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Concept
Closed set
Summary
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation.
This should not be confused with a closed manifold.
By definition, a subset of a topological space is called if its complement is an open subset of ; that is, if A set is closed in if and only if it is equal to its closure in Equivalently, a set is closed if and only if it contains all of its limit points. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points.
Every subset is always contained in its (topological) closure in which is denoted by that is, if then Moreover, is a closed subset of if and only if
An alternative characterization of closed sets is available via sequences and nets. A subset of a topological space is closed in if and only if every limit of every net of elements of also belongs to In a first-countable space (such as a metric space), it is enough to consider only convergent sequences, instead of all nets. One value of this characterization is that it may be used as a definition in the context of convergence spaces, which are more general than topological spaces. Notice that this characterization also depends on the surrounding space because whether or not a sequence or net converges in depends on what points are present in
A point in is said to be a subset if (or equivalently, if belongs to the closure of in the topological subspace meaning where is endowed with the subspace topology induced on it by ).
Because the closure of in is thus the set of all points in that are close to this terminology allows for a plain English description of closed subsets:
a subset is closed if and only if it contains every point that is close to it.
Official source
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