= Closure
{disambiguate=topology}
{title2=$\overline A$}
{wiki}
= Closure
{synonym}
The <closure> of a <subset> $A$ of a <topological space> $X$ is the smallest <closed set> containing $A$. Equivalently, it is the <intersection> of all <closed sets> containing $A$, or the <set> of points whose every <neighbourhood> in $X$ meets $A$. The notation $\operatorname{cl}_X A$ makes the ambient <topological space> explicit. This definition works without a <metric> or any appeal to <sequences>. https://math.mit.edu/~hrm/palestine/munkres-topology.pdf[James Munkres, Topology, closure and interior of a set]{ref}
For $A=[0,1)$ in the usual <real line>, $\overline A=[0,1]$: every <neighbourhood> of $1$ meets $A$, while a point outside $[0,1]$ has an <open ball> disjoint from $A$. Passing to the <closure> adds the missing endpoint and retains all original points, including <isolated points> when present. In general, the <closure> is $A$ together with its <limit points>; its individual points are the <adherent points> of $A$.
The <closure> turns an arbitrary <subset> into a <closed set> with no unnecessary added points. It satisfies $A\subseteq\overline A$ and $\overline{\overline A}=\overline A$; a <subset> is <closed> exactly when $A=\overline A$. In a <metric space>, $x\in\overline A$ exactly when some <sequence> of points of $A$ has <limit of a sequence> $x$: choose its $n$th point in $A\cap B(x,1/n)$. For a general <topological space>, keep the <neighbourhood> definition; <sequences> need not detect every point of the <closure>.
Back to article page
