is called a of then there exist some point in that is close to (although not an element of ), which is how it is possible for a subset to be closed in but to be closed in the "larger" surrounding super-space If and if is topological super-space of then is always a (potentially proper) subset of which denotes the closure of in indeed, even if is a closed subset of (which happens if and only if ), it is nevertheless still possible for to be a proper subset of However, is a closed subset of if and only if for some (or equivalently, for every) topological super-space of Closed sets can also be used to characterize continuous functions: a map is continuous if and only if for every subset ; this can be reworded in plain English as: is continuous if and only if for every subset maps points that are close to to points that are close to Similarly, is continuous at a fixed given point if and only if whenever is close to a subset then is close to More about closed sets 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. Whether a set is closed depends on the space in which it is embedded. However, the compact Hausdorff spaces are "absolutely closed", in the sense that, if you embed a compact Hausdorff space in an arbitrary Hausdorff space then will always be a closed subset of ; the "surrounding space" does not matter here. 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. Furthermore, every closed subset of a compact space is compact, and every compact subspace of a Hausdorff space is closed. Closed sets also give a useful characterization of compactness: a topological space is compact if and only if every collection of nonempty closed subsets of with empty intersection admits a finite subcollection with empty intersection. A topological space is disconnected if there exist disjoint, nonempty, open subsets and of whose union is Furthermore, is totally disconnected if it has an open basis consisting of closed sets. Properties A closed