Collar neighbourhood

Template:One source In topology, a branch of mathematics, a collar neighbourhood of a manifold with boundary $M$ is a neighbourhood of its boundary $\partial M$ 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 $x\in \partial M$, $f(x,0)=x$.Template:Rp Since $[0,1)$ is diffeomorphic to $[0,\mathrm{\infty })$, it is equivalent to take a diffeomorphism $f:\partial M\times [0,\mathrm{\infty })\to U$.Template:Rp

Every differentiable manifold has a collar neighbourhood.Template:RpTemplate:Rp

References

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