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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).

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I think the answer is simple: recall that each Tychonoff space can be embedded in a Tychonoff cube of the same weight. Less simple case concerns closed embeddings into powers of the real line. Spaces admitting such embeddings are exactly realcompact spaces. To them is devoted a small section 3.11 of Engelking’s “General topology” (2nd edn.). I attached its first page below.

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    $\begingroup$ Thank you very much. I knew this cube and by "nice" I actually mean finite dimensional space...I will edit my question. $\endgroup$ – 1830rbc03 Jan 22 at 5:49

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