Manifold with boundary:
An $n$-dimensional manifold with boundary is a second countable Hausdorff space in which every point has a neighborhood homeomorphic either to an open subset of $\mathbb{R^n}$, or to an open subset of $\mathbb{H^n}$.
Neighborhood:
If $X$ is a topological space and $p$ is a point in $X$, a neighborhood of $p$ is a subset $V$ of $X$ that includes an open set $U$ containing $p$.
Can someone please exhibit for me the following triple:
- a manifold $M$ with non empty boundary,
- a point $x$ of the boundary $\partial M$ of $M$,
- a neighborhood of $x$ that is not $M$ itself.
I'm probably confused between topological and manifold boundaries, but it seems to me the above does not exist. I'm somewhat convinced the only open subset of $M$ containing $x$ is $M$ itself.
But this also seems absurd as every boundary chart would have to be a global homeomorphism. Hence defying the whole "locally xxx" stuffs of manifolds.
EDIT: Please also be explicit the topology your are using to define a neighborhood, as for in the first answer below I do not believe the defined set to be open...
If $M=D$ the closed unit disk, and $x$ is on the boundary of $D$, then if $U$ is any open disk centered at $x$, then $U\cap D$ is an open neighborhood of $x$ in the sense of $D$. (Neighborhoods like this are homeomorphic to half-spaces.)