Continuous injections of nice spaces into $\ell^2$

Question

If X is a separable, first countable Hausdorff space, then does there exist a continuous map from X to $\ell^2$?

Intuitively put, can you always fill in the holes of nice spaces?

Answer 1

Let $X$ be the Double Arrow space, $[0,1] \times \{0,1\}$ in the lexicographic order topology. Then $X$ is a non-metrisable, compact, first-countable, separable, Hausdorff (even hereditarily normal, being an ordered topological space) and if a 1-1 continuous function $f: X \to Y$ existed where $Y$ is metrisable, then $f$ would be a homeomorphism (as $Y$ is Hausdorff and so $f$ is closed etc.) and hence metrisable too. So from $X$ there can be no injective map into $\ell^2$.