Topological Product of Compact Spaces/Finite Product

Theorem
Let $T_1, T_2, \ldots, T_n$ are topological spaces.

Let $\displaystyle \prod_{i \mathop = 1}^n T_i$ be the topological product of $T_1, T_2, \ldots, T_n$.

Then $\displaystyle \prod_{i \mathop = 1}^n T_i$ is compact iff both $T_1$ and $T_2$ are compact.

Proof
This can be shown by induction.

Also see

 * Tychonoff's Theorem, where this result is extended to the topological product of any infinite number of topological spaces.