>>11081636I might have as well explained what it is.
manifold = a space which locally looks like a portion of the Euclidean space. surface of the Earth is a manifold. the universe is probably a manifold.
manifold with boundary = there is an edge. for example a closed interval, a closed square or a closed disc are manifolds with boundary. at the points of the boundary the manifold doesn't look like an Euclidean space, but rather like an half-space.
compact manifold = it doesn't stretch into infinity, but also in the sense that there are no places where you can get infinitely close somewhere but never reach it. for example the plane is non-compact, but an open interval is also non-compact, because of the missing endpoints.
compact manifold without boundary = no physical edges and also no "infinite edges". if you keep walking for a very long time, you must eventually start walking in circles, since you won't reach an edge, but you cannot escape to infinity either.
compact manifolds without boundary are the nicest possible spaces. sphere, torus, surface with genus g all classify as such. a plane does not, because it's not compact. a square does not, because it has a bounday (or is not compact if the boundary is removed).
