Product of two compact spaces ($A,B\text{ compact}⟹A\times B\text{ compact}$)

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.