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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.

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  • $\begingroup$ I can't find a reference to " locally upper-Euclidean"; what does this mean? Do you mean locally homeomorphic to the upper-half space? $\endgroup$ – AIM_BLB Mar 25 at 9:50
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    $\begingroup$ 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. $\endgroup$ – kaba Mar 25 at 13:12
  • $\begingroup$ Thanks kaba, just wanted to be sure I wasn't missing something important :) $\endgroup$ – AIM_BLB Mar 25 at 13:15

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