Definition of a Manifold

[littlemathquark]

TL;DR Summary

Why do we need second countable and Hausdorff conditions for manifold definition?

Why do we need second countable and Hausdorff conditions for manifold definition?

 

[fresh_42]

TL;DR Summary: Why do we need second countable and Hausdorff conditions for manifold definition?

Why do we need second countable and Hausdorff conditions for manifold definition?

My first thought was: We need a reasonable partition of unity in which the sums are indexed by the natural numbers. This would probably require even normal topologies. The two specific properties (second countability aka complete separability and Hausdorff) both occur in Urysohn's metrization theorem:

One of the first widely recognized metrization theorems was Urysohn's metrization theorem. This states that every Hausdorff second-countable regular space is metrizable.

However, ...

It follows that every such space is completely normal as well as paracompact.

... and here we are, back at the partition of unity.

Both are important tools we do not want to lose.

 

[dextercioby]

"Manifold" is improper usage. One has "topological manifolds" and a very important subset of all of these, the "complex or real differential manifolds".

 

[fresh_42]

"Manifold" is improper usage. One has "topological manifolds" and a very important subset of all of these, the "complex or real differential manifolds".

Indeed. I was so used to reading Riemannian and even smooth, nice charts, metrics, and spheres that I almost forgot the topological aspect. The two concepts of Urysohn are somehow the minimum we want to have.

 

[martinbn]

TL;DR Summary: Why do we need second countable and Hausdorff conditions for manifold definition?

Why do we need second countable and Hausdorff conditions for manifold definition?

That is a strange question! Why do you need the other conditions in the definition of a manifold? You can study whatever objects you want, why is a question only you can answer. Some people do look at non Hausdorff manifolds. https://en.wikipedia.org/wiki/Non-Hausdorff_manifold

 

[littlemathquark]

In the definition of a topological manifold, the conditions that the topological space X be Hausdorff and second countable are not always included. Is this because these two properties can be transferred via a homeomorphism from R^n, since they are topological properties?

 

[fresh_42]

In the definition of a topological manifold, the conditions that the topological space X be Hausdorff and second countable are not always included. Is this because these two properties can be transferred via a homeomorphism from R^n, since they are topological properties?

Yes, but these are all local properties. It gives you information about open sets at a certain location. The atlas patches all these local information. The result is a book (an atlas) of maps (the homeomorphisms). But this is literally a patchwork. Urysohn's results (partition of unity and metrization) are global properties for the entire manifold.

 

[littlemathquark]

I thought of counter examples like the "line with two origins" and the "long line." Although these are topological manifolds, one is not Hausdorff, and the other is not second countable. However, I believe these are pathological examples.

 

[fresh_42]

However, I believe these are pathological examples.

I am the wrong person to answer that. I believe that particularly topology is a huge collection of pathological examples. It can trick someone's intuition in so many cases! E.g. I cannot really imagine how we can color a square with a pen that has no width. As I mentioned above, the "nice" manifolds are Riemannian, and smooth, and the only question is whether they can be embedded in $\mathbb{R}^{n+1}$ or $\mathbb{R}^{2n}.$

I was even surprised how many topological spaces are not Hausdorff:
https://en.wikipedia.org/wiki/Hausdorff_space#Examples_of_Hausdorff_and_non-Hausdorff_spaces
https://de.wikipedia.org/wiki/Hausdorff-Raum#Beispiele (right click on Chrome translates it into English)

Especially the example of a local Euclidean, nevertheless non-Hausdorff space was interesting.

 

[mathwonk]

"Although these are topological manifolds, one is not Hausdorff, and the other is not second countable. However, I believe these are pathological examples"

didn't you just answer your own question? i.e. in your opinion, as soon as a manifold fails to be Hausdorff and second countable, it is pathological.

 

[jbergman]

One of the motivations that Lee mentions in the topological case is that with second countability and Hausdorff every manifold is Homeomorphic to a subset of Euclidean space.