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Note that this notion depends on the concept of "outside", the surrounding space with respect to which the complement is taken. For instance, the unit interval [0,1] is closed in the real numbers, and the set [0,1] ∩ Q of rational numbers between 0 and 1 (inclusive) is closed in the space of rational numbers, but [0,1] ∩ Q is not closed in the real numbers. Some sets are neither open nor closed, for instance the half-open interval [0,1) in the real numbers.

The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces, as well as for other spaces that carry topological structures, such as metric spaces, differentiable manifolds, uniform spaces, and gauge space s.

An alternative characterization of closed sets is available via sequences and nets. A subset A of a topological space X is closed in X if and only if every limitA limit can be: Limit (mathematics), including: Limit of a function Limit of a sequence Net (topology) Limit (category theory) A constraint (mathematical, physical, economical, legal, etc. in the form of an inequality, such as: Chandrasekhar limit Greisen of every net of elements of A also belongs to A. In a first-countable spaceIn topology, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable local base. That is, for each x ∈ X there exists a sequence (such as a metric space), it is enough to consider only sequences, instead of all nets. One value of this characterisation is that it may be used as a definition in the context of convergence space s, which are more general than topological spaces. Notice that this characterisation also depends on the surrounding space X, because whether or not a sequence or net converges in X depends on what points are present in X.

Any intersectionIn mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. This article uses mathematical symbols. The intersecti of arbitrarily many closed sets is closed, and any unionAbstract algebra Algebra In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. This article uses mathematical symbols. Basic definition If of finitelyIn mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2,. n with n isin N . It is a theorem that a set is finite if and only if there exists no bijection between the set and any of its prop many closed sets is closed. In particular, the empty setAbstract algebra Algebra Set theory In mathematics, the empty set is the set with no elements. Notation The standard notation for denoting the empty set, invented by Nicholas Bourbaki, is the symbol , also written as or ∅, and sometimes approximated and the whole space are closed. In fact, given a set X and a collection F of subsets of X that has these properties, then F will be the collection of closed sets for a unique topology on X. The intersection property also allows one to define the closureIn topology and mathematical analysis, the closure of a subset of a topological space is the smallest closed subset of which contains. This can be constructed by intersecting all closed supersets of in. Notation The closure of is written as or. If there i of a set A in a space X, which is defined as the smallest closed subset of X that is a superset of A. Specifically, the closure of A can be constructed as the intersection of all of these closed supersets.

We have seen twice that whether a set is closed is relative; it depends on the space that it's embedded in. However, the compact Hausdorff spaces are "absolutely closed" in a certain sense. To be precise, if you embed a compact Hausdorff space K in an arbitrary Hausdorff space X, then K will always be a closed subset of X; the "surrounding space" does not matter here. In fact, this property characterizes the compact Hausdorff spaces.

A manifold is called closed if it has no boundary and is compact. This is a somewhat different notion from the one discussed above.

In dynamical systems, an orbit is called closed if it has a finite number of elements. This is also different from the general notion of a closed set.

In film, a closed set is a sound stage to which no visitors are admitted.