# Paracompact iff Countable Atlas

I am a physics student trying to study differential geometry. I am trying to work out the following exercise. Please give me some help.

Let $M$ be be a connected Hausdorff space which is locally Euclidean. Show that $M$ is paracompact iff $M$ has a countable atlas.

Currently, I can show that $M$ has has a countable atlas iff $M$ is $2^{nd}$ countable. So it seems that I need to prove that $M$ is $2^{nd}$ countable iff $M$ is paracompact. But this still seems very hard.

1. Every open cover of $X$ has a locally finite open refinement. (i.e. $X$ is paracompact)
2. Ever open cover of $X$ has a $\sigma$-locally finite open refinement.
It's easy to show 2. for a second countable space (every open cover has a refinement by base elements so a countable refinement, which is trivially $\sigma$-locally finite.).
• I don't know what $\sigma$-finite is, but I will read the book. Thank you very much. – Xiaoyi Jing Sep 23 '17 at 7:13
• A collection $\mathcal{B}$ is $\sigma$-locally finite if we can write $\mathcal{B} = \cup_{n \in \mathbb{N}} \mathcal{B}_n$ where each $\mathcal{B}_n$ is locally finite. Munkres calls it "countably locally finite", which is non-standard. – Henno Brandsma Sep 23 '17 at 8:03