Combination Theorem for Continuous Mappings/Topological 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 ring.


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


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

From Inverse Rule for Continuous Mappings to Topological Group, $- g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ is a continuous mapping.

$\blacksquare$

Also see