Which topological spaces admit embeddings into Euclidean spaces I'm interested in the dual question to:
continuous images of open intervals, about surjections onto open intervals.
Namely, if $X$ is a topological space, when can we guarantee that there exists a topological embedding of $X$ into some Euclidean space?
 A: There's an old theorem of Deák that gives an interesting characterization. 
Given a topological space $X$, define a relation on subsets of $X$ as follows: we write $U \sqsubseteq V$ if and only if $\overline{U} \subseteq V$.
Theorem (Deák): A separable metrizable space is homeomorphic to a subset of $\mathbb R^n$ if and only if its topology has a subbasis generated by $\leq n+1$ collections of open sets, each totally ordered by $\sqsubseteq$. 
For example, to generate the topology of $\mathbb R^2$ with $3$ collections of this kind, think of how $3$ families of open half-planes can be used to form a small open triangle around every point of the plane.
This theorem appears in

J. Deák, "A new characterization of the class of subspaces of a Euclidean space," Studia Sci. Math. Hungar. 11 (1980), pp. 253-258.

I don't have a link to this paper, but the result is discussed in section 2 of this paper, which contains a few other related things as well.
My favorite part of this theorem is that by changing it a little, one arrives at an interesting (to me) conjecture:
Conjecture: A separable metrizable space is homeomorphic to a subset of $\mathbb R^n$ if and only if its topology has a subbasis generated by $\leq n+1$ collections of open sets, each totally ordered by $\subseteq$. 
The conjecture is true for $n=1$, but I don't think it's known for larger $n$. (Does the Klein bottle have a subbasis generated by $4$ nested collections of open sets? Even this special case does not seem trivial to me.)
