# Combination Theorem for Continuous Mappings/Topological Division Ring/Translation Rule

## Theorem

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

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

Let $\lambda \in R$.

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

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

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

Then

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

## Proof

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

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

$\blacksquare$