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This can be constructed by intersecting all closed supersets of in .
The closure of is written as or . If there is more than one topology on (say and ), then the different topologies may give rise to different closures; this can be indicated in the notation by a subscript, as in "". If the topology is itself defined by some other structure, such as a metric , then "" can be placed in the subscript instead of "".
In a metric space (such as the -dimensional Euclidean space) the closure is the set of all points in whose distance from is 0. Here, is defined as the infimum of the set .
In a first-countable spaces (such as a metric space), is the set of all limits of all convergent sequences of points in . For a general topological space, this statement remains true if one replaces "sequence" by " netTopology In mathematics the term net has at least two meanings. See the glossary of Riemannian and metric geometry for its meaning for metric spaces. This article is about its meaning in topology, where the concept of a net is a generalization of that of".
Another characterization of is as follows: an element of belongs to if and only if every neighborhood of contains an element of . In other words, iffIn mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if . It is often, not always, written italicized: iff''. Although "P iff Q" is most standard, common alternative phrases include "P or is a limit pointTopology General topology In mathematics, informally speaking, a limit point (or cluster point of a set S in a topological space X is a point x in X that can be "approximated" by points of S other than x as well as one pleases. This concept profitably gen of .
The closure of the open intervalTopology In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. Interval notation is where the permitted values for a variable are expressed as ranging over an in (0,1) in the real numbers is the closed interval [0,1]. If denotes the set of all rational numbers greater than the square root of 2, then the closure of in the rational numbers is ; the closure of in the real numbers is the set of all real numbers greater than or equal to .
In the trivial topology, the closure of any non-empty set is the whole space. In the discrete topology, the closure of any set is that set itself.
The set is closed if and only if . In particular, the closure of the empty set is the empty set, and the closure of itself is . 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.
The closure operation can be characterized by the Kuratowski closure axioms; in particular, this operation is an example of a closure operator.
The closure of the set is equal to the complement of the interior of the complement of .
If is a subspace of containing , then the closure of computed in is equal to the intersection of and the closure of computed in : . In particular, is dense in iff is a subset of .
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