Box Topology may not form Categorical Product in the Category of Topological Spaces
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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:
- $\displaystyle X := \prod_{i \mathop \in I} X_i$
Let $\tau$ be the box topology on $X$.
Then $\tau$ may not be the categorical product in the category of topological spaces.
Proof
$\blacksquare$