Combination Theorem for Continuous Mappings/Topological Division Ring/Negation 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 $-g : S \to R$ be the mapping defined by:

$\forall x \in S: \map {\paren {-g} } x = -\map g x$


Then:

$-g: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is continuous.


Proof

By definition of a topological division ring:

$\struct {R, +, *, \tau_{_R} }$ is a topological ring.

From Negation Rule for Continuous Mappings to Topological Ring:

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

$\blacksquare$


Also see