\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\groupAdd}[1] { +_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \require{mathtools} \)

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 [...].

ε-adherent point

Let \( X \) be a subset of \( \mathbb{R} \), let \( \varepsilon > 0 \) be a real and \( x \in \mathbb{R} \) be another real. We say that \( x \) is ε-adherent to \( X \) iff [...].

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 [...].

Closure

Let \( X \) be a subset of \( \mathbb{R} \). The closure of \( X \), sometimes denoted as \( \overline{X} \), is defined to be [...].