I read that a map is 'visually' a homeomorphism if you don't have to fold or tear the object. Thus, I was wondering what the problem with folding is? I guess that in this statement they don't assume the classical procedure of folding, but more or less a superposition where you merge different parts of the object?
Also, I have a question regarding this homeomorphism where the donut is transformed into a mug. See here
Why is taking the volume out of the filled mug different from reducing a line to a cylinder?
From my experience I would say that a line is not homeomorphic to a cylinder, but it is also just some sort of extension, as is filling the mug or taking some volume out of it?
EDIT: In the comments it was suggested that a mug that is closed at the bottom and at the top is homeomorphic to a mug that is just closed at the bottom.
Now if you think about this in a plane, then the two things are clearly different, as the first one has a trivial fundamental group and the latter one a fundamental group that is given by the integers(due to homeomorphy to the circle). Thus, I still don't understand why moving an additional layer up to the top is a homeomorphism?
Here we go, a post with an image. I'll keep all of my comments included in here to avoid having many cross references.
Firstly, folding, as I said in my comment the map on the box
$$\{(x,y)\in\Bbb R^2 : \lVert (x,y)\rVert_\infty\le 1\}$$
can be folded in half via
$$(x,y)\mapsto (x,|y|)$$
which is not injective, hence not a homeomorphism.
For the mug to donut question, we have the picture of the mug. As they mean it, even though visually you cannot see the hollow inside. This is depicted below in a cross-section, so you can see the empty space. Also, I've colored the corresponding part you're having issues with red so it is more explicit where things are going and which part of the bottom becomes the top. Notice there is just more empty space in the middle, so we're inflating it, not really "filling" it with anything.
Notice how the first mug has the empty space between the bottom of the inside and the bottom of the outside. It's that part that gets lifted up to the top of the mug, NOT a whole new cap added.
Additionally, if you have an actual cap added, you get the same old object, only with a sphere glued on, as illustrated here