Collar neighbourhood

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

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 $$x \in \partial M$$, $$f (x, 0) = x$$. Every differentiable manifold has a collar neighbourhood.

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