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Concept
Finite topological space
Summary
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 can
lend good insight to a variety of questions".
Let be a finite set. A topology on is a subset of (the power set of ) such that
and .
if then .
if then .
In other words, a subset of is a topology if contains both and and is closed under arbitrary unions and intersections. Elements of 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 which includes both the bottom element and the top element .
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 {a}. Here the open sets are ∅ and {a}. 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 {a}, namely the constant function to a. In the language of the empty space serves as an initial object in the while the singleton space serves as a terminal object.
Let X = {a,b} be a set with 2 elements.
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