Collar neighbourhood

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

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 ${\displaystyle x\in \partial M}$, ${\displaystyle f(x,0)=x}$.