Finite topological space

In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space for which there are only finitely many points.

While topology has mainly been developed for infinite spaces, finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures. William Thurston has called the study of finite topologies in this sense "an oddball topic that can lend good insight to a variety of questions."

A topology on a set X is defined as a subset of P(X), the power set of X, which includes both ∅ and X and is closed under finite intersections and arbitrary unions.

Since the power set of a finite set is finite there can be only finitely many open sets (and only finitely many closed sets). Therefore, one only need check that the union of a finite number of open sets is open. This leads to a simpler description of topologies on a finite set.

Let X be a finite set. A topology on X is a subset τ of P(X) such that

A topology on a finite set is therefore nothing more than a sublattice of (P(X), ⊂) which includes both the bottom element (∅) and the top element (X).

Every finite bounded lattice is complete since the meet or join of any family of elements can always be reduced to a meet or join of two elements. It follows that in a finite topological space the union or intersection of an arbitrary family of open sets (resp. closed sets) is open (resp. closed).

Topologies on a finite set X are in one-to-one correspondence with preorders on X. Recall that a preorder on X is a binary relation on X which is reflexive and transitive.

...
Wikipedia