TITLE: A topology on the natural numbers: a set is closed if it contains the divisors of all its elements
QUESTION [7 upvotes]: Is there a name of or a reference to the following topology on $X=\mathbb N$: 
$A\subseteq X$ is closed if and only if
$n\in A\wedge m|n\implies m\in A$?

REPLY [11 votes]: Given a poset $(P,\leq)$, an up-set is a subset $U\subseteq P$ such that if $a\in U$ and $a\leq b$, then $b\in U$. A down-set is a subset $D\subseteq P$ such that if $a\in D$ and $b\leq a$, then $b\in D$.
The Alexandrov topology on $P$ is the topology whose open sets are exactly the up-sets (equivalently, its closed sets are exactly the down-sets). The Alexandrov topology has the unusual property that an arbitrary intersection of open sets is open (equivalently, an arbitrary union of closed sets is closed). In fact, a $T_0$ topology with this property is always the Alexandrov topology on a poset (take the specialization order on its points). 
Your example is the Alexandrov topology on the divisibility order $(\mathbb{N},|)$. A quick google search for "Alexandrov topology divisibility" led me to this paper, where this topology on $\mathbb{N}$ is considered, and this paper, where the generalization to arbitrary integral domains is considered. I'm sure there are many others.