big list - Applications of compactness

Any continuous function on a compact space is bounded and admits a maximum. This is perhaps the most important application to compactness.

Also pretty important in my opinion is the fact that a bijective continuous map $f:Xrightarrow Y$, X,Y Hausdorff topological spaces, is automatically an homeomorphism if X is compact.

Another important application of compactness is the Stone-Weierstrass theorem : assuming X compact, a subalgebra of $C^0(X,R)$ is dense iff it separates points.

Now something a bit more fancy :

Let G be a compact group. Then the semi-group generated by an element is dense in the groupe generated by that element.

All maximal connected compact subgroups of Lie groups are conjuguated to each other.

Let $f_n$ a sequence of continuous functions on a compact space that converges uniformly to $f$. Then for all neighborhood $V$ of $f^{-1}(0)$, and any sufficiently big n, $f_n^{-1}(0)$ is contained in $V$.

Compact manifolds (more generally ANR) have finitely generated homology groups.

On a compact smooth riemannian manifold, there is always infinitely many geodesics connecting two points.

The list goes on forever. Let me end with some famous conjecture on compact spaces (Kaplansky). Let X be a Hausdorff compact space. Are all algebra homomorphisms from $C^0(X)$ to a Banach algebra A , continuous ?