Tychonoff's Theorem/General Case

Theorem
Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of non-empty topological spaces, where $I$ is an arbitrary index set.

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

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

Proof
First assume that $X$ is compact.

From Projection from Product Topology is Continuous, the projections $\pr_i : X \to X_i$ are continuous.

It follows from Continuous Image of Compact Space is Compact that the $X_i$ are compact.

Assume now that each $X_i$ is compact.

By Equivalent Definitions of Compactness it is enough to show that every ultrafilter on $X$ converges.

Thus let $\FF$ be an ultrafilter on $X$.

From Image of Ultrafilter is Ultrafilter, for each $i \in I$, the image filter $\map {\pr_i} \FF$ is an ultrafilter on $X_i$.

Each $X_i$ is compact by assumption.

So by Equivalent Definitions of Compactness, each $\map {\pr_i} \FF$ converges.

By Filter on Product Space Converges iff Projections Converge, $\FF$ converges.

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.