Some definitions: Manifold:  A manifold is an object defined by a set of overlapping charts (called an 'atlas'), which are functions from real Euclidean space into the manifold (as a set of points). Every point of an n-manifold must have some neighborhood homeomorphic to an open ball in Rn. In topology, we say two manifolds are the same if there exists a homeomorphism between them (a continuous bijection having continuous inverse). So to a topologist, a donut is no different than a coffee cup!
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