= Product of two compact spaces
{title2=$A,B\text{ compact}\ \Longrightarrow\ A\times B\text{ compact}$}
The <product topology> of two <compact spaces> is <compact>, without a <Hausdorff condition>. Refine an <open cover> locally by basis rectangles. For fixed $a\in A$, <compactness> of $B$ selects finitely many rectangles whose second factors cover $B$. Their first factors have an <open> finite <intersection> $U_a$ containing $a$, and the finitely many original covering members cover $U_a\times B$. Use <compactness> of $A$ to choose finitely many $U_a$ covering $A$. The associated finitely many original members form a <finite subcover> of $A\times B$. Repeating the argument proves the finite-factor case of <Tychonoff's theorem>.
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