this is the exercice 2-2 of Do Carmo's page 67
1- is the set $\{(x,y,z)\in \mathbb{R}^3 ; \;z=0 \text{ and } \;x^2+y^2\leq 1 \}$ a regular surface ?
2-is the set $\{(x,y,z)\in \mathbb{R}^3 ; \;z=0 \text{ and }\;x^2+y^2<1 \}$ a regular surface ?
my answer
1- the closed disk is not a regular surface,if it's the case the closed disk wich is compacte in $\mathbb{R}^3 $ is homeomorphic to an open set of $\mathbb{R}^2$ and this is contradiction
2- the open disk is a regular surface because the paramtrisation $X$from the open uniatry disk from $\mathbb{R}^2$ to the open disk in $\mathbb{R}^3 $ verify the definition of a regular sufraces $(x(u,v)=(u,v,0))$
For the second i 'am sur but for my first answer i have some doubt what do you think ?
You are correct that the closed unit disk is not a regular surface (aka a 2-manifold), but the reasoning is not quite correct. For example, the 2-sphere (embedded in $\mathbb{R}^3$, if you like) is a regular surface which is not homeomorphic to an open subset of $\mathbb{R}^2$ due to the compactness obstruction that you mention.
To argue that the closed disk is not a regular surface you will have to find a point in it around which there cannot possibly be a neighbourhood homeomorphic to an open subset of $\mathbb{R}^2$. Given what you know about the open disk (namely, that it is a regular surface), this point will clearly have to be a boundary point.
For your reference, the closed disk is an example of an object called a "manifold with boundary" (which is exactly what it sounds like). The charts on such spaces are homeomorphisms from (relatively) open sets of the space to (relatively) open subsets of $\mathbb{H}^n$ (the closed upper half plane).
Good luck!