Take any ordered subcollection $C\subset\mathcal{A}$ , and the union of its sets $U$ . We want to show that $U\in\mathcal{A}$ . Take any finite subset $B$ of $U$ . For each point in $B$ , there is a set in $C$ containing it, and since the number of points is finite and $C$ is ordered, there is a set in $C$ that contains all these points, i.e. it contains $B$ (if one set contains $b_{1}$ and another contains $b_{2}$ , then one of them has to contain both $b_{1}$ and $b_{2}$ , etc.). Since the set containing $B$ is in $C\subset\mathcal{A}$ , all its finite subsets, including $B$ , are in $\mathcal{A}$ , i.e. $B\in\mathcal{A}$ . So, any finite subset of $U$ is in $\mathcal{A}$ , and therefore, $U\in\mathcal{A}$ . Now apply Kuratowski lemma.