Manifold with boundary

See @baez1994gauge page 115.
We define a n-dimensional manifold with boundary to be a topological space M equipped with charts (in the sense of a topological manifold) of the form

φα:Uα→Rn

or

φα:Uα→Hn,

where$$
H^n = {(x^1, ..., x^n): x^n \ge 0}.

andwhere$Uα$areopensetscovering$M$,suchthatthetransitionfunction$φα∘φβ−1$issmoothwhereitisdefined.(Wealsoassumesometechnicalconditions,namelythat$M$isHausdorffandparacompact.)Notethataplainold[[01CONCEPTS/manifold∥manifold]]isautomaticallyamanifoldwithboundary.Wehavetoworryabitaboutthefactthatwehavenotyetdefinedwhatitmeansforafunctionon$Hn$tobesmooth!Wewantsuchfunctionstobesmooth′uptoandincludingtheboundary′.Perhapsthesimplestwaytosaythisisthatafunctionon$Hn$issmoothifitextendstoasmoothfunctiononthemanifold

{(x^1, ..., x^n): x^n > -\epsilon}

forsome$ϵ>0$.Here′sacorrectedandmorepolishedversionofyourtext:If$M$isamanifoldwithboundary,wedefinetheboundaryof$M$tobethesetofpoints$p∈M$suchthatthereexistsachart$φ:Ua→Hn$mapping$p$toapointintheboundaryof$Hn$:

\begin{align*}
\partial H^n &= {(x^1, \dots, x^n) \in \mathbb{R}^n \mid x^n = 0}.
\end

Wewrite$∂M$fortheboundaryof$M$.