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The definition of Differential Manifold or Smooth Manifold include $\text{Second countability}$ and $\text{Hausdorffness condition}$.

My question is why we include Second countability and Hausdorffness in the defintition of Differentiable manifold?

I got something from a online article that Second countability is used to exclude large space .

But I could not mean it.

Can someone explain why we take Second Countability and Hausdorffness condition ?

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  • $\begingroup$ @user549397, it is extra property that the manifold is locally homeomorphic to $\mathbb{R}^n$, which is second countable and housdorff. But why we are taking the second count ability and Housdorffness condition separately? $\endgroup$ Commented May 6, 2019 at 8:03
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    $\begingroup$ Not sure about what you say @user549387. I mean the line with two origins is locally homeomorphic to $\mathbb R^n$ but it isn't Hausdorff. Here I attach you a link. $\endgroup$ Commented May 6, 2019 at 9:39
  • $\begingroup$ @Dog_69 Yep, you are right! Another helpful post I found: math.stackexchange.com/q/709777/549397 $\endgroup$ Commented May 6, 2019 at 10:25
  • $\begingroup$ See: Why is important for a manifold to have countable basis? math.stackexchange.com/q/2131530/549397 $\endgroup$ Commented May 6, 2019 at 10:29
  • $\begingroup$ @user549397 Really interesting :) $\endgroup$ Commented May 6, 2019 at 11:24

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