Collar neighbourhood

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

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

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