For smooth $n$-manifolds, we know that they can always be embedded in $\mathbb R^{2n}$ via a differentiable map.  However, is there any corresponding theorem for the topological category?  (i.e. Can every topological manifold embed continuously into some $\mathbb R^N$, and do we get the same bound for $N$?)
I'm not sure about $\mathbb{R}^{2n}$, but you can embed them in $\mathbb{R}^{2n+1}$ using dimension theory.  The theorem is that every compact metric space whose covering dimension is $n$ can be embedded in $\mathbb{R}^{2n+1}$.  The example of non-planar graphs (which are $1$-dimensional) shows that this is the best you can do in general.

The classic source for this is Hurewitz-Wallman's beautiful book "Dimension Theory", which I recall being pretty readable to me when I was an undergraduate, though I haven't looked at it in a while.  There is also a nice discussion of this in Munkres's book on point-set topology -- when I last taught a point-set topology class, I used this as one of the capstone theorems in the course.
