Closure (topology)

In topology, the closure of a subset S of a topological space X, denoted by cl(S), clos(S), or sometimes S̄, is the smallest closed set containing S. Intuitively, the closure of a set includes all the points in the set itself, as well as all the limit points of the set.

More formally, the closure of S can be defined in several equivalent ways:

• Intersection of Closed Sets: The closure of S is the intersection of all closed sets in X that contain S. Since the intersection of any collection of closed sets is itself a closed set, the closure defined in this way is guaranteed to be closed.

• Union with Limit Points: The closure of S is the union of S and its set of limit points (also known as accumulation points). A point x in X is a limit point of S if every neighborhood of x contains a point of S other than x itself.

• Adherent Points: The closure of S is the set of all adherent points of S. A point x in X is an adherent point of S if every neighborhood of x contains a point of S. Note that limit points are adherent points, but isolated points of S are also adherent points.

Properties of Closure:

The closure operator satisfies several important properties:

• Idempotence: cl(cl(S)) = cl(S). Applying the closure operator twice results in the same set as applying it once.
• Increasing: S ⊆ cl(S). The closure of a set always contains the original set.
• Monotonicity: If S ⊆ T, then cl(S) ⊆ cl(T). If one set is contained in another, then the closure of the first set is contained in the closure of the second.
• Closure of Union: cl(S ∪ T) = cl(S) ∪ cl(T). The closure of the union of two sets is the union of their closures.
• cl(∅) = ∅. The closure of the empty set is the empty set.
• cl(X) = X. The closure of the entire space is the entire space.

Interior, Boundary, and Exterior:

The concept of closure is closely related to the concepts of interior, boundary, and exterior of a set:

• Interior: The interior of a set S, denoted int(S), is the largest open set contained in S.
• Boundary: The boundary of a set S, denoted ∂S, is the set of points that are in the closure of both S and its complement. That is, ∂S = cl(S) ∩ cl(X \ S).
• Exterior: The exterior of a set S, denoted ext(S), is the interior of the complement of S. That is, ext(S) = int(X \ S).

These four sets (interior, boundary, exterior, and the set itself) partition the topological space X.

Importance:

The closure operator is a fundamental concept in topology. It provides a way to "complete" a set by adding its limit points, and it is used to define other important topological concepts, such as density, continuity, and connectedness. The properties of the closure operator are essential for proving many topological theorems.