Closure, definition 

To define a closure, we will utilize ε-adherent points and adherent points. Sets of reals have adherent points analogous to sequences of reals having limit points.

ε-adherent point

Let $X$ be a subset of $\mathbb{R}$, let $\epsilon >0$ be a real and $x\in \mathbb{R}$ be another real. We say that $x$ is ε-adherent to $X$ iff there exists a $y\in X$ which is ε-close to $x$ (i.e. $|x-y|\le \epsilon$ ).

Adherent point

Let $X$ be a subset of $\mathbb{R}$, and let $x\in \mathbb{R}$ be a real. We say that $x$ is an adherent point of $X$ iff it is ε-adherent to $X$ for every $\epsilon >0$.

Closure

Let $X$ be a subset of $\mathbb{R}$. The closure of $X$, sometimes denoted as $\bar{X}$, is defined to be the set of all adherent points of $X$.