Is a Manifold Defined by Sheaves Always Hausdorff?

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Discussion Overview

The discussion centers on the properties of manifolds defined by sheaves, specifically questioning whether such manifolds are necessarily Hausdorff. The scope includes theoretical considerations related to the definitions of manifolds and the implications of these definitions in topology and differential geometry.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that an ##n##-manifold is defined as a locally ringed space locally isomorphic to a subset of ##(\mathbb{R}^n, C^0)## but questions the Hausdorff property of such manifolds.
  • Another participant asserts that Hausdorff and second countable conditions must be demanded separately, as they are global conditions.
  • A later reply acknowledges this clarification, indicating a shift in understanding regarding the necessity of these conditions.
  • Additional commentary introduces the concept of diffeological spaces, suggesting they may offer advantages over traditional differential geometry in the context of locally ringed spaces.

Areas of Agreement / Disagreement

Participants express differing views on whether the Hausdorff property is guaranteed by the definition of manifolds via sheaves. There is no consensus on this issue, and the discussion remains unresolved.

Contextual Notes

The discussion highlights the distinction between local and global properties in topology, particularly in relation to the definitions of manifolds and the implications for differential geometry.

Mandelbroth
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While reading about sheaves, I came across a beautiful definition of a manifold. An ##n##-manifold is simply a locally ringed space which is locally isomorphic to a subset of ##(\mathbb{R}^n, C^0)##. However, I don't see how this guarantees a manifold to be Hausdorff. Would someone please explain this?
 
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You need to demand Hausdorff and second countable separately since they are global conditions.
 
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micromass said:
You need to demand Hausdorff and second countable separately since they are global conditions.
Alright. That makes more sense. Thank you!
 
By the way, if you're interested in this, check out this book:

https://www.amazon.com/dp/0821837028/?tag=pfamazon01-20

Also, it needs to be said that differential geometry doesn't really fit well in the theory of locally ringed spaces for several reasons. One thing that is very interesting is that of diffeological spaces. A diffeological space is to a differentiable manifolds as a topological space is to a topological manifold. Diffeological spaces behave way better under categorical constructions. See http://en.wikipedia.org/wiki/Diffeology The references below the wiki article are very good.
 
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