Combination Theorem for Continuous Mappings/Topological Ring/Sum 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 $f,g : \struct{S, \tau_{_S}} \to \struct{R, \tau_{_R}}$ be continuous mappings.


Let $f + g : S \to R$ be the mapping defined by:

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


Then

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

Proof

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

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

$\blacksquare$

Also see