Open and Closed Sets
TWO QUESTIONS ASKED OF THE SAME SUBSET
A topology picks out some subsets of \(M\); those are the ones called open. A subset is called closed when its complement is open. They are two separate questions put to the same subset, and it can answer yes to both, no to both, or yes to exactly one.
Closed is a condition on the complement
Let \((M, \mathcal{O})\) be a topological space.
A subset \(U \subseteq M\) is open if and only if \(U \in \mathcal{O}\).
A subset \(V \subseteq M\) is closed if and only if its complement \(M \setminus V\) is in \(\mathcal{O}\).
Openness is a question about the subset in front of you. Closedness is the same question asked of the complement.
Closed is not the negation of open
On \(\mathbb{R}\) with its standard topology, all four combinations occur. \((0,1)\) is open, and not closed: its complement holds \(0\), and every interval around \(0\) reaches into \((0,1)\). \([0,1]\) is closed and not open, failing at \(0\). \([0,1)\) is neither, failing at \(0\) for openness and at \(1\) for closedness. And \(\emptyset\) is both.
The empty set and the whole set are always both
In any topological space, \(\emptyset\) and \(M\) are open by the first condition, and each is the other's complement, so both are closed as well. A set that is both is called clopen.
Neither is a property the subset carries alone
Both words describe a subset judged against a topology, so the same subset changes its answers when the topology changes underneath it. Under the discrete topology every subset is open, and every complement is too, so every subset is also closed. Under the chaotic topology no subset other than \(\emptyset\) and \(M\) is either.