Collar neighbourhood

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

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

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

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