Normed Division Ring Operations are Continuous/Corollary

Theorem
Let $\struct {R, +, \times, \norm{\,\cdot\,}}$ be a normed division ring.

Let $d$ be the metric induced by the norm $\norm{\,\cdot\,}$. Let $\tau$ be the topology induced by the metric $d$.

Then:
 * $\struct {R, \tau}$ is a topological division ring.

Proof
Let $d_\infty$ be the Chebyshev distance metric on $R \times R$.

Let $\tau^\times$ be the product topology on $R \times R$.

By $p$-Product Metric Induces Product Topology then $\tau^\times$ is the topology induced by the metric $d_\infty$.

Let $R^* = R \setminus \set{0}$

Let $d^*$ be the restriction of $d$ to $R^*$.

Let $\tau^*$ be the subspace topology on $R^*$.

By Metric Subspace Induces Subspace Topology then $\tau^*$ is the topology induced by the metric $d^*$

By Normed Division Ring Operations are Continuous and Continuous Mapping is Continuous on Induced Topological Spaces then the mappings:
 * $\phi : \struct {R \times R, \tau^\times} \to \struct{R, \tau} : \phi \tuple {x,y} = x + y$
 * $\theta : \struct {R ,\tau} \to \struct{R,\tau} : \theta \tuple {x} = -x$
 * $\psi : \struct {R \times R, \tau^\times} \to \struct{R, \tau} : \psi \tuple {x,y} = xy$
 * $\xi : \struct {R^* ,\tau^*} \to \struct{R,\tau} : \xi \tuple {x} = x^{-1}$

are continuous.

By the definition of a topological division ring then the result follows.