$S$ unit sphere of an $\infty$-dim Banach isn't compact. Pf: for $H$ closed hyperplane, $\bigcap(S\cap H)\neq\varnothing$ (Hahn-Banach), but no finite subintersection is empty.

Note: $\infty$, $\bigcap$, $\cap$, $\neq$, and $\varnothing$ are unicode characters, so this is actually tweetable!

$S$ unit sphere of an $\infty$-dim Banach isn't compact. Pf: for $H$ closed hyperplane, $\bigcap(S\cap H)=\varnothing$ (Hahn-Banach), but no finite subintersection is empty.

Note: $\infty$, $\bigcap$, $\cap$, $\neq$, and $\varnothing$ are unicode characters, so this is actually tweetable!