Product Topology is Coarsest Topology such that Projections are Continuous

Theorem
Let $\mathbb X = \left \langle {\left({X_i, \tau_i}\right)}\right \rangle_{i \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.

Let $X$ be the cartesian product of $\mathbb X$:
 * $\displaystyle X := \prod_{i \mathop \in I} X_i$

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

For each $i \in I$, let $\operatorname {pr}_i : X \to X_i$ be the corresponding projection which maps each ordered tuple in $X$ to the corresponding element in $X_i$:


 * $\forall x \in X: \operatorname {pr}_i \left({x}\right) = x_i$

Then $\tau$ is the coarsest topology on $X$ such that all the $\operatorname {pr}_i$ are continuous.

Proof
The result follows from the definition of the Tychonoff topology and Equivalence of Definitions of Initial Topology.