Topological spaces · Definition
Closed set
Before you start you need to know
Closed sets are the other face of open sets: they describe the sets that contain every point they can approach.
Definition
A subset \(C\) of a topological space \((X,\mathcal{T})\) is closed when its complement in \(X\) is an open set of the topology \(\mathcal{T}\).
“Closed” is not the opposite of “open”. A set can be both open and closed, like \(\varnothing\) and \(X\) in any topology, or neither, like \([0,1)\) in \(\mathbb{R}\). In the discrete topology every subset is both open and closed.
By De Morgan's laws, closed sets behave dually to open sets: \(\varnothing\) and \(X\) are closed, the intersection of any family of closed sets is closed, and the union of finitely many closed sets is closed. Infinite unions fail: \(\bigcup_{n\ge1}[1/n,1]=(0,1]\) is not closed in \(\mathbb{R}\). In fact, a topology can be defined by giving directly its family of closed sets with those three properties.
In a metric space, \(C\) is closed exactly when it contains the limits of all its convergent sequences; thus \([a,b]\) is closed and \((a,b)\) is not. In a general topological space this idea is formalized with closure and limit points.
A typical mistake is to think that having or lacking a “boundary” decides whether a set is closed: the only thing that matters is whether the complement is open in the chosen topology. The same set can be closed in one topology and not in another.
Think about it
Is \(\mathbb{Q}\) a closed subset of \(\mathbb{R}\) with the usual topology?
No: its complement, the irrationals, is not open, since every interval contains rationals. Equivalently, some sequences of rationals converge to an irrational.
In \(X=\{a,b\}\) with \(\mathcal{T}=\{\varnothing,\{a\},X\}\), what are the closed sets?
The complements of the open sets: \(X\), \(\{b\}\) and \(\varnothing\).
Exercises 10
- In \(X=\{1,2,3\}\) with \(\mathcal{T}=\{\varnothing,\{1\},\{1,2\},X\}\), which of these sets is closed?
- How many closed sets does \(X=\{1,2,3\}\) with \(\mathcal{T}=\{\varnothing,\{1\},\{1,2\},X\}\) have?
- In the discrete topology on a set \(X\), which subsets are closed?
- In the trivial topology on \(X=\{1,2,3\}\), which of these sets is closed?
- In \(\mathbb{R}\) with the usual topology, which of these sets is closed?
- In any topological space \((X,\mathcal{T})\), which statement is true?
- Let \((X,\mathcal{T})\) be a topological space. Which statement is true?
- Let \(X=\{1,2,3,4\}\) with \(\mathcal{T}=\{\varnothing,\{1\},\{2\},\{1,2\},\{1,2,3\},X\}\). How many closed subsets does it have?
- In \(X=\{1,2,3,4\}\) with \(\mathcal{T}=\{\varnothing,\{1\},\{2\},\{1,2\},\{1,2,3\},X\}\), which of these sets is NOT closed?
- Can a subset of a topological space be both open and closed?
References
- James R. Munkres, Topology, 2nd edition, Pearson, 2000. §17
- Bert Mendelson, Introduction to Topology, 3rd edition, Dover Publications, 1990. cap. 3