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

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Theorem

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

Let $\struct{R, +, *, \tau_{_R}}$ be a topological division ring.


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


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

Let $g^{-1} : U \to R$ be 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

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

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

By definition of a topological division ring:

$\phi: \struct{R^*, \tau^*} \to \struct{R, \tau_{_R}}$ such that $\forall x \in R^*: \map \phi x = x^{-1}$ is a continuous mapping


Let $g^* : \struct{U, \tau_{_U}} \to \struct{R^*, \tau^*}$ be the restriction of $g$ to $U \subseteq R$.

From Restriction of Continuous Mapping is Continuous, $g^*$ is a continuous mapping.


From Composite of Continuous Mappings is Continuous, the composition $\phi \circ g^* : \struct{U, \tau_{_U}} \to \struct{R, \tau_{_R}}$ is continuous.


Now

\(\, \displaystyle \forall x \in U : \, \) \(\displaystyle \map {\paren {g^{-1} } } x\) \(=\) \(\displaystyle \map g x^{-1}\) Definition of $g^{-1}$
\(\displaystyle \) \(=\) \(\displaystyle \map \phi {\map g x}\) Definition of $\phi$
\(\displaystyle \) \(=\) \(\displaystyle \map \phi {\map {g^*} x}\) Since $\map g x \neq 0$
\(\displaystyle \) \(=\) \(\displaystyle \map {\paren {\phi \circ g^*} } x\) Definition of composition of mappings

From Equality of Mappings, $g^{-1} = \phi \circ g^*$.

The result follows.

$\blacksquare$