Tychonoff's Theorem/General Case

Theorem
Let $I$ be an indexing set.

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

Let $\ds X = \prod_{i \mathop \in I} X_i$ be the corresponding product space.

Then $X$ is compact each $X_i$ is compact.

Also see
Tychonoff's Theorem for Hausdorff Spaces, a weaker result requiring only BPI instead of the full AoC.

Tychonoff's Theorem Without Choice, a weaker result that holds in pure ZF theory.