In topology, a branch of mathematics, a collar neighbourhood of a manifold with boundary M {\displaystyle M} is a neighbourhood of its boundary ∂ M {\displaystyle \partial M} that has the same structure as ∂ M × [ 0 , 1 ) {\displaystyle \partial M\times [0,1)} . Formally, if M {\displaystyle M} is a differentiable manifold with boundary, U ⊂ M {\displaystyle U\subset M} is a collar neighbourhood of M {\displaystyle M} whenever there is a diffeomorphism f : ∂ M × [ 0 , 1 ) → U {\displaystyle f:\partial M\times [0,1)\to U} such that for every x ∈ ∂ M {\displaystyle x\in \partial M} , f ( x , 0 ) = x {\displaystyle f(x,0)=x} . Since [ 0 , 1 ) {\displaystyle [0,1)} is diffeomorphic to [ 0 , ∞ ) {\displaystyle [0,\infty )} , it is equivalent to take a diffeomorphism f : ∂ M × [ 0 , ∞ ) → U {\displaystyle f:\partial M\times [0,\infty )\to U} .
Every differentiable manifold has a collar neighbourhood.
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