# Closed set

The complement of an open subset a topological space. It contains all points that are "close" to it.

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.[1][2] 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.

a subset is closed if and only if it contains every point that is close to it.

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.

Furthermore, every closed subset of a compact space is compact, and every compact subspace of a Hausdorff space is closed.

Sets that can be constructed as the union of countably many closed sets are denoted Fσ sets. These sets need not be closed.