Projection from Product Topology is Continuous/General Result

Theorem
Let $\family {T_i}_{i \mathop \in I} = \family {\struct{S_i, \tau_i}}_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.

Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be the corresponding product space.

Let $\tau$ denote the Tychonoff topology on $S$.

Let $\pr_i: S \to S_i$ be the corresponding projection from $S$ onto $S_i$.

Then $\pr_i$ is continuous for all $i \in I$.