Real Set Closures

Definition

Suppose $A \subset \R $. A point $x \in \R $ is a limit point of $A$ if there exists a sequence $a: \N \to A$ so that $a_n \to x$ In other words, the limit points of $A$ are the points which are the limits of some sequences in $A$.

It is possible that $x \not\in A$ but $x$ is a limit point. The closure of $A$ is the set containing $A$ and all limit points of $A$.

Notation

The closure of $A \subset \R $ is denoted $\bar{A}$. Other notation includes $\cl(A)$.