Connectedness Through Clopen Sets
A space is connected when it cannot be split into two disjoint, nonempty open pieces. That definition asks us to consider pairs of sets. An equivalent test uses just one set: look for a proper subset that is both open and closed. This equivalence is useful because openness and closedness can often be checked directly from a topology’s list of sets.
Throughout, \((X,\mathcal{T})\) is a topological space. A subset is proper if it is not all of \(X\). Recall that a set is closed when its complement in \(X\) is open. A separation of \(X\) is a pair of disjoint, nonempty open sets whose union is \(X\).
A set need not be either open or closed: it may be neither. The criterion below does not say that every open set is clopen. It says that connectedness is determined by whether any open set other than \(\varnothing\) and \(X\) also has an open complement.
The Clopen Criterion
Proof. First suppose \(X\) is connected. We show that a clopen subset \(C\) must be either \(\varnothing\) or \(X\). If \(C\) were nonempty and proper, then \(X\setminus C\) would also be nonempty. Both \(C\) and \(X\setminus C\) are open: \(C\) is open because it is clopen, and \(X\setminus C\) is open because \(C\) is closed. They are disjoint and their union is \(X\). They would therefore be a separation, contradicting connectedness. So no such \(C\) exists. The sets \(\varnothing\) and \(X\) are always clopen, since both they and their complements are open.
Conversely, suppose the only clopen subsets are \(\varnothing\) and \(X\). If \(X\) were not connected, there would be a separation \(U,V\): the sets would be disjoint, nonempty, open, and satisfy \(U\cup V=X\). Since \(V=X\setminus U\) is open, \(U\) is closed as well as open. Thus \(U\) is a clopen set. It is nonempty and proper, because \(U\ne\varnothing\) and \(V\ne\varnothing\), contradicting the assumption. Therefore \(X\) is connected. \(\square\)
The proof gives an exact correspondence. A proper nonempty clopen set \(C\) produces the separation \(C,X\setminus C\). Conversely, either side of any separation is a proper nonempty clopen set. The criterion is therefore a reformulation of the definition, not an additional condition requiring stronger hypotheses such as a metric or a separation axiom.
Checking Clopen Sets in Finite Spaces
For a finite space, one direct method is to list the open sets, form their complements to find the closed sets, and identify which sets occur in both lists. The topology still matters: the same underlying finite set can have different collections of open sets and different connectedness behavior.
Worked Example: A Finite Space with a Separation
Let \(X=\{a,b,c\}\) and let \(\mathcal{T}=\{\varnothing,\{a\},\{b,c\},X\}\). This is a topology: it contains \(\varnothing\) and \(X\); the union of \(\{a\}\) and \(\{b,c\}\) is \(X\); and their intersection is \(\varnothing\). Unions or intersections involving \(\varnothing\) or \(X\) also remain in the listed collection.
To find the closed sets, take complements in \(X\): \(X\setminus\varnothing=X\), \(X\setminus\{a\}=\{b,c\}\), \(X\setminus\{b,c\}=\{a\}\), and \(X\setminus X=\varnothing\). Thus the closed-set list is also \(\{\varnothing,\{a\},\{b,c\},X\}\). Intersecting the open-set and closed-set lists gives all four sets, so the proper nonempty clopen sets are \(\{a\}\) and \(\{b,c\}\).
Conclusion and check: The set \(\{a\}\) and its complement \(\{b,c\}\) are both nonempty and open, and their union is \(X\). They form a separation, so \(X\) is disconnected. This agrees with the clopen criterion: \(\{a\}\) is a proper nonempty clopen subset.
Worked Example: A Finite Space That Is Connected
Let \(X=\{a,b\}\) have the topology \(\mathcal{T}=\{\varnothing,\{a\},X\}\). Its closed sets, obtained by taking complements of the open sets, are \(X,\{b\},\varnothing\), respectively. Compare the two lists: \(\{a\}\) is open but not closed, because its complement \(\{b\}\) is not open; \(\{b\}\) is closed but not open. The only sets in both lists are \(\varnothing\) and \(X\).
Conclusion and check: By the clopen criterion, \(X\) is connected. Directly, there can be no separation: the only nonempty open sets are \(\{a\}\) and \(X\). The set \(X\) cannot be one side of a separation because the other side would have to be empty, and \(\{a\}\) cannot be one side because its complement \(\{b\}\) is not open.
Indiscrete Spaces
The indiscrete topology on a set \(X\) is \(\{\varnothing,X\}\). It is the smallest topology on \(X\): it declares no proper nonempty subset open. The clopen criterion immediately settles connectedness, as the next result states.
Proof. In the indiscrete topology the only open sets are \(\varnothing\) and \(X\). A clopen set must in particular be open, so it can only be one of those two sets. Both are clopen: their complements are \(X\) and \(\varnothing\), which are open. Thus the only clopen subsets are \(\varnothing\) and \(X\), and the clopen criterion implies that the space is connected. If \(X=\varnothing\), these two names designate the same subset, and there are no nonempty sets that could form a separation. The argument still applies. \(\square\)
Worked Example: An Indiscrete Space on Three Points
Let \(X=\{a,b,c\}\) with \(\mathcal{T}=\{\varnothing,X\}\). The closed sets are also \(\varnothing\) and \(X\), because each is the complement of the other. A direct check of all eight subsets of \(X\) gives the same conclusion: \(\varnothing\) and \(X\) are clopen; \(\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\}\) are not open and hence are not clopen.
Conclusion and check: There is no proper nonempty clopen subset, so the space is connected. Equivalently, a separation would require two nonempty open sets, but the only nonempty open set is \(X\). Two copies of \(X\) cannot be disjoint, so no separation exists.
A Common Pitfall: Open Does Not Mean Clopen
To use the criterion, it is not enough to find a proper nonempty open set. Its complement must also be open. In the two-point example above, \(\{a\}\) is open, but its complement \(\{b\}\) is not open; therefore \(\{a\}\) does not witness disconnectedness.
Common mistake. One might argue, “\(\{a\}\) is a proper nonempty open set, so the space is disconnected.” The correction is to check the complement: \(X\setminus\{a\}=\{b\}\), and \(\{b\}\notin\mathcal{T}\). Thus \(\{a\}\) is not closed, is not clopen, and does not yield a separation. In general, an open set \(U\) gives a separation only when \(X\setminus U\) is also open and both sets are nonempty.
This criterion is useful whenever a topology is specified by listing open sets or by a concise rule for deciding which subsets are open. Instead of searching through every possible pair of open sets, one can test whether any proper nonempty open set has an open complement. In finite spaces, the lists of open and closed sets make that test explicit.
Check Your Understanding
Use the clopen criterion, and check that each proposed set is both open and closed when you use it as evidence.
- In a space with topology \(\{\varnothing,\{p\},\{q,r\},\{p,q,r\}\}\), list the proper nonempty clopen subsets and decide whether the space is connected.
- For the topology \(\{\varnothing,\{p\},\{p,q\}\}\) on \(\{p,q\}\), which of the proper nonempty subsets are open, which are closed, and is the space connected?
- Explain why a proper nonempty open subset by itself does not prove that a space is disconnected.
- Show directly from the definition of separation that the empty space is connected.
- Can an indiscrete space with more than one point be Hausdorff? Explain using the definition of Hausdorff separation by neighborhoods.