= Interior
{disambiguate=topology}
{title2=$A^\circ=\operatorname{int}_X A$}
{wiki}
= Interior
{synonym}
= Interior of a set
{synonym}
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 $A^\circ=\operatorname{int}_X A$ 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 $A$. https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/mit18_100af20_basic_analysis.pdf[Jiří Lebl, Basic Analysis, Definition 7.2.23]{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>, $\operatorname{int}_X X=X$, whereas $\operatorname{int}_{\mathbb R}X=(0,1)$. 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 $\operatorname{int}_X A=X\setminus\operatorname{cl}_X(X\setminus A)$ links <interior> directly to <closure>.
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