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.



When is a manifold boundary a deformation retract of its open neighborhood?
The following theorem and example give (partial) answers. By an *$n$-manifold* we understand a Hausdorff space whose any point has a neighborhood, homeomorphic to an open subspace of $\mathbb R^n\_+=\{(x\_1,\dots,x\_n)\in\mathbb R^n:x\_n\ge 0\}$.


**Theorem.** For each 1-manifold $M$ the boundary $\partial M$ is a deformation retract of some open set $U\subseteq M$ that contains $\partial M$.


*Proof.* By the definition of a 1-manifold, for every $x\in\partial M$ there exists a neighborhood $U\_x\subseteq M$, homeomorphic to $[0,1)$. Fix a homeomorphism $h\_x:U\_x\to[0,1)$ such that $h\_x(x)=0$. It can be shown that for distinct $x,y\in\partial M$ the sets $U\_x,U\_y$ are disjoint. Then $U=\bigcup\_{x\in \partial M}U\_x$ is an open neighborhood of $\partial M$ and the function $H:U\times [0,1]\to U$, $H:(u,t)=h\_x^{-1}(t\cdot h\_x(u))$ where $u\in U\_x$, is a deformation retraction of $U$ onto $\partial M$.


**Example.** There exists a separable 2-manifold $M$ whose boundary $\partial M$ is homeomorphic to $\mathbb R\times\mathfrak c$. Being non-separable, the boundary $\partial M$ cannot be a retract of a (necessarily separable) neighborhood $U$ of $\partial M$ in $M$.


This example is described by Peter Nyikos as Example 3.6 of his survey paper "[The theory of nonmetrizable manifolds](https://www.sciencedirect.com/science/article/pii/B9780444865809500173?via%3Dihub)" in the Handbook of Set-Theoretic Topology.


