Combination Theorem for Continuous Mappings/Normed Division Ring/Inverse Rule

Theorem

Let $\struct{S, \tau_{_S}}$ be a topological space.

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

Let $\tau_{_R}$ be the topology induced by the norm $\norm{\,\cdot\,}$.

Let $g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ be continuous mappings.

Let $U = S \setminus \set{x : \map g x = 0}$

Let $g^{-1} : U \to R$ denote the mapping defined by:

$\forall x \in U : \map {g^{-1}} x = \map g x^{-1}$

Let $\tau_{_U}$ be the subspace topology on $U$.

Then:

$g^{-1} : \struct{U, \tau_{_U}} \to \struct{R, \tau_{_R}}$ is continuous.

Proof

From Corollary to Normed Division Ring Operations are Continuous, $\struct{R, +, *, \tau_{_R}}$ is a topological division ring.

From Inverse Rule for Continuous Mappings to Topological Division Ring, $g^{-1} : \struct{U, \tau_{_U}} \to \struct{R, \tau_{_R}}$ is a continuous mapping.

$\blacksquare$