martinbn said:
May be, depends on what you mean by "the points". Do you mean all points i.e. the whole manifold? Then, no, it doesn't have to be homeomorphic to ##\mathbb R^n##.
Please see below.
lavinia said:
yes but what do you mean by a boundary?
Well, I mean something that is in agreement with our notion of a end point or a barrier, for familiar geometrical objects, like the Earth, a sphere, a disk. At the same time, this definition has to be correct for any kind of manifold we want to apply it. So, given
a manifold ##M## and a point ##x \in M##,
an open subset ##U## of ##M## containing ##x##;
Suppose we can find an homeomorphism ##\Psi: U \rightarrow V \subseteq \mathbb{R}^n##. If this can happen for every ##x \in M## we say ##M## have no boundaries.
For the case when it's possible to define only a half region around ##x## where there are a homeomorphism as above, then we say ##x## is in the boundary of ##M##. Thus ##M## have a boundary. All points on the boundary will get mapped to ##y \in \mathbb{R}^n = (0,y_1,...,y_n)## in the
local mapping.
We see that by this way we can recover our notions of a boundary for the objects that I mentioned above (The sphere, etc...)
I've constructed a diagram showing these concepts for the case of the closed 1-ball, namely the disk. I'm not sure if that diagram is actually correct in everything, however.
I wrote some notes in my language, not in English. So I'm going to translate it
1 - Here we see the disk from far away. The disk consists of all points that lie within the red circle.
2 - Now we zoom into the border of the disk and we see something like this
3 - Here is the result of the map of several points of the border of the disk in ##\mathbb{H}^2##
Of course objects have physical limitations, so the above diagram is only a representation.
