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If you have some object, then compactness allows you to extend results that you know are true for all finite sub-objects to the object itself.

The main result used to prove this kind of thing is the fact that if $X$ is a compact space, and $(K_\alpha)_{\alpha\in A}$ is a family of closed sets with the finite intersection property (no finite collection has empty intersection) then $(K_\alpha)$ has non-empty intersection. For if $(K_\alpha)$ has empty intersection then the complements of the $K_\alpha$ form an open cover of $X$, which then has to have a finite subcover $(X\setminus K_{\alpha(i)})_{i=1}^n$, and so the $(K_{\alpha(i)})_{i=1}^n$ is a finite collection of the $K_\alpha$ with empty intersection.

For example, the De Bruijn-Erdős Theorem in graph theory states that an infinite graph $G$ is $n$-colourable if all its finite subgraphs are $n$-colourable (i.e., you can colour the vertices with $n$ colours in such a way that no two vertices connected by an edge are the same colour). You can prove this by noting that the space $X$ of all colourings of the vertices of $G$ with $n$ colours (for which vertices of the same colour may share an edge) is a compact topological space (since it is the product of discrete spaces). Then, for each finite subgraph $F$, let $X_F$ be the set of all colourings of $G$ that give an $n$-colouring of $F$. It can be checked that the $X_F$ are closed and have the finite intersection property, so they have non-empty intersection, and any member of their intersection must $n$-colour the whole of $G$.

In general, if you have some property that you know is true for finite sub-objects, then you can often encode that in a collection of closed sets in a topological space $X$ that have the finite intersection property. Then, if $X$ is compact, you can show that the closed sets have non-empty intersection, which normally tells you that the result is true for the object itself (sorry that this is all so imprecise!)

A very closely-related example is the compactness theorem in propositional logic: an infinite collection of sentences is consistent if every finite sub-collection is consistent. This can be proved using topological compactness, or it can be proved using the completeness theorem: if the collection is inconsistent, then it must be possible to derive a contradiction using finitely many finite statements, so some finite collection of sentences must be inconsistent. Either way you look at it, though, the compactness theorem is a statement about the topological compactness of a particular space (products of compact Stone spaces).