Proof. By Tychonoff, since $[0,1]$ is compact (Heine-Borel) and $\mathbbR$ is any index set, the product is compact. (Note: In product topology, not in box topology.) □
Let $X$ be compact metric, $Y$ complete metric. Show $C(X,Y)$ is complete in uniform metric. munkres topology solutions chapter 5
Let $X$ be a Tychonoff space. Show that if $f: X \to \mathbbR$ is bounded and continuous, then $f$ extends to $\beta X$. Proof. By Tychonoff
Example: Show that if $X$ is compact Hausdorff, then $\beta X \cong X$. $Y$ complete metric. Show $C(X