Enumerating Topologies on a Two-Point Set

Why a Two-Point Set Is a Useful Test Case

A topology is a collection of subsets, so even a small underlying set can produce several possible collections to check. On a two-point set, there are only four subsets, and the topology axioms sharply restrict which collections can occur. Enumerating them gives a complete, finite example of how to organize a classification: first identify the choices that determine a candidate, then check that each candidate satisfies the axioms.

Let \(X=\{a,b\}\), where \(a\ne b\). Its power set is \(\mathcal{P}(X)=\{\varnothing,\{a\},\{b\},X\}\). A topology \(\mathcal{T}\) on \(X\) must contain \(\varnothing\) and \(X\), must be closed under arbitrary unions, and must be closed under finite intersections. In particular, an arbitrary union may be indexed by an empty family, with value \(\varnothing\); an intersection of an empty finite family has value \(X\). These conventions are consistent with the required inclusion of \(\varnothing\) and \(X\).

The Four Possibilities

Since \(\varnothing\) and \(X\) are required to be open, the only decisions left are whether \(\{a\}\) and \(\{b\}\) are open. There are four possible patterns of answers. The topology axioms determine the collection in each pattern, and the two patterns in which exactly one singleton is open require a check that the resulting collections really are topologies.

Proof. Let \(\mathcal{T}\) be any topology on \(X\). The sets \(\varnothing\) and \(X\) belong to \(\mathcal{T}\) by the first topology axiom. As the only other subsets of \(X\) are \(\{a\}\) and \(\{b\}\), the topology is determined by which of these two singletons belong to \(\mathcal{T}\). If neither belongs, then \(\mathcal{T}=\{\varnothing,X\}\). If both belong, then \(\mathcal{T}=\mathcal{P}(X)\). If \(\{a\}\) belongs and \(\{b\}\) does not, then \(\mathcal{T}=\{\varnothing,\{a\},X\}\); if \(\{b\}\) belongs and \(\{a\}\) does not, then \(\mathcal{T}=\{\varnothing,\{b\},X\}\). Thus every topology must be one of the four displayed collections. It remains to check that the two collections with exactly one singleton are topologies; those checks are given below. The other two are the indiscrete and discrete topologies, respectively, and satisfy the axioms: unions and finite intersections of their members remain in the same collections. \(\square\)

The four collections are distinct as collections of subsets of the labelled set \(X\). In the first, neither singleton is open; in the second, both are open; in the third, only \(\{a\}\) is open; and in the fourth, only \(\{b\}\) is open. The last two are the intermediate cases: they lie between the indiscrete and discrete topologies in the sense that each contains \(\{\varnothing,X\}\) and is contained in \(\mathcal{P}(X)\).

Checking the Two Intermediate Topologies

To verify a topology, it is not enough to check that the empty set and whole space are included. One must also check closure under unions of arbitrarily many open sets and intersections of finitely many open sets. Here, those checks can be organized by noting all possible results of the operations.

Proof. First consider \(\mathcal{T}_a=\{\varnothing,\{a\},X\}\). It contains \(\varnothing\) and \(X\). Take any family \((U_i)_{i\in J}\) of members of \(\mathcal{T}_a\), where \(J\) may be empty. If \(J\) is empty, its union is \(\varnothing\). If one of the \(U_i\) is \(X\), the union is \(X\). If neither case applies, every \(U_i\) is either \(\varnothing\) or \(\{a\}\); their union is \(\varnothing\) if all are empty, and is \(\{a\}\) if at least one equals \(\{a\}\). Thus every such union belongs to \(\mathcal{T}_a\).

Now take a finite family of members of \(\mathcal{T}_a\). The intersection of an empty family is \(X\). For a nonempty family, if any member is \(\varnothing\), the intersection is \(\varnothing\). If no member is empty but at least one is \(\{a\}\), the intersection is \(\{a\}\), since intersecting \(\{a\}\) with either \(\{a\}\) or \(X\) leaves \(\{a\}\). If every member is \(X\), the intersection is \(X\). In all cases the intersection belongs to \(\mathcal{T}_a\). This proves all three topology axioms for \(\mathcal{T}_a\).

For \(\mathcal{T}_b=\{\varnothing,\{b\},X\}\), the same checks can be stated directly with the possible results. Any union of a family in \(\mathcal{T}_b\) is \(\varnothing\) if every member is empty, is \(\{b\}\) if no member is \(X\) and at least one is \(\{b\}\), and is \(X\) if some member is \(X\). For finite intersections, the empty intersection is \(X\); a nonempty intersection is \(\varnothing\) if some member is empty, is \(\{b\}\) if no member is empty and some member is \(\{b\}\), and is \(X\) if all members are \(X\). These outcomes all belong to \(\mathcal{T}_b\), which also contains \(\varnothing\) and \(X\). Hence \(\mathcal{T}_b\) is a topology. \(\square\)

A Reliable Enumeration Method

The classification can be reused as a short procedure whenever a proposed topology on a two-point set is encountered. First record the required sets \(\varnothing\) and \(X\). Then ask two separate yes-or-no questions: is \(\{a\}\) open, and is \(\{b\}\) open? The answers determine one of four candidate collections. Finally, use the union and intersection axioms to confirm the candidate. The verification above establishes that every candidate produced this way is valid.

Common pitfall. A collection is not a topology just because it contains the empty set and is closed under some unions or intersections. For instance, \(\{\varnothing,\{a\}\}\) fails immediately because it omits \(X\). Conversely, a collection containing both singleton sets but omitting \(X\) fails the union axiom as well: \(\{a\}\cup\{b\}=X\). A careful check includes the required sets and both closure axioms; on a finite set, writing out the possible outcomes makes that check explicit.

The enumeration also illustrates a useful general strategy for finite classification problems. Rather than guessing collections one by one, identify the choices that determine them and then verify the axioms for each resulting case. Here the two singleton membership choices give four cases, and the topology axioms confirm that none of those cases must be discarded. Therefore there are exactly four topologies on the labelled two-point set.