Collar neighbourhood

In topology, a branch of mathematics, a collar neighbourhood of a manifold with boundary $M$ is a neighbourhood of its boundary $M$ that has the same structure as $\partial M\times [0,1)$ .

Formally if $M$ is a differentiable manifold with boundary, $U\subset M$ is a collar neighbourhood of $M$ whenever there is a diffeomorphism $f:\partial M\times [0,1)\to U$ such that for every $x\in \partial M$ , $f(x,0)=x$ .^[1]^{: p. 222} Every differentiable manifold has a collar neighbourhood.^[1]^{: th. 9.25}

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

References

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

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[Lee_1988-1] Lee, John (2012), Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218, Springer, ISBN 9781441999825

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