Let me include an example of compactness which is a bit farther away from analysis and geometry.

Given a set $F = \{ \phi_i \}$ of symbols of prepositions, assume that you form some composed prepositions connecting those with the symbols $\wedge$, $\vee$, $\neg$ and parenthesis (let me be not really precise here). A valid concatenation is for instance $(\phi \vee \psi) \wedge (\neg \rho)$. Take any set $X$ of composed propositions. We say that $X$ is non-contradictory if one can assign a truth value to all $\phi_i$ in such a way that, using ordinary rules for connectives, all sentences in $X$ are true.

Theorem: If any every finite subset of $X$ is non-contradictory, $X$ is non-contradictory as well.

The proof is simple. The set of possible choices for truth values is just $Y = \{0, 1\}^F = \prod_{\phi_i \in F} \{0, 1 \}$. Topologize $Y$ with the product topology, using the discrete topology on $\{0, 1\}$. Then $Y$ is compact by Tychonoff's theorem. For each composed preposition $\psi \in X$, the set $\{\psi \text{ is true} \}$ is a finite intersection of closed subset, hence a closed subset of $Y$.

The hypothesis says that the intersection of finitely many of these closed sets is non-empty; by compactness, the intersection of all these closed subsets is not empty, which is the thesis.

Let me include an example of compactness which is a bit farther away from analysis and geometry.

Given a set $F = \{ \phi_i \}$ of symbols of prepositions, assume that you form some composed prepositions connecting those with the symbols $\wedge$, $\vee$, $\neg$ and parenthesis (let me be not really precise here). A valid concatenation is for instance $(\phi \vee \psi) \wedge (\neg \rho)$. Take any set $X$ of composed propositions. We say that $X$ is non-contradictory if one can assign a truth value to all $\phi_i$ in such a way that, using ordinary rules for connectives, all sentences in $X$ are true.

Theorem: If any finite subset of $X$ is non-contradictory, $X$ is non-contradictory as well.

The proof is simple. The set of possible choices for truth values is just $Y = \{0, 1\}^F = \prod_{\phi_i \in F} \{0, 1 \}$. Topologize $Y$ with the product topology, using the discrete topology on $\{0, 1\}$. Then $Y$ is compact by Tychonoff's theorem. For each composed preposition $\psi \in X$, the set $\{\psi \text{ is true} \}$ is a finite intersection of closed subset, hence a closed subset of $Y$.

The hypothesis says that the intersection of finitely many of these closed sets is non-empty; by compactness, the intersection of all these closed subsets is not empty, which is the thesis.