When (why) did we allow manifolds to be non-Hausdorff and/or non-second countable?

Question

I was reading David Carchedi's answer for a question on Grothendieck topology for a non-small category. It "reads" like people "choose" if they allow manifolds to be Hausdorff and/or second countable. When I came across the notion of smooth manifolds for the first time, by definition, smooth manifolds are Hausdorff and second countable.

When (why) did we allow manifolds to be non-Hausdorff and/or non-second countable?

I have observed this when I am reading about stacks.

Is it because of the constructions when we do in the set up of manifolds (stacks) which resulted in spaces that are "same as" manifolds but are not Hausdorff or not second countable?

Is it because of the influence of algebraic geometry?

Answer 1

The étale space construction produces non-Hausdorff and nonparacompact spaces (e.g., smooth manifolds) in many practical examples that have nothing to do with algebraic geometry.

The étale space is often non-Hausdorff because two germs can coincide on some nontrivial part of the domain without being equal. For example, germs of continuous real-valued functions around 0 can be equal on (−ε,0), but different on (0,ε) for arbitrarily small ε>0.

Nonparacompactness arises for the same reason. For connected manifolds, paracompactness coincides with second countability. The étale space, even if connected, is often not second countable because there are uncountably many disjoint open subsets, e.g., germs of constant real-valued functions.

Answer 2

Hmm, I don't know anything about stacks and all that, but below is a situation where one cares about non-Hausdorff 1-manifolds.

If one considers a codimension-1 foliation of a manifold, a very useful thing to do in dimension 3, it is often fruitful to study the leaf space of the universal cover of this manifold. That is to lift the foliation to the universal cover and examine the quotient space you get by collapsing every leaf to a point.

The generic case of this space is a non-Hausdorff, second countable, locally Euclidean topological space. The leaf space is homeomorphic to $\mathbb{R}$ if and only if it is Hausdorff. When the leaf space is $\mathbb{R}$ the foliation is called $\mathbb{R}$-covered.

The leaf space contains a remarkable amount of information about the manifold, the original foliation, and its fundamental group. This comes by way of studying how the fundamental group naturally acts on the leaf space, which is by deck transformations before quotienting. Non-Hausdorff 1-manifolds are in the literature often referred to as $\mathbb{R}$-trees, and perhaps less commonly as order trees.

Answer 3

Non-Hausdorffness shows up in several contexts when dealing with Lie groupoids: the integration (Lie's 3rd Theorem) for Lie algebroids to Lie groupoids will typically produce a non-Hausdorff one, if it works at all. So there seems to be a large part of oid-geometry involved with non-Hausdorff manifolds.

However, to abandon second countability is a more serious step, at least in my eyes. It is not so much the existence of partition of unities (which requires paracompactness) but the second countability itself which is extremely useful. Consider the following statement:

A bijective immersion is a diffeomorphism

This is a theorem in differential geometry which you definitely want to be true and it relies directly (well, a bit hidden) on second countability. To see that it fails right on the nose if you drop second countability, consider a manifold $M$ in the usual sense of positive dimension and $M_{\mathrm{discrete}}$ as a zero-dimensional manifold with uncountably many connected components, each of which is paracompact (it's a point...) Then $\mathrm{id}\colon M_{\mathrm{discrete}} \longrightarrow M$...

Now why is this theorem important: Lie theory depends strongly on it. Any group would be a zero dimensional Lie group otherwise. In particular, the manifold structure of a Lie group would not be determined by the group structure. This would also imply that a transitive smooth group action of a Lie group on a manifold is not the same thing as a homogeneous space $G/H$ and many more problems...

So before asking why one should abandon a property, it might be good to understand what it is good for. For Hausdorffness the situation seems to be very different to second countability.

Answer 4

If you're asking a historical question, it is probably because of Whitney, who set out a clear account of differentiable manifolds and proved the embedding theorem, that the Hausdorff and second countability assumptions have been regarded as "standard." Nevertheless, violations of these assumptions had been contemplated prior to Whitney's work, notably the non-second-countable Prüfer surface, which dates back to the 1920s.

In addition to the examples mentioned by others, note that people studying general relativity have sometimes considered non-Hausdorff spacetime manifolds (see here for a non-paywalled version). The consensus, however, seems to be that these are mathematical curiosities that are not physically significant. Luc and Placek amusingly quote Penrose as saying, "I must … return firmly to sanity by repeating three times: spacetime is a Hausdorff differentiable manifold; spacetime is a Hausdorff …"