Can someone explain me why the set
$A:=\{(x,y)\in \mathbb R^2: x\geq 0\}$
is not a submanifold?
I also got another (easy) question:
In our lecture we are always talking about submanifolds. We never defined the meaning of manifolds. So what is there difference? Is there even a difference?
Thanks
On
It's a manifold with boundary. https://en.m.wikipedia.org/wiki/Manifold#Manifold_with_boundary
Minor distinction.
A $d$-dimensional topological manifold is simply any Hausdorff space in which each point has a neighborhood that is homeomorphic to the open unit ball in $\mathbb R^d$.
(Your $A$ does not have that property, because no neighborhood of $(0,0)$ is homeomorphic to an open disk. It may not be entirely obvious how to prove that, but one way would be that the open disk has the property that removing any single point from it will make it not simply connected, whereas for every simply connected neighborhood of $(0,0)$ in $A$ we can remove $(0,0)$ itself without losing its simple connectedness).
That's pretty simple -- but often we want to speak of properties that the simple definition of "topological manifold" cannot guarantee. For example, tangent spaces or curvature don't make much sense for topological manifolds in general. Working with these things requires us to go to differentiable manifolds, smooth manifolds, or perhaps Riemannian manifolds, and the accepted definitions of those concepts are fairly abstract and are often hard to penetrate for newcomers.
It turns out to be a lot easier and more intuitive to work with differentiable (or smooth) submanifolds of $\mathbb R^k$. The definitions can be made much simpler simpler that way, and the intuitive meaning of many of the concepts is clearer. There are some technical downsides to this too, but they are minor enough that many educators feel that this makes it worthwhile to develop as much of the theory as one can in this more-or-less familiar setting before expecting students to handle the more abstract case of a general differentiable manifold.