A Finite-Space Test for Connectedness
For a finite space, connectedness can be decided by checking subsets directly. The key is that a separation of the space is equivalent to a nonempty proper subset that is both open and closed. In a finite set there are only finitely many subsets to check, so this gives a concrete test even when the topology is not familiar.
Let \((X,\mathcal{T})\) be a topological space. A subset \(C\subseteq X\) is closed if \(X\setminus C\) is open. A subset is clopen if it is both open and closed. A separation of \(X\) is a pair of disjoint, nonempty open sets whose union is \(X\). The space is connected if it has no separation. In particular, the empty space and a one-point space are connected: neither has two disjoint nonempty subsets that cover it.
Proof. Suppose first that \(X\) is disconnected. By definition, there are disjoint, nonempty open sets \(U,V\) with \(U\cup V=X\). Since \(V=X\setminus U\) is open, \(U\) is closed as well as open. Also \(U\ne\varnothing\) and \(U\ne X\), because \(U\) and \(V\) are both nonempty. Thus \(U\) is a nonempty proper clopen subset.
Conversely, suppose \(A\subseteq X\) is nonempty, proper, and clopen. Then \(A\) is open, and \(X\setminus A\) is open because \(A\) is closed. The two sets \(A\) and \(X\setminus A\) are disjoint, nonempty, and have union \(X\). They form a separation, so \(X\) is disconnected. This proves both directions. \(\square\)
How to Apply the Test in a Finite Space
If \(X\) has \(n\) points, it has \(2^n\) subsets, including \(\varnothing\) and \(X\). To test connectedness, list the nonempty proper subsets \(A\) and check whether both \(A\) and \(X\setminus A\) belong to \(\mathcal{T}\). Finding one such subset proves that the space is disconnected. If none exists, the clopen test proves that the space is connected.
Use the given topology \(\mathcal{T}\); a subset counts as open only if it is on this list.
For each candidate \(A\), test whether \(A\in\mathcal{T}\) and \(X\setminus A\in\mathcal{T}\).
If both tests succeed for some \(A\), it is clopen and gives a separation. If no candidate succeeds, the space is connected.
For this test, the topological information is in the open-set list, not in how many points a subset contains. A subset with many points need not be open, and a one-point subset need not be closed. The examples below show how much the answer can change when the topology changes while the underlying finite set stays the same.
Worked Example: A Three-Point Discrete Space
Let \(X=\{a,b,c\}\) with the discrete topology \(\mathcal{T}=\mathcal{P}(X)\), so every subset is open. Determine whether \(X\) is connected.
Each subset \(A\subseteq X\) is open. Its complement \(X\setminus A\) is also a subset of \(X\), so it too is open. Thus every subset is clopen. In particular, take \(A=\{a\}\). Then \(A\ne\varnothing\), \(A\ne X\), and \(X\setminus A=\{b,c\}\). Both sets are open, so \(A\) is a nonempty proper clopen subset.
Conclusion and check: The clopen test shows that \(X\) is disconnected. Indeed, \(\{a\}\) and \(\{b,c\}\) are disjoint, nonempty open sets, and their union is \(\{a,b,c\}=X\), so they explicitly form a separation.
Worked Example: A Three-Point Indiscrete Space
Let \(X=\{a,b,c\}\) have the indiscrete topology \(\mathcal{T}=\{\varnothing,X\}\). Determine whether \(X\) is connected.
The only open subsets are \(\varnothing\) and \(X\). A clopen subset must be open, so it must be one of these two sets. Both are indeed clopen: their complements are \(X\) and \(\varnothing\), respectively, and both complements are open. There is no nonempty proper clopen subset.
Conclusion and check: By the clopen test, \(X\) is connected. Directly, a separation would require two nonempty open sets. But the only nonempty open set is \(X\), and two copies of \(X\) are not disjoint. Therefore no separation exists.
Connected and Disconnected Non-Discrete Examples
Worked Example: A Connected Non-Discrete Space
Let \(X=\{a,b,c\}\) and \(\mathcal{T}=\{\varnothing,\{a\},\{a,b\},X\}\). First, this is a topology: the listed sets form a chain under inclusion, so the intersection of any two is the smaller one and their union is the larger one. An arbitrary union of listed sets is one of the listed sets (or \(\varnothing\)), and the same holds for finite intersections. Determine whether \(X\) is connected.
The open sets are precisely \(\varnothing,\{a\},\{a,b\},X\). Their complements show that the closed sets are \(X,\{b,c\},\{c\},\varnothing\). To find the clopen sets, compare these lists. The only sets occurring in both lists are \(\varnothing\) and \(X\). For example, \(\{a\}\) is open but not closed, because its complement \(\{b,c\}\) is not open. The set \(\{b,c\}\) is closed but not open.
Conclusion and check: There is no nonempty proper clopen subset, so the clopen test gives that \(X\) is connected. The space is not discrete, since, for instance, \(\{b\}\notin\mathcal{T}\). This verifies that connected finite spaces need not have the indiscrete topology.
Worked Example: A Disconnected Non-Discrete Space
Let \(X=\{a,b,c\}\) and \(\mathcal{T}=\{\varnothing,\{a,b\},\{c\},X\}\). This is a topology: its open sets are all unions of the two blocks \(\{a,b\}\) and \(\{c\}\). Unions of these blocks are \(\varnothing,\{a,b\},\{c\},X\), and intersections of listed sets remain listed. Determine whether \(X\) is connected.
The complement of \(\{c\}\) is \(\{a,b\}\), which is open, and \(\{c\}\) itself is open. Thus \(\{c\}\) is clopen. It is nonempty and proper, so the clopen test shows that \(X\) is disconnected. The complementary set \(\{a,b\}\) is also clopen: it is open and its complement \(\{c\}\) is open.
Conclusion and check: The pair \(\{a,b\},\{c\}\) consists of disjoint nonempty open sets, and \(\{a,b\}\cup\{c\}=X\). It is a separation. This topology is not discrete, since neither \(\{a\}\) nor \(\{b\}\) is open.
A Common Mistake: Closed Does Not Mean Clopen
In the connected non-discrete example, \(\{b,c\}\) is closed but not open. A common incorrect test is to notice that \(\{b,c\}\) is closed and conclude that the space is disconnected. That does not establish a separation: the proposed complementary set is \(X\setminus\{b,c\}=\{a\}\), which is open, but \(\{b,c\}\) itself is not open. The pair therefore fails the requirement that both pieces be open.
The correction is to test both conditions, or equivalently to check whether a subset and its complement are both open. For a finite space, this is a complete test, not just a way to find some separations: if no nonempty proper subset passes both checks, the clopen theorem proves connectedness. The four examples also show why the underlying set alone cannot decide connectedness. The same three points support a connected indiscrete topology, a connected non-discrete topology, and disconnected topologies.
| Topology on the finite set | Clopen subsets | Connected? |
|---|---|---|
| Discrete topology | Every subset | No, if the set has at least two points |
| Indiscrete topology | \(\varnothing\) and \(X\) | Yes |
| \(\{\varnothing,\{a\},\{a,b\},X\}\) | \(\varnothing\) and \(X\) | Yes |
| \(\{\varnothing,\{a,b\},\{c\},X\}\) | All four listed open sets | No |
For the discrete row, the qualification matters: if \(X\) is empty or has one point, it has no nonempty proper subset and is connected. If \(X\) has at least two points, a singleton is nonempty and proper, and both it and its complement are open, so it is disconnected.
Check Your Understanding
For each question, use the open-set definition and the clopen test; distinguish carefully between open and closed.
- On \(X=\{p,q\}\), let \(\mathcal{T}=\{\varnothing,\{p\},X\}\). List the closed sets and determine whether \(X\) is connected.
- Prove that every finite discrete space with at least two points is disconnected, and explain why the empty and one-point discrete spaces are connected.
- In the topology \(\{\varnothing,\{a\},\{a,b\},X\}\) on \(\{a,b,c\}\), which listed sets are open but not closed?
- Give a nonempty proper clopen subset that witnesses disconnectedness for the topology \(\{\varnothing,\{a,b\},\{c\},X\}\).
- Why does finding a nonempty proper closed subset alone not prove that a space is disconnected?