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.

Also see

 * Inverse Rule for Continuous Mappings to Topological Division Ring