Finite topological space explained

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 which has only finitely many elements.

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 canlend good insight to a variety of questions".^[1]

Topologies on a finite set

Let

X be a finite set. A topology on X is a subset \tau of P(X) (the power set of X ) such that\varnothing \in \tau and X\in \tau .

1. if

U, V \in \tau then U \cup V \in \tau .

1. if

U, V \in \tau then U \cap V \in \tau .

In other words, a subset

\tau of P(X) is a topology if \tau contains both \varnothing and X and is closed under arbitrary unions and intersections. Elements of \tau are called open sets. 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. Here, that distinction is unnecessary. Since the power set of a finite set is finite there can be only finitely many open sets (and only finitely many closed sets).

A topology on a finite set can also be thought of as a sublattice of

(P(X), \subset) which includes both the bottom element \varnothing and the top element X .

Examples

0 or 1 points

There is a unique topology on the empty set ∅. The only open set is the empty one. Indeed, this is the only subset of ∅.

Likewise, there is a unique topology on a singleton set . Here the open sets are ∅ and . This topology is both discrete and trivial, although in some ways it is better to think of it as a discrete space since it shares more properties with the family of finite discrete spaces.

For any topological space X there is a unique continuous function from ∅ to X, namely the empty function. There is also a unique continuous function from X to the singleton space, namely the constant function to a. In the language of category theory the empty space serves as an initial object in the category of topological spaces while the singleton space serves as a terminal object.

2 points

Let X = be a set with 2 elements. There are four distinct topologies on X:

Notes and References

1. Thurston . William P. . William Thurston . April 1994 . On Proof and Progress in Mathematics . . 30 . 2 . 161–177 . math/9404236 . 10.1090/S0273-0979-1994-00502-6 .