Topological manifold with boundary (${\mathbb{H}}^n=\{x\in {\mathbb{R}}^n:x_n\ge 0\}$)

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.

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.