Concerning topological manifolds: Are paracompact and connected locally euclidian Hausdorff spaces always second-countable?

There are different definitions for topological manifolds, sometimes second-countability or paracompactness are added to being locally euclidian Hausdorff. (Sometimes Hausdorff is also left out, but I won't do that here.) The two additional conditions lead to two new definitions not equivalent to the one without, for example the long line is locally euclidian Hausdorff, but neither second-countable nor paracompact. (Six examples for the independence of the conditions when second-countable is added can be found here.) I was wondering if the two new definitions (secound-countable locally euclidian Hausdorff as well as paracompact locally euclidian Hausdorff) are equivalent? We indeed have an implication:

Lemma 1: Second-countable and locally compact spaces are $$\sigma$$-compact (See here).

Lemma 2: Locally compact and $$\sigma$$-compact spaces are paracompact (See here).

For a second-countable locally euclidian Hausdorff space, we have: It is locally euclidian and therefore locally compact. According to lemma 1 it is $$\sigma$$-compact and according to lemma 2 it is paracompact.

Can the backwards direction, that paracompact locally euclidian Hausdorff spaces are second-countable also be proven or can someone give a counterexample?

Since Sassatelli Guilio has provided a counterexample to this, but it is not connected: What about when connectedness is added?

• The fact is that the disjoint union of any family of paracompact spaces is paracompact, so you can take $\coprod_{i\in \Bbb \aleph_1} X_i$ with $X_i=\Bbb R^n$ and this is a paracompact, locally Euclidean metric space, but not second countable. Jul 18, 2022 at 13:29
• I don't know of a connected example, and apparently neither does pi base (though it may be due to the fact that the database does not have its spaces tagged by local Euclideanity). Jul 18, 2022 at 13:46
• Thanks for the counterexample, I have included connectedness as a new question. Jul 18, 2022 at 15:04
• See my answer at mathoverflow.net/a/237/75. Jul 18, 2022 at 15:12

1. A connected, locally compact, paracompact Hausdorff space is $$\sigma$$-compact. See here.
2. Let $$X$$ be a paracompact and connected locally euclidian Hausdorff space. Cover $$X$$ by open subsets $$B_\alpha$$ which are open euclidean balls. Since $$X = \bigcup_n K_n$$ with compact $$K_n$$, countably many $$B_k = B_{\alpha_k}$$ suffice to cover $$X$$. Thus $$X$$ is a countable union of second countable open subsets. Therefore $$X$$ is second countable.