Dimension of a manifold which is a non-empty, finite subset of Euclidean space

Question

$\mathbb{R}^n$ is an $n$-manifold of course. In my notes on smooth manifolds, it says that we also allow $n=0$, so a point is a $0$-manifold. This is clear. I am confused by the next part where it says "in fact, any non-empty finite subset of any $\mathbb{R}^n$ is a $0$-manifold". Is this a typo, and meant to say just "manifold"? Surely a $2$-manifold, such as $S^2$, embedded in $\mathbb{R}^3$ is a finite, non-empty subset which is not a $0$-manifold. Is this correct, or am I missing something fundamental?

Answer 1

You are confusing the word finite with the word compact. Finite means a finite amount of elements in the set for example $ \{a_1,...,a_n\} $. $S^2$ is an infinite set since there are infinitely many points on $S^2$ but it is compact in the sense that it is closed and bounded (when thought of as a subset of $\mathbb{R}^3$). Hope this helps.