Functorality of the Alexandrov topology

A number of set systems associated to structured sets either preserve or reflect substructures and they therefore form functors to categories of hypergraphs or topologies. The Alexandrov topology consisting of all lower sets (respectively upper sets) of a poset is a basic example.

Lemma 1. Let $f: A \to B$ be a monotone map of posets. Let $L$ be a lower set of $B$, then $f^{-1}(L)$ is a lower set of $A$.

Proof. suppose that $a \in f^{-1}(L)$ then $f(a) \in L$ and further suppose that $b \subseteq a$. By the fact that $f$ is monotone we have $f(a) \subseteq f(b)$ and now by the fact that $f(b) \in L$ and $L$ is a lower set, we have that $f(a) \in L$. Which implies that $a \in f^{-1}$ so in short $b \subseteq a \text{ and } a \in f^{-1}(L)$. Therefore, $f^{-1}(L)$ is a lower set.

The dual proposition that monotone maps preserve lower sets is trivially false, consider $F : 1 \to 3$ that maps the singleton poset to the middle element of an ordered triple. Then this map preserves neither upper sets or lower sets. So the Alexandrov topology is not functorial to the category of topologies and open maps, but it is for the more relevent category $Top$ of topologies and continuous maps.

Theorem 1. $F: Ord \to Top$ is a covariant functor from the category of thin categories to the category of topological spaces and continuous maps.

Proof. (1) let $P$ be a thin category then $F(P)$ is the topology generated by the map of singletons that produces all predecessor elements and this is trivially an Alexandrov topology (2) and by lemma 1 for any functor $f : A \to B$ we have that $F(f) : F(A) \to F(B)$ is a continuous map. Finally (3) $F$ preserves composition by preserving underlying functions so $F$ is functorial. $\square$

The fact that monotone maps reflect lower sets (respectively upper sets) is simply a long line of a group of theorems dealing with the preservation and reflection of open sets. For example, the fact that the pre image of a prime ideal is a prime ideal is used in commutative algebra to describe the functoriality of the topology $Spec(R)$. We see that order ideals are not that different from ring ideals.