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

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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.

$\blacksquare$


Also see