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In this question different people understood different things when talking about topological manifolds. Some argued they they have to be Hausdorff, some that they have to be second countable and some, both.

When I studied them, my teacher showed us examples of non-Hausdorff (the line with two origins) and non-second countable (the long line) manifolds. For me, a topological manifold is a locally Euclidean topological space.

What are the different definitions of a topological manifold you know? What it depends on? What author you read when you studied them? Who was your teacher?

EDIT: What properties have the topological manifolds if we define them as second-countable and Hausdorff that they don't have if they are only locally Euclidean?

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You need Hausdorff and second countability to get Paracompactness, and paracompactness and partitions of unity are very important for local to global constructions and in homotopy related questions : proving that homotopic maps induce isomorhpic pullbacks for instance. – Olivier Bégassat Aug 11 '11 at 17:07
For me a manifold is something that you can embeds nicely in $\mathbb{R}^n$ for some $n$ large enough. For this, the countability and separation axioms are necessary. – Mark Aug 11 '11 at 17:14
I'm tempted to close as subjective and argumentative. What do you expect to do with an answer to this question? – Qiaochu Yuan Aug 11 '11 at 17:29
@Qiaochu : I added another question that is not subjective or argumentative in any way. – dan232 Aug 11 '11 at 18:15
It depends on where you encounter them and what you want to do with them. As a set-theoretic topologist I find non-metric manifolds more interesting than metric manifolds -- things that embed in some $\mathbb{R}^n$ are way too ‘nice’! -- but that’s very much a minority view. – Brian M. Scott Aug 11 '11 at 19:33
up vote 6 down vote accepted

Hausdorffness is a necessary condition for continuous real-valued functions to separate points. If continuous real-valued functions don't separate points on your space, you're dealing with a weird space (which in particular does not embed into $\mathbb{R}^n$). Of course weird spaces exist, but it's harder to prove nice theorems about them.

As Mark Schwarzmann says in the comments, any manifold that embeds into $\mathbb{R}^n$ is necessarily both Hausdorff and second-countable.

It is also common to see paracompactness as a condition, since this is equivalent (in the presence of the other axioms) to both metrizability and the existence of partitions of unity, which I understand to be quite useful.

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The last two sentences of the first paragraph are awfully subjective: weirdness is very much in the eye of the beholder. I’d say that the real value of Hausdorffness here is to make convergence behave; that it also happens to give you functional separation of points is a consequence of other properties. – Brian M. Scott Aug 11 '11 at 19:25
@Brian: sure; I think that's a fine reason to require Hausdorffness in general. For the particular case of manifolds, it's known that you can completely recover a locally compact Hausdorff space $X$ (any Hausdorff manifold is such a space) from the Banach algebra $C_b(X)$ of continuous bounded functions $X \to \mathbb{R}$. I like being able to replace topology with algebra, so for me this is an important motivation for specifically requiring manifolds to be Hausdorff. – Qiaochu Yuan Aug 11 '11 at 19:34
Different backgrounds and interests: the last thing I want to do is replace topology with algebra! – Brian M. Scott Aug 11 '11 at 19:38

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