Normed Division Ring Operations are Continuous

Theorem

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

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

Let $p \in \R_{\ge 1} \cup \set \infty$.

Let $d_p$ be the $p$-product metric on $R \times R$.

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

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

Then the following results hold:

Addition is Continuous

The mapping:

$+ : \struct {R \times R, d_p} \to \struct{R,d}$

is continuous.

Negation is Continuous

The mapping:

$\eta: \struct {R, d} \to \struct {R, d}: \map \eta x = -x$

is continuous.

Multiplication is Continuous

The mapping:

$* : \struct {R \times R, d_p} \to \struct {R, d}$

is continuous.

Inversion is Continuous

The mapping:

$\iota : \struct {R^* ,d^*} \to \struct {R, d} : \map \iota x = x^{-1}$

is continuous.

Corollary

Let $\tau$ be the topology induced by the metric $d$.

Then:

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

.

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.3$: Topology: Problem $43$