Collar neighbourhood explained

M is a neighbourhood of its boundary M that has the same structure as \partial M x [ 0 , 1 ) .

Formally if

M is a differentiable manifold with boundary, U \subset M is a collar neighbourhood of M whenever there is a diffeomorphism f : \partial M x [ 0 , 1 ) \to U such that for every x \in \partial M , f (x, 0 ) = x .Every differentiable manifold has a collar neighbourhood.

Formally if

M is a topological manifold with boundary, U \subset M is a collar neighbourhood of M whenever there is an homeomorphism f : \partial M x [ 0 , 1 ) \to U such that for every x \in \partial M , f (x, 0 ) = x