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".
The general description of topological spaces requires that a topology be closed under arbitrary (finite or infinite) unions of open sets, but only under intersections of finitely many open sets.
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.
Given a (not necessarily finite) topological space X we can define a preorder on X by where cl{y} denotes the closure of the singleton set {y}.
Now if X is finite, the converse is also true: every upper set is open in X.
The equivalence between preorders and finite topologies can be interpreted as a version of Birkhoff's representation theorem, an equivalence between finite distributive lattices (the lattice of open sets of the topology) and partial orders (the partial order of equivalence classes of the preorder).
If a finite topological space is T1 (in particular, if it is Hausdorff) then it must, in fact, be discrete.
Therefore, any finite topological space which is not discrete cannot be T1, Hausdorff, or anything stronger.
In general, two points x and y are topologically indistinguishable if and only if x ≤ y and y ≤ x, where ≤ is the specialization preorder on X.
Similarly, a space is R0 if and only if the specialization preorder is an equivalence relation.
Since the partition topology is pseudometrizable, a finite space is R0 if and only if it is completely regular.
The excluded point topology on any finite set is a completely normal T0 space which is non-discrete.
Connectivity in a finite space X is best understood by considering the specialization preorder ≤ on X.
We can associate to any preordered set X a directed graph Γ by taking the points of X as vertices and drawing an edge x → y whenever x ≤ y.
In other words, this single set forms a local base at x.
Each such component is both closed and open in X. Finite spaces may have stronger connectivity properties.
A finite space X is For example, the particular point topology on a finite space is hyperconnected while the excluded point topology is ultraconnected.
Perhaps surprisingly, there are finite topological spaces with nontrivial fundamental groups.
A simple example is the pseudocircle, which is space X with four points, two of which are open and two of which are closed.
More generally it has been shown that for any finite abstract simplicial complex K, there is a finite topological space XK and a weak homotopy equivalence f : |K| → XK where |K| is the geometric realization of K. It follows that the homotopy groups of |K| and XK are isomorphic.
In fact, the underlying set of XK can be taken to be K itself, with the topology associated to the inclusion partial order.
As discussed above, topologies on a finite set are in one-to-one correspondence with preorders on the set, and T0 topologies are in one-to-one correspondence with partial orders.
The table below lists the number of distinct (T0) topologies on a set with n elements.
Let T(n) denote the number of distinct topologies on a set with n points.
There is no known simple formula to compute T(n) for arbitrary n. The Online Encyclopedia of Integer Sequences presently lists T(n) for n ≤ 18.