Box Topology may not be Coarsest Topology such that Projections are Continuous

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 $X$.

For each $i \in I$, let $\pr_i: X \to X_i$ denote the $i$th projection on $X$:
 * $\forall \family {x_j}_{j \mathop \in I} \in X: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$

Then $\tau$ may not be the coarsest topology on $X$ for which the projections $\family{\pr_i} _{i \mathop \in I}$ are continuous.