Redirected from Closed subset
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[?].
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 limit of every net of elements of A also belongs to A. In a first countable space (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 spaces[?], 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 intersection of arbitrarily many closed sets is closed, and any union of finitely many closed sets is closed. In particular, the empty set 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 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. 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. StoneČech 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.
A manifold is called closed if it has no boundary and is compact. This is a somewhat different notion from the one discussed above.
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