A topology on a set X is a set O of subsets of X satisfying the following two conditions: firstly, every union of sets of O is a member of O; secondly, every finite intersection of sets of O belongs to O.

A topology on X is thus a subset of the power set of X.

How many topologies are there on a finite set X?

In the case, where X is empty or has a single element, the answer to this question is easy. It is however tedious and complex, when the cardinal of X is large. To illustrate this assessment, we make in this paper the enumeration of all the possible topologies on a set with two or three elements.

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