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.

$\blacksquare$


Also see