I'm interested in the dual question to:
[continuous images of open intervals](https://mathoverflow.net/questions/168084/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?



Which topological spaces admit embeddings into Euclidean spaces
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](https://wrbrian.files.wordpress.com/2012/01/neight2.pdf) 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.)


