# When is a manifold boundary a deformation retract of its open neighborhood?

For this question a manifold is a locally upper-Euclidean Hausdorff space; paracompactness or second-countability is not assumed, and boundary may be present.

Let $$M$$ be a manifold. What are sufficient conditions for there to exist open $$U \subset M$$ such that $$\partial M \subset U$$ and $$U$$ (strongly?) deformation retracts onto $$\partial M$$? (Could this be true for every manifold?)

Clearly $$\partial M$$ being collared in $$M$$ is a fairly general sufficient condition (this includes paracompact manifolds). However, I'd be interested in whether there is a more general sufficient condition than this.

• I can't find a reference to " locally upper-Euclidean"; what does this mean? Do you mean locally homeomorphic to the upper-half space? – AIM_BLB Mar 25 at 9:50
• Yes. For each point there exists an open neighborhood homeomorphic to an open subset of $\mathbb{R}^{n - 1} \times \mathbb{R}_+$, where $\mathbb{R}_+ = \{x \in \mathbb{R} : x \geq 0\}$, i.e. $M$ is a manifold-with-boundary but not necessarily metrizable. – kaba Mar 25 at 13:12
• Thanks kaba, just wanted to be sure I wasn't missing something important :) – AIM_BLB Mar 25 at 13:15