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A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.
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uniqueness of regular/tubular neighborhood with equivariant boundary
Let's consider the smooth case. The basic idea is to solve the problem on the free quotient, and then to lift the solution by covering space theory. For the external reference below, I'm going to ass …