Closure (topology) - Examples

Examples

• In any space, .
• In any space X, X = cl(X).
• If X is the Euclidean space R of real numbers, then cl((0, 1)) = .
• If X is the Euclidean space R, then the closure of the set Q of rational numbers is the whole space R. We say that Q is dense in R.
• If X is the complex plane C = R2, then cl({z in C : |z| > 1}) = {z in C : |z| ≥ 1}.
• If S is a finite subset of a Euclidean space, then cl(S) = S. (For a general topological space, this property is equivalent to the T₁ axiom.)

On the set of real numbers one can put other topologies rather than the standard one.

• If X = R, where R has the lower limit topology, then cl((0, 1)) = [0, 1).
• If one considers on R the topology in which every set is open (closed), then cl((0, 1)) = (0, 1).
• If one considers on R the topology in which the only open (closed) sets are the empty set and R itself, then cl((0, 1)) = R.

These examples show that the closure of a set depends upon the topology of the underlying space. The last two examples are special cases of the following.

• In any discrete space, since every set is open (closed), every set is equal to its closure.
• In any indiscrete space X, since the only open (closed) sets are the empty set and X itself, we have that the closure of the empty set is the empty set, and for every non-empty subset A of X, cl(A) = X. In other words, every non-empty subset of an indiscrete space is dense.

The closure of a set also depends upon in which space we are taking the closure. For example, if X is the set of rational numbers, with the usual subspace topology induced by the Euclidean space R, and if S = {q in Q : q2 > 2}, then S is closed in Q, and the closure of S in Q is S; however, the closure of S in the Euclidean space R is the set of all real numbers greater than or equal to