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:

for $x \in X$.