Is a connected separable locally euclidean Hausdorff topological space second countable? 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 $k\in\mathbb N$ .
 A: I answer my our question: Separability of a connected locally euclidean Hausdorff topological space does not imply second countability, or any of the equivalent conditions (1), ... (6) given in the question. A counterexample is given in Example 5 on page 15 in 
David Gauld's preprint. There is constructed a separable Hausdorff topological space, which is not second countable. The space can be equipped with a compatible analytic atlas modelled on $\mathbb R^2$ . One such is $\lbrace{\rm id\ }S\rbrace\cup\lbrace\phi_{\eta,\zeta}:\eta,\zeta\in\mathbb R\rbrace$ , where $\phi_{\eta,\zeta}$ is given by $(0,\eta,z)\mapsto(0,z-\zeta)=(0,v)$ when $|v|<1$ , and $(x,y)\mapsto(x,|x|^{-1}(y-\eta)-\zeta)=(u,v)$ when $0<|x|<1$ and $|v|<1$ . So Gauld's space does not satisfy any of the conditions (1), ... (6).
A: Indeed, check the paper by Gauld. Your (4) implies his condition hemicompact.
His example at p15, that you saw refutes the just separable condition (7). Note that (1)-(6) imply imply metrisability for just continuous manifolds, so it still might be that the situation vis à vis separability is different for smooth manifolds instead of continuous ones, though I suspect not.
A: See my article on nonmetrizable manifolds in the Handbook of Set-Theoretic Topology. Examples 3.7 and 3.9 include separable, nonmetrizable manifolds; neither is due to me and the second is due to R.L. Moore in a 1942 paper. After reading the descriptions, you might try going back to Gauld's smooth example.
