book and it says:

Example 1.2 (Spheres) Let $\mathbb S^n$ denote the (unit) n-sphere, which is the set of unit-length vectors in $\mathbb{R}^{n+1}$:

$\mathbb{S}^n = \{x \in \mathbb{R}^{n+1}: |x|=1\}.$

It is Hausdorff and second countable because it is a subspace of $\mathbb{R}^n$.

Why can we say that $\mathbb S^n$ is a subspace of $\mathbb R^n$? It should be of $\mathbb R^{n+1}$ because the way it is defined we have the elements in $\mathbb R^{n+1}$.

Answer 1

This is a typo. It's listed (along with all the other mistakes I know about) on my correction list, which you probably should download and keep handy as you read. (Note that the version you quoted is the first edition, so be sure to get the correction list for that edition.)

Answer 2

Indeed, if you read the rest of the question, it is clear that $S^n$ is a subset of $\Bbb R ^{n+1}$. What you have correctly noticed is just a typing mistake, doin't let it stop you from your studying.