Euclidean spheres and closed balls as subspaces of $\mathbb{R}^n$

Let $n \in \mathbb{N}$ with $n \ge 1$. Give $\mathbb{R}^n$ its Euclidean norm $\lVert\cdot\rVert_2$ and its induced Euclidean metric $d_2$ ($\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it, A norm on a real vector space, the induced metric, and the dictionary with the metric axioms). For $c \in \mathbb{R}^n$ and $r>0$, put

$\overline B_2(c,r):=\{x\in\mathbb{R}^n:\lVert x-c\rVert_2\le r\},\qquad S_2(c,r):=\{x\in\mathbb{R}^n:\lVert x-c\rVert_2=r\}.$

These are respectively the Euclidean closed ball and Euclidean sphere with centre $c$ and radius $r$. They carry the subspace topology inherited from $\mathbb{R}^n$ (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace). Since $d_2(c,x)=\lVert x-c\rVert_2$, they are precisely the closed ball and sphere $\bar B(c,r)$ and $S(c,r)$ of the metric-space definition (Open ball, closed ball and sphere in a metric space).

For the unit sphere centred at the origin write

$S^{n-1}:=S_2(0,1).$

The exponent is notation for this particular sphere, not a claim that a dimension theory has been developed here.