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Suppose $M^n$ is a topological manifold, then $M^n$ locally looks like $\mathbb{R}^n$. $M^n$ is locally Hausdorff, since $\mathbb{R}^n$ is Hausdorff and Hausdorff-ness is a topological invariant.

All I want to understand is:

1) Does locally Hausdorff-ness imply Hausdorff-ness? (I can not imagine a locally Hausdorff topological space that is not globally Hausdorff)

2) Why do we need Hausdorff-ness in definition of the topological manifold? Is locally Hausdorff-ness not sufficient? If not, why?

Can anyone say anything that might be helpful?

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There is the Line with two origins which is even locally euclidean and second-countable, but only locally Hausdorff. –  Stefan Hamcke Aug 21 '13 at 18:15
In the book by J. Lee "Introduction to Topological manifolds" there is a section "Remarks on the Definition of Manifolds" where the author carefully goes over all properties we require the $n$-manifold to have, and in particular the Hausdorff property. May be of interest to you. –  Ilya Dec 4 '14 at 11:56

3 Answers 3

There are non-Hausdorff spaces that are locally Euclidean; some people include them in the class of manifolds, and some prefer to exclude them by requiring a manifold to be Hausdorff. The line with two origins is a simple example of a non-Hausdorff manifold: it is locally Hausdorff, since it has a base of open sets homeomorphic to $(0,1)$, but the two origins cannot be separated by disjoint open sets.

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I know three main reasons we require manifolds to be Hausdorff (and 2nd countable):

  1. Make classification of 1-dimensional manifolds possible. Without such classification, classifying (or even understanding) manifolds in higher dimensions is pretty hopeless.

  2. One would like to be able to embed manifolds in some higher-dimensional Euclidean spaces. Again, it would be impossible without requiring both conditions.

  3. Theory of manifolds did not come from nowhere, its origins are in analysis (primarily complex analysis) and differential geometry. Riemann first defined Riemann surfaces using multivalued complex-analytic functions, since he was interested in making sense of those. Later, about 100 years ago, Weyl gave the first rigorous definition of an abstract manifold, again in the context of Riemann surfaces. Since Riemann surfaces were the primary motivation, he required them to be both 2nd countable and Hausdorff. Then, I think, differential geometers jumped onto the bandwagon, since they realized that Weyl gave a definition which served them perfectly well. Much later, they realized that there are natural settings where one should weaken "Hausdorfness", such generalizations were primarily motivated by the theory of foliations (and, maybe quantum mechanics, but I am unsure about this).

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I think the main reason behind the hausdorffness condition in the definition of topological manifold is that one needs to be able to do calculus on manifolds. More to the point, one needs to deal with convergent sequences on manifolds and here comes the necessity of including the hausdorffness condition.

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