Combination Theorem for Continuous Mappings/Topological Ring/Product 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 semigroup.

From Product Rule for Continuous Mappings to Topological Semigroup:

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

$\blacksquare$


Also see