Combination Theorem for Continuous Mappings/Topological Ring/Translation 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 $\lambda \in R$.
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 ring:
- $\struct {R, +, \tau_{_R} }$ is a topological group.
From Multiple Rule for Continuous Mappings to Topological Group:
- $\lambda + f: \struct {S, \tau_{_S} } \to \struct {R, \tau_{_R} }$ is a continuous mapping.
$\blacksquare$