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

From ProofWiki
Jump to navigation Jump to search

Theorem

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

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

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

$\blacksquare$

Also see