Collar neighbourhood

In topology, a branch of mathematics, a collar neighbourhood of a manifold with boundary ${\displaystyle M}$ is a neighbourhood of its boundary ${\displaystyle \partial M}$ 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} Since ${\displaystyle [0,1)}$ is diffeomorphic to ${\displaystyle [0,\infty )}$, it is equivalent to take a diffeomorphism ${\displaystyle f:\partial M\times [0,\infty )\to U}$.^[2]: §6

Every differentiable manifold has a collar neighbourhood.^{[1]: Th. 9.25 [2]: Th. 4.6.1}