Box Topology may not form Categorical Product in the Category of Topological Spaces

Theorem

Let $\family {\struct{X_i, \tau_i}}_{i \mathop \in I}$ be an $I$-indexed family of topological spaces.

Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$, that is:

$\ds X := \prod_{i \mathop \in I} X_i$

Let $\tau$ be the box topology on $X$.

Then $\tau$ may not be the categorical product in the category of topological spaces.

Proof

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$\blacksquare$