Product of Continuous Functions on Topological Ring is Continuous
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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 $x \in X$.
$\blacksquare$