By general manifold I mean Hausdorff differential manifold not necessarily second-countable. By standard manifold I mean Hausdorff, second-countable differential manifold.

So my question is, we have an injective immersion, ie with a non singular derivative, $\iota\colon M\to N$. Where $M$ is a connected general manifold and $N$ is a standard manifold. Is it true that $M$ is second-countable.

Note that if we take away the connection hypothesis then it is false. If we consider $M$ to be $(0,1)\times (0,1)$ with the horizontal topology, ie open basis of the form ${a}\times U$ with U open set of $(0,1)$; $N$ to be $(0,1)\times (0,1)$ with the standard topology, and $\iota$ to be the identity then it is false.

If we take away the injective hypothesis I think it is also false. I'm not really sure if it can be done, but if it is possible to wind up the long line around $\mathbb S^1$ similar to how we do with $\mathbb R$ it would be a counterexample.

I've been told it can be done with Riemann manifolds. Something like: $N$ has a riemann metric because it is second-countable, $\iota(M)$ has a riemann metric induced by $N$, taking it back to $M$ by $\iota$, you get a metric in $M$ so it has to be second-countable. However I don't know anything about Riemann Geometry so I'd prefer a pure topological/differential proof.

I'm interested because it allows you in the proof of the Frobenius theorem not to check about second-countable in the manifold you obtain.