Take the 2-minute tour ×  Suppose that we have
$\{X_{\alpha}\}_{\alpha \in J}$, an indexed family of
topological spaces. Let $X := \prod_{\alpha \in J}X_{\alpha}$.
When we have a map $f_{\alpha} : A \rightarrow X_{\alpha}$ with
a topological space $A$. Define $f : A \rightarrow \prod_{\alpha
\in J}X_{\alpha}$ by $a \mapsto (f_{\alpha}(a))_{\alpha \in J}$.
Property. We know that $f$ is continuous if and only if each
$f_{\alpha}$ is continuous, once we are given product topology
on $X$. We further know that this property does not generally
hold with box topology on $X$.  After reviewing the definition,
it seems to me that this property seems rather trivial (or
natural) since product topology only collects finite
intersections of inverse images under projections as open sets.
Can someone give more examples outside this property that
explains why we prefer product topology? A good list of evident
examples will be much preferred than statement of theorems.
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