Product of Continuous Functions on Topological Ring is Continuous

Theorem

Let $X$ be a topological space.

Let $R$ be a topological ring.

Let $f, g : X \to R$ be continuous functions.

Then $f \cdot g : X \to R$ is continuous.

Proof

Equip the Cartesian product $R \times R$ with its product topology.

Define $h : X \to R \times R$ by:

$\map h x = \tuple {\map f x, \map g x}$

for each $x \in X$ and $p : R \times R \to R$ by:

$\map p {x, y} = x y$

for each $\tuple {x, y} \in R \times R$.

From Continuous Mapping to Product Space, $h$ is continuous.

From the definition of a topological ring, $p$ is continuous.

Then from Composite of Continuous Mappings is Continuous, $p \circ h : X \to R$ is continuous, while:

\(\ds \map {\paren {p \circ h} } x\)

\(=\)

\(\ds \map p {\map f x, \map g x}\)

\(\ds \)

\(=\)

\(\ds \map f x \map g x\)

for $x \in X$.

$\blacksquare$