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 $\parallel \cdot {\parallel }_2$ and its induced Euclidean metric $d_2$ (${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ 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

${\bar{B}}_2(c,r):=\{x\in {\mathbb{R}}^n:\parallel x-c{\parallel }_2\le r\},S_2(c,r):=\{x\in {\mathbb{R}}^n:\parallel x-c{\parallel }_2=r\}\mathrm{.}$

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 $d_2(c,x)=\parallel x-c{\parallel }_2$, 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

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

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

${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it

A norm on a real vector space, the induced metric, and the dictionary with the metric axioms

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

Open ball, closed ball and sphere in a metric space