TITLE: Is a connected separable locally euclidean Hausdorff topological space second countable?
QUESTION [6 upvotes]: 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 $k\in\mathbb N$ .

REPLY [3 votes]: 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.