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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?

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    $\begingroup$ There is ambiguity between the title question and body question. (“Embedding” usually means that $f$ is a homeomorphism $X\to f(X)$ with subspace topology on the latter.) $\endgroup$
    – Francois Ziegler
    Commented Apr 14, 2020 at 13:09
  • $\begingroup$ Another thing which should be clarified is whether $d$ represents finite number or whether products of infinitely many copies are allowed, too. $\endgroup$
    – Martin Sleziak
    Commented Apr 14, 2020 at 13:10
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    $\begingroup$ @MartinSleziak should be finite. $\endgroup$
    – ABIM
    Commented Apr 14, 2020 at 13:14
  • $\begingroup$ It's fine I'll accept the answer but would be open to other posts (purely for scientific interest) $\endgroup$
    – ABIM
    Commented Apr 14, 2020 at 13:52
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    $\begingroup$ @New_Topologist_On_The_Block Then I suggest changing the question & title question back to “embedding” so that it matches the answer you accepted. $\endgroup$
    – Francois Ziegler
    Commented Apr 14, 2020 at 14:21

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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.)

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  • $\begingroup$ Note that the question was originally about the existence of a continuous injection, which is a priori weaker (for non-separable metrizable spaces it's weaker, just considering an uncountable discrete space of at most continuum cardinal). (But the question might be amended retroactively...) $\endgroup$
    – YCor
    Commented Apr 14, 2020 at 14:28

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