Continuous Mapping to Product Space/Corollary

Theorem
Let $T = T_1 \times T_2$ be a topological product of two topological spaces $T_1$ and $T_2$.

Let $T'$ be a topological space.

Let $f: T' \to T_1$ be a mapping.

Let $g: T' \to T_2$ be a mapping.

Let $f \times g : T’ \to T$ be the mapping defined by:
 * $\forall x \in T’ : \map {\paren {f \times g}} x = \tuple{ \map f x, \map g x}$

Then $f \times g$ is continuous $f$ and $g$ are continuous.

Proof
Let $\operatorname{pr}_1: T \to T_1$ and $\operatorname{pr}_2: T \to T_2$ be the first and second projections from $T$ onto its factors.

From Continuous Mapping to Topological Product, $f \times g$ is continuous $\operatorname{pr}_1 \circ \paren {f \times g}$ and $\operatorname{pr}_2 \circ \paren {f \times g}$ are continuous.

Now:

From Equality of Mappings, $\operatorname{pr}_1 \circ \paren {f \times g} = f$.

Similarly, $\operatorname{pr}_2 \circ \paren {f \times g} = g$.

The result follows.