This question arose from considering for a connected smooth Hausdorff manifold the (possible) equivalence of the following properties:

(1) paracompact,

(2) metrizable,

(3) second countable,

(4) countable at infinity,

(5) $\sigma$−compact,

(6) Lindelöf,

(7) separable.

I know proofs for the equivalence of the first six, and that they imply (7), but it is problematic, whether this implies the others. By *countable at infinity* I mean existence of a sequence of compact sets $K_i$ whose union covers the space and which satisfies $K_i\subseteq{\rm Int\ }K_{i+1}$ . Of course, *locally euclidean* means that each point has a neighbourhood homeomorphic to $\mathbb R^k$ with the standard topology and $k\in\mathbb N$ .

separable but not metrizable. I have to think it through carefully. $\endgroup$