# Is there an embedding theorem for non-second-countable manifolds?

It is well known although I don't know how to prove that any second-countable topological manifold of dimension $$n$$ can be embedded into $$\mathbb{R}^{2n}$$(we consider Hausdorff manifolds only). I wonder if there is a similar result, whether positive or negative, for non-second-countable manifolds? Of course there are some very strange non-second-countable manifolds like Prüfer surface which I don't want to count in, so I quote a definition from Handbook of Set-theoretic Topology:

Chapter 14, Definition 5.1: A space is $$\omega$$-bounded if every countable subset has compact closure.

Denote the long line by $$L$$. Many basic examples of non-second-countable manifolds, such as $$L$$ and $$L\times L$$ are $$\omega$$-bounded. In the same chapter the structure theorem for $$\omega$$-bounded surfaces(Theorem 5.14, "The Bagpipe Theorem") is proved. So it seems that $$\omega$$-boundedness is a suitable condition.

Question: Is there a "nice" space(a finite dimensional $$\omega$$-bounded manifold, if possible) such that every $$\omega$$-bounded manifold of dimension $$n$$ can be embedded into it?

I have proved that the space obtained by gluing the diagonal line and $$x$$-axis of the first octant of $$L\times L$$ cannot be embedded into $$L^{n}$$ for any $$n$$, so possibly $$L^{n}$$ does not work(but the proof is ugly and may be flawed).