Most astonishing applications of compactness theorem outside logic The compactness theorem has a lot of applications to logic and model theory. I'm looking for applications. I'm looking for theorems in other areas of mathematics which seem at first sight to have nothing to do with logic but which allow a fairly simple proof with Compactness. For example, the existence of the algebraic closure can easily be proved in that manner (see JDH's answer in this MO question), and Marker shows in his book on model theory that every injective polynomial map $\Bbb C^n \rightarrow \Bbb C^n$ is surjective.
Do you know any other examples? I would be especially grateful for an application outside of algebra, if there is any.
 A: Similar to Carl Mummert's example, if the Four Color Theorem holds for all finite maps in the plane, then it holds for all infinite maps in the plane.
A: The compactness theorem is routinely used in Ramsey theory and combinatorial graph theory to give a link between finitary versions and infinitary versions of theorems. For example, if we know that every infinite graph has either an infinite complete subgraph or an infinite antichain, we can deduce that for each $k$ there is an $n$ such that every graph of $n$ vertices has either a complete subgraph of size $k$ or an antichain of size $k$. I believe that people in the field view these implications as somewhat routine. 
A: Yet another version of these 'extension' examples: if a set of figures (polyominos, Wang tiles, etc.) tiles arbitrarily large regions of the plane (or, for instance, if it tiles a quarter-infinite plane) then it tiles the whole plane.  This is a straightforward application of Konig's theorem, and compactness is another way of framing the argument; see https://math.stackexchange.com/a/38751/785 for the basic details of this approach.
A: The compactness theorem is equivalent to various other results in mathematics, including


*

*the ultrafilter lemma,

*Stone's representation theorem,

*Tychonoff's theorem for compact Hausdorff spaces, and

*the Banach-Alaoglu theorem (see this math.SE post).

A: There is a very nice proof from compactness that every set can be linearly ordered.
For a set $A$ define the language $\mathcal L_A$ to have the binary relation symbol $<$, and a constant symbol $c_a$, for every $a\in A$.
Let $T$ be the theory stating that $<$ is a linear order, and for every distinct $a,b\in A$ add an axiom $c_a\neq c_b$. It is not hard to show that every finite $T_0\subseteq T$ has a model. By compactness $T$ has a model $M$, which $<^M$ linearly orders. The function $a\mapsto c_a^M$ is an injection, therefore $A$ can be linearly ordered.
