Closure (topology)

In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" 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.

Contents

Definitions

Point of closure

For S a subset of a 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 neighborhood 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 (or both).

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), \scriptstyle\bar{S} or \scriptstyle 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, which still make sense when applied to other types of closures (see below).

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" or "filter".

Note that these properties are also satisfied if "closure", "superset", "intersection", "contains/containing", "smallest" and "closed" are replaced by "interior", "subset", "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 refers to the set-theoretic difference.

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.

Facts about closures

The set S is closed if and only if Cl(S) = S. In particular, the closure of the empty set is the empty set, and the closure of X itself is X. The closure of an intersection of sets is always a subset of (but need not be equal to) the intersection of the closures of the sets. In a union of finitely many sets, the closure of the union and the union of the closures are equal; the union of zero sets is the empty set, and so this statement contains the earlier statement about the closure of the empty set as a special case. The closure of the union of infinitely many sets need not equal the union of the closures, but it is always a superset of the union of the closures.

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 if and only if A is a subset of ClX(S).

Categorical interpretation

One may elegantly define the closure operator in terms of universal arrows, as follows.

The powerset of a set X may be realized as a partial order category P in which the objects are subsets and the morphisms are inclusions A \to B whenever A is a subset of B. Furthermore, a topology T on X is a subcategory of P with inclusion functor I: T \to P. The set of closed subsets containing a fixed subset A \subseteq X can be identified with the comma category  (A \downarrow I). This category — also a partial order — then has initial object Cl(A). Thus there is a universal arrow from A to I, given by the inclusion A \to Cl(A).

Similarly, since every closed set containing X \ A corresponds with an open set contained in A we can interpret the category  (I \downarrow X \setminus A) as the set of open subsets contained in A, with terminal object int(A), the interior of A.

All properties of the closure can be derived from this definition and a few properties of the above categories. Moreover, this definition makes precise the analogy between the topological closure and other types of closures (for example algebraic), since all are examples of universal arrows.

See also

  • Closure algebra

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