Closed_set

Definition

A set $F \subseteq \mathbb{R}$ is closed if it contains its limit_points.

The adjective “closed” appears in several other mathematical contexts and is usually employed to mean that an operation on the elements of a given set does not take us out of the set. In linear algebra, for example, a vector_space is a set that is “closed” under addition and scalar multiplication. In analysis, the operation we are concerned with is the limiting operation. Topologically speaking, a closed set is one where convergent sequences within the set have limits that are also in the set.

Theorems

(i)

The union of a finite collection of closed sets is closed.

(ii)

The interinterion of an arbitrary collection of closed sets is closed