Interior (topology, $A^{\circ }={\mathrm{int}}_{X}A$)

The interior of a subset $A$ of a topological space $X$ is the union of all open sets contained in $A$, equivalently the largest open subset of $A$. A point lies in $A^{\circ }={\mathrm{int}}_{X}A$ exactly when it has an open neighbourhood contained in $A$. In a metric space, this says that some open ball $B(x,r)$, with $r>0$, is contained in $A$. The interior therefore identifies the points with room to move locally while staying in $A$. ^[ref]

For the half-open interval $A=[0,1)$ in the real line, $A^{\circ }=(0,1)$. Every point strictly between the endpoints has a sufficiently small open ball inside $A$, but every open ball about zero also contains negative points. Although $0\in A$, it is not in the interior; it is a boundary point instead. Together, the interior and closure bracket the original subset: $A^{\circ }\subseteq A\subseteq \overline{A}$.

The ambient topological space matters: for $X=[0,1]$ with its subspace topology, ${\mathrm{int}}_{X}X=X$, whereas ${\mathrm{int}}_{\mathbb{R}}X=(0,1)$. The diagram accompanying boundary of a set illustrates the relative open ball at zero. A subset is open exactly when $A=A^{\circ }$, and taking the interior twice has the same effect as taking it once. The complement identity ${\mathrm{int}}_{X}A=X\setminus {\mathrm{cl}}_X(X\setminus A)$ links interior directly to closure.