Let $C$ be the Cantor set and $S:=C\times C$. Let $f(x,y):S\to \mathbb{R}$ be defined by $f(x,y)=\lVert (x,y)\rVert$, so $f$ is continuous.. Here I am negating the generalized Extreme value theorem: Let $V$ be a normed vector space, $f:V\to \mathbb{R}$ be a continuous function; if $S$ is not compact, $f$ will not attain its sup or inf values on $S$. Suppose $S$ is not compact. But $f(x,y)= \sqrt{x^2+y^2}\le \sqrt{2}$ and $f(x,y) =\sqrt{x^2+y^2}\ge 0$ (since $C\subset [0,1]$). Thus $f$ attains both inf and sup on $S$, a contradiction. Hence, $C\times C$ is compact.
Please let me know if my proof looks fine.