For this question a manifold-with-boundary is a topological space which is Hausdorff and locally upper-Euclidean. Every metrizable manifold-with-boundary has a collared boundary, as shown in "Locally flat imbeddings of topological manifolds", Morton Brown, 1962. Let $M$ be a *non-metrizable* manifold-with-boundary. Does $M$ have a collared boundary?



Collared boundary of a non-metrizable manifold
A nice recent reference for questions about non-metrisable manifolds is David Gauld's book aptly named "non-metrisable manifolds". For instance it is shown that any metrisable component of the boundary of a manifold (metrisable or not) is collared (Corollary 3.11 on page 44, it is an almost immediate consequence of R. Connely proof of M. Brown's result). 


This is not true if the component is non-metrisable: Example 1.29 on p. 16 (which is originally due to P. Nyikos) gives a description of a manifold whose interior is $\mathbb{R}^2$ and whose boundary is the open long ray $\mathbb{L}\_{+}$. This boundary component cannot be collared because there is no embedding sending $\mathbb{L}\_{+}\times[0,1]$ into the manifold.


It is not clear to me whether the usual definition of `collared' is only for connected components of the boundary or if there is a global one asking for an embedding from $\partial M\times[0,1]$ into the manifold $M$. In that case a simpler counter example is the Prüfer surface (dating back to Rado), which is also detailed in D. Gauld's book in Example 1.25.


**EDIT:** Comments by the OP (and silly ones by myself) made me realize that something was a bit unclear in the statement of Theorem 3.10 in Gauld's book, from which Corollary 3.11 follows. Here are some details.


Theorem 3.10 states that if a closed subset $B$ of a Hausdorff space $X$ is locally collared and strongly paracompact, then $B$ is collared. But actually, what is needed is a bit stronger: that $B$ is strongly paracompact **in $X$**, that is, given a cover $\mathcal{U}$ of $B$ by open sets of $X$, there is another cover $\mathcal{V}$ of $B$ by open sets of $X$ such that each member of $\mathcal{V}$ is contained in a member of $\mathcal{U}$ and $\mathcal{V}$ is star-finite **in $X$**. (Star-finite means each member intersects finitely many members.) I actually asked D. Gauld about it, and he agreed that he has been a little bit careless in the statement.


This requirement is stronger, since for instance any manifold whose components are metrisable is strongly paracompact. The boundary $\partial P$ of the Prüfer surface $P$ is a discrete union of continuum many real lines and hence is strongly paracompact (in itself, so to say). But $\partial P$ is *not* strongly paracompact in $P$, and actually $\partial P$ is not collared in $P$.


A consequence of Theorem 3.10 (amended) is that if the boundary $\partial M$ of a manifold $M$ is made of (at most) countably many metrisable components, then it is collared. The proof is by using the fact that Lindelöfness is equivalent to metrisability and strong paracompactness for connected manifolds to cover $\partial M$ by countably many euclidean sets whose union yields a strongly paracompact submanifold of $M$ also having $\partial M$ as a boundary.


