Give an example of a subset of \(\RR\) having uncountably many connected components. Can such a subset be open? Closed?
Solution
Let \(C\subset\RR\) be the middle-thirds Cantor set.
1. The Cantor set \(C\) has uncountably many connected components.
The Cantor set is uncountable. It is also totally disconnected: if \(x<y\) are distinct points of \(C\), then at some finite stage of the Cantor construction they lie in two different surviving closed intervals, separated by one of the deleted open middle thirds. Hence no connected subset of \(C\) contains both \(x\) and \(y\).
Therefore every connected component of \(C\) is a singleton. Since \(C\) is uncountable, it has uncountably many connected components.
2. Such a subset can be closed.
The Cantor set \(C\) is an intersection of finite unions of closed intervals, hence closed in \(\RR\), and step 1 shows that it has uncountably many connected components.
3. Such a subset cannot be open.
Every connected component of an open subset \(U\subseteq\RR\) is an open interval, possibly unbounded, and distinct components are disjoint. Each nonempty open interval contains a rational number. Choosing one rational from each component gives an injection from the set of connected components of \(U\) into \(\QQ\). Since \(\QQ\) is countable, \(U\) has at most countably many connected components.
4. Q.E.D.
Step 1 gives the example \(C\). Step 3 shows that such a subset cannot be open, and step 2 shows that it can be closed.