Collar neighbourhood

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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 M} that has the same structure as  M × [ 0 , 1 ) {\displaystyle 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} .[1]: p. 222 Every differentiable manifold has a collar neighbourhood.[1]: th. 9.25

Formally if  M {\displaystyle M} is a topological manifold with boundary,  U ⊂ M {\displaystyle U\subset M} is a collar neighbourhood of  M {\displaystyle M} whenever there is an homeomorphism 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} .

## References

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1. ^ a b Lee, John (2012), Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218, Springer, ISBN 9781441999825

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