= Topological manifold with boundary
{title2=$\mathbb H^n=\{x\in\mathbb R^n:x_n\geq0\}$}
For $n\geq1$, 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.<Topology I: smooth manifolds, University of Toronto MAT1300HF, 2012/Week 2 course notes>[Topology I: smooth manifolds, Week 2, Section 1.3, equation (21), Definition 1.21 and Remark 1.24, page 7]{ref}
The <closed interval> $[0,1]$ illustrates the distinction: its interior points have <open interval> charts, while $0$ and $1$ have charts modeled on $[0,\varepsilon)\subseteq\mathbb H^1$, 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.
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