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The boundary of a planar domain $D$ is its boundary of a set. It is distinct from the boundary of a parametrizing interval: a curve can begin on $\partial D$ while all its positive-time points lie inside $D$.

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.

The manifold boundary of a topological manifold with boundary consists of the points which, in a half-space coordinate chart, have last coordinate zero. These are precisely the points without a neighbourhood homeomorphic to an open set in the full Euclidean space of the same dimension. It is an intrinsic property, independent of the coordinate chart. It differs from the boundary of a set in an ambient topological space: a circle has empty manifold boundary, but its boundary of a set in the Euclidean plane is the entire circle.

For $n\ge 1$, a topological $n$-manifold with boundary is a Hausdorff second-countable space locally homeomorphic to relatively open sets in the half-space ${\mathbb{H}}^n$. The openness here is in the subspace topology of ${\mathbb{H}}^n$, not necessarily in ${\mathbb{R}}^n$. Its intrinsic boundary $\partial M$ consists of points sent to $x_n=0$; the remaining points have ordinary coordinate charts with images open in Euclidean space. The intrinsic boundary is independent of the chosen chart.^[ref]

The closed interval $[0,1]$ illustrates the distinction: its interior points have open interval charts, while $0$ and $1$ have charts modeled on $[0,\epsilon )\subseteq {\mathbb{H}}^1$, using $x$ and $1-x$. Its intrinsic boundary is $\{0,1\}$. The circle has empty intrinsic boundary even though its boundary of a set as a subset of the Euclidean plane is the whole circle. A topological manifold without boundary is therefore a special case with $\partial M=\varnothing$, and a smooth atlas on the half-space charts gives a manifold with boundary in the smooth sense.