Closed set

In mathematics, a set is called closed if its complement is open, i.e.: if you are outside the set, and you "wiggle" a little bit, you will still be outside the set. 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. The rational numbers between 0 and 1 (inclusive) are closed in the space of rational numbers, but not 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 can be defined for all topological spaces and in particular for all metric spaces, again as the complements of open sets. An alternative characterization of a closed set A is as follows: any convergent sequence (or net) with elements in A has its limit in A. This definition also depends on the surrounding space, because whether a sequence converges or not depends on what points are present in that space.

In general, an arbitrary intersection of closed sets is closed, and a finite union of closed sets is closed. The empty set and the whole space are also closed. This allows to define the closure of a given set: the smallest closed set containing the given one.

In a certain sense, the compact Hausdorff spaces are "absolutely closed": if you embed such a space in an arbitrary Hausdorff space, it will always be a closed subset of that Hausdorff space; the "surounding 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 if it is compact. This is a different notion from the one discussed above.