Closed set

:For closed manifolds, see Closed manifold.

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In topology and related branches of mathematics, a closed set is a set whose complement is open. This implies that a closed set contains its own boundary. Intuitively, if you are outside the set, and you "wiggle" a little bit, you will still be outside the set.

Related Topics:
Topology - Mathematics - Set - Complement - Open - Boundary

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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.

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For instance, the unit interval is closed in the real numbers, and the set  ∩ Q of rational numbers between 0 and 1 (inclusive) is closed in the space of rational numbers, but  ∩ Q is not closed in the real numbers.

Related Topics:
Unit interval - Real number - Rational number

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Some sets are neither open nor closed, for instance the half-open interval [0,1) in the real numbers.

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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 spaces.

Related Topics:
Topological space - Metric space - Differentiable manifold - Uniform space - Gauge space

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An alternative characterization of closed sets is available via sequences and nets.

Related Topics:
Sequence - Net

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A subset A of a topological space X is closed in X if and only if every limit of every net of elements of A also belongs to A.

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In a first-countable space (such as a metric space), it is enough to consider only sequences, instead of all nets.

Related Topics:
First-countable space - Sequence

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One value of this characterisation is that it may be used as a definition in the context of convergence spaces, which are more general than topological spaces.

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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.

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Any intersection of arbitrarily many closed sets is closed, and any union of finitely many closed sets is closed.

Related Topics:
Intersection - Union - Finitely

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In particular, the empty set and the whole space are closed.

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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.

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The intersection property also allows one to define the closure of a set A in a space X, which is defined as the smallest closed subset of X that is a superset of A.

Related Topics:
Closure - Superset

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Specifically, the closure of A can be constructed as the intersection of all of these closed supersets.

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We have seen twice that whether a set is closed is relative; it depends on the space that it's embedded in.

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However, the compact Hausdorff spaces are "absolutely closed" in a certain sense.

Related Topics:
Compact - Hausdorff space

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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.

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In fact, this property characterizes the compact Hausdorff spaces.

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Stone-Čech compactification, a process that turns a completely regular Hausdorff space into a compact Hausdorff space, may be described as adjoining limits of certain nonconvergent nets to the space.

Related Topics:
Stone-Čech compactification - Completely regular

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