Let $\mathcal{U} \subseteq \wp\left ( X \right )$ be a family of sets that contains both $\emptyset$ and $X$, is closed under arbitrary intersections and is closed under arbitrary unions of chains of its elements under inclusion. Formally, let $\mathcal{U}$ satisfy the following:
- $\emptyset, X \in \mathcal{U}$
- $ \left \{U_i \right \}_{i \in I} \subseteq \mathcal{U} \Rightarrow \bigcap_{i \in I}^{} U_i \in \mathcal{U}$
- $\left \{ U_i \right \}_{i \in I} \subseteq \mathcal{U} : \forall \alpha, \beta \in I : U_{\alpha}\subseteq U_{\beta} \vee U_{\beta} \subseteq U_{\alpha} \Rightarrow \bigcup_{i \in I}^{} U_i \in \mathcal{U}$
We define a closure operator for every $S \subseteq X$ as:
$\mathrm{cl}\left ( S \right )=\bigcap \left \{U : U\supseteq S \quad \mathrm{and} \quad U \in \mathcal{U} \right \}$
How can I prove that $\mathrm{cl}: \mathcal{P}(X) \rightarrow \mathcal{P}(X)$ is a finitary closure operator? That is, how can I prove that for every $S \subseteq X$ the following is satisfied?
$\mathrm{cl} \left ( S \right )=\bigcup \left \{\mathrm{cl}(U) : U\subseteq S \quad \mathrm{and} \quad U \; \mathrm{finite} \right \}$
This result is (without proof) stated in https://math.berkeley.edu/~gbergman/245/2.4/Ch.5.pdf as Lemma 5.3.6.