Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 is mostly interesting only 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).

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