Closed set

A subset of a topological space is closed when its complement is open. Closedness depends on the topology, not just on the underlying set.

Topological space: A set equipped with a collection of subsets designated as open, satisfying axioms for unions and finite intersections. Closedness is defined relative to the topology on this space.
Closure (topology): The closure of a subset is the smallest closed set containing it. A subset is closed exactly when it equals its closure.
Closed interval: An interval of real numbers that includes both of its finite endpoints. In the usual topology on the real line, every closed interval is a closed set.
Clopen set: A subset that is both open and closed in a topological space. Closedness does not imply openness; clopen sets satisfy both conditions.
Open set: A subset belonging to the specified collection of open sets in a topological space. A set is closed precisely when its complement is open.
Limit point: A point is a limit point of a set if every neighborhood of it meets the set at a point other than itself. In a topological space, a set is closed if it contains all its limit points.
Cantor set: The subset of the real line obtained by repeatedly removing middle thirds from the unit interval. It is a closed set with empty interior and uncountably many points.
Compact space: A topological space is compact if every open cover has a finite subcover. Closed subsets of compact spaces are compact, but closedness alone does not imply compactness.
Complement (set theory): The complement of a subset consists of elements of the ambient set that are not in that subset. Taking the complement converts the defining condition into openness.
Neighborhood (topology): A neighborhood of a point is a set containing an open set that contains that point. Neighborhoods express the local condition used to identify limit points.

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