Closure (topology)

In mathematics, the closure of a set S consists of all points which are intuitively "close to S". A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.

Definitions

Point of closure

For S a subset of an Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S. (This point may be x itself.)

This definition generalises to any subset S of a metric space X. Fully expressed, for X a metric space with metric d, x is a point of closure of S if for every r > 0, there is a y in S such that the distance d(x, y) < r. (Again, we may have x = y.) Another way to express this is to say that x is a point of closure of S if the distance d(x, S) := inf{d(x, s) : s in S} = 0.

This definition generalises to topological spaces by replacing "open ball" or "ball" with "neighbourhood". Let S be a subset of a topological space X. Then x is a point of closure of S if every neighbourhood of x contains a point of S. Note that this definition does not depend upon whether neighbourhoods are required to be open.

Limit point

The definition of a point of closure is closely related to the definition of a limit point. The difference between the two definitions is subtle but important — namely, in the definition of limit point, every neighbourhood of the point x in question must contain a point of the set other than x itself.

Thus, every limit point is a point of closure, but not every point of closure is a limit point. A point of closure which is not a limit point is an isolated point. In other words, a point x is an isolated point of S if it is an element of S and if there is a neighbourhood of x which contains no other points of S other than x itself.

For a given set S and point x, x is a point of closure of S if and only if x is an element of S or x is a limit point of S.

Closure of a set

The closure of a set S is the set of all points of closure of S. The closure of S is denoted cl(S), Cl(S), or S. The closure of a set has the following properties.

• cl(S) is a closed superset of S.
• cl(S) is the intersection of all closed sets containing S.
• cl(S) is the smallest closed set containing S.
• A set S is closed if and only if S = cl(S).
• If S is a subset of T, then cl(S) is a subset of cl(T).
• If A is a closed set, then A contains S if and only if A contains cl(S).

Sometimes the second or third property above is taken as the definition of the topological closure.

In a first-countable space (such as a metric space), cl(S) is the set of all limits of all convergent sequences of points in S. For a general topological space, this statement remains true if one replaces "sequence" by "net".

Note that these properties are also satisfied if "closure", "intersection", "contains/containing", "smallest" and "closed" are replaced by "interior", "union", "contained in", "largest", and "open". For more on this matter, see closure operator below.

Examples

• In any space, the closure of the empty set is the empty set.
• In any space X, X = cl(X).
• If X is the Euclidean space R of real numbers, then cl((0, 1)) = [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 $\sqrt2$.

Closure operator

The closure operator is dual to the interior operator o, in the sense that

S = X \ (X \ S)o

and also

So = X \ (X \ S)

where X denotes the topological space containing S, and the backslash denotes the complement of a set.

Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be easily translated into the language of interior operators, by replacing sets with their complements.

If A is a subspace of X containing S, then the closure of S computed in A is equal to the intersection of A and the closure of S computed in X: $Cl_A(S) = A\cap Cl_X(S)$. In particular, S is dense in A iff A is a subset of ClX(S).