Closure (Analysis)

analysis topology

Definition

Closure

The closure of a set $D$, denoted by $\bar{D}$, is the set of all adherent points of $D$. A point $x$ is an adherent point of $D$ if every neighbourhood of $x$ contains at least one point of $D$.

$$\bar{D}=\{x\mid \forall \epsilon >0:U\_\epsilon (x)\cap D\ne \varnothing \}$$

Equivalently, the closure is the union of the interior and the boundary of the set: $\bar{D}=\mathring{D}\cup \partial D$.

Properties

• Closedness: The closure $\bar{D}$ is always a closed set. In fact, it is the smallest closed set containing $D$.
• Idempotency: Taking the closure of a closure does not change the set: $\overline{\stackrel{ˉ}{D}}=\bar{D}$.
• Containment: $D\subseteq \bar{D}$. A set is closed if and only if $D=\bar{D}$.
• Limit Points: In a metric space, the closure of a set consists of the set itself plus all of its limit points.