The mystery of non-Hausdorff manifolds


The basic definition of a manifold $M$ involves a family of coordinate charts, $\{ (\phi_i, U_i) \}$, such that

\begin{equation} \phi_i : U_i \subset M \to O_i \subset \mathbb{R}^n \end{equation}

This covers a fair number of weird manifolds that are pretty seldomly studied, such as the complete feather, the long line or the Prüfer manifold. To mostly get reasonable manifolds, the following conditions are usually thrown in the definition as well.

If we drop the requirement of the Hausdorff property, we can get non-Hausdorff manifolds, defined by their negation of that property
A manifold is non-Hausdorff if and only if there exists two points $p, q \in M$ for which there exists no neighbourhoods of $p$ and $q$ such that $U_p \cap U_q = \varnothing$.

In other words, a manifold is non-Hausdorff if there exists at least two points which are not well separated.

Non-Hausdorff manifolds aren't terribly studied, mostly due to the breaking of many theorems that generally hold for Hausdorff manifolds and the lack of concrete applications for them. The only texts that mention them in some details are the reviews of non-metrizable manifolds by Gould[1] and Mardani[2], Hick's Notes on differential geometry[3], Haefliger and Reeb's paper[4], the few papers on Hajicek's for non-Hausdorff manifolds[5][6] and a handful of papers on the topic of branching spacetimes. The main contexts where they pop up are

1. Gluing manifolds

The most common method of building non-Hausdorff manifolds is gluing two existing Hausdorff manifolds. This is a slightly different type of gluing than the usual type used in connected sums and other such processes (where the gluing is done along boundaries), as those will give us Hausdorff manifolds.

Given two manifolds $M_1$ and $M_2$, with two open sets $N_1$ and $N_2$, and a diffeomorphism $\phi : N_1 \to N_2$, the gluing of those manifolds is the new manifold produced by considering the space

\begin{equation} M = M_1 \sqcup M_2 / \sim_\phi \end{equation}

where $p \sim_\phi q$ if $p \in N_1, q \in N_2$ and $\phi(p) = q$ (or vice versa, if $p \in N_2$ and $q \in N_1$). As $N_1$ and $N_2$ are identified here, we can simply refer to them as $N \subset M$. Any points outside of $N$ are already covered by the appropriate charts, so to show that $M$ is a manifold, we must simply find the charts on $N$.

The charts of $N_1$ and $N_2$ are already fine to cover $N$, and have the transition maps with $M_1$ and $M_2$, but they do not possess the transition maps between each other, as well as between $N_1$ and $M_2$, and $N_2$ and $M_1$. This is where our gluing map comes in.

$$\tau_{N_1, N_2} = $$

2. A few non-Hausdorff manifolds

There are many examples of non-Hausdorff manifolds, in fact they do not even form a set as its class has too large a cardinality, even in one dimension (just consider a collection of $\mathbb{R}$ indexed by some class $I$ to be glued together, for instance, and we can construct different manifolds for each cardinality of $I$). The 4 most well-known non-Hausdorff manifolds are the line with two origins, the branching real line, the lasso and the feather. While there are many other different manifolds, those are a good cross-sections of the various behaviours we can encounter from non-Hausdorff manifolds.

The line with two origins

The simplest non-Hausdorff manifold, it is constructed by considering two copies of the real line $\mathbb{R} \sqcup \mathbb{R}$ with the equivalence relation $x_1 \sim x_2$ if $(x_1, x_2) \in (\mathbb{R} \setminus \{0\}) \sqcup (\mathbb{R} \setminus \{0\})$ and $x_1 = x_2$. Then we can denote it as

$$\mathbb{R}_\div = \mathbb{R} \sqcup \mathbb{R} / \sim$$ Line with two origins 0 0*

This easily generalizes with any number of points for any values of the real line, including uncountable numbers of points.

The branching real line

Similarly to the line with two origins, the branching real line is made of two copies of the real line under the following equivalence relation :

$$\sim = \{ (x_1, x_2) | x_1 = x_2 \text{ and } x_1 < 0 \}$$ Line with two origins 0 0*

The lasso

The lasso is an interesting case since it is an example of a non-orientable manifold in one dimension, meaning that it is possible to visualize its tangent bundle.

Its manifold structure is made of two charts, $U_1 = (-1, +\infty)$ and $(-1, 1)$, along with the transition maps

The feather

The feather is simply a generalization of the branching real line, where the real line branches at every point.

Topological properties

From its definition, a non-Hausdorff manifold's most important feature will be a set of points with always overlapping neighbourhoods. This will be defined by the notion of adjacency.

Definition : For a cardinal number $\alpha$, two points $p, q \in M$ are $\alpha$-adjacent (noted $p \operatorname{Y}^\alpha q$) if for every neighbourhood $U_p \ni p$ and $U_q \ni q$, we have $\operatorname{Card}(U \cap V) \geq \alpha$. If $\alpha = 1$, we also say that $p \operatorname Y q$.
We will call the set of all adjacent points to a point $p$ $$\operatorname{adj}(p) = \{ q | p \operatorname Y q \}$$ While the neighbourhoods of points may not be separated in a non-Hausdorff manifold, there will at least always be a neighbourhood that will not contain the other point. In other word, every non-Hausdorff manifold is still a $T_1$ space [1] :

Theorem : Every manifold is $T_1$.

Proof : Take any $p,q \in M$, $p \neq q$, with a coordinate patch $U \ni p$. If $q \notin U$, then we are done. If $q \in U$, consider the points $\phi(p), \phi(q)$ in $\mathbb R^n$. Since $\mathbb R^n$ is itself $T^1$, there is an open set $O \subset \mathbb R^n$ that contains $\phi(p)$ but not $\phi(q)$. Hence $\phi^{-1} (O)$ contains $p$ but not $q$.

Theorem : If $p$ and $q$ are adjacent, there is no coordinate patch that contain them both of them.

Proof : If there is a coordinate patch $(U, \phi)$ which contains both points, consider $O \subset \mathbb{R}^n$ such that $\phi(U) = O$. Take the two images $\phi(p)$ and $\phi(q)$. As $\mathbb{R}^n$ is Hausdorff, so is $O$ and therefore there exists two neighbourhoods $O_p$, $O_q$ of $p$ and $q$ such that $O_p \cap O_q = \varnothing$. But then $\phi^{-1}(O_p) \cap \phi^{-1}(O_q) = \phi^{-1}(O_p \cap O_q) = \varnothing$, in contradiction with their adjacentness.

This theorem will be quite useful for many applications.

Theorem : The relation of adjacency is symmetric and reflexive, but not transitive.

Proof :

  • Reflexive : Since $p$ will always be present in its own neighbourhoods, $p$ is always adjacent to itself (for two neighbourhoods, $p \in U_p \cap U'_p$).
  • Symmetric : Since the intersection of two sets is commutative, if $p \operatorname Y q$, then $U_p \cap U_q = U_q \cap U_p \neq \varnothing$, meaning that $q \operatorname Y p$
  • Not transitive : Consider the manifold defined by three copies of $\mathbb R$, with $\{ x > 0 \}$ identified for $\mathbb{R}_1$ and $\mathbb{R}_2$, and $\{ x < 0 \}$ identified for $\mathbb{R}_2$ and $\mathbb{R}_3$. The open sets $(-\varepsilon_1, \varepsilon_1)$ and $(-\varepsilon_2, \varepsilon_2)$, as well as $(-\varepsilon_2, \varepsilon_2)$ and $(-\varepsilon_3, \varepsilon_3)$ always intersect, but not $(-\varepsilon_1, \varepsilon_1)$ and $(-\varepsilon_3, \varepsilon_3)$.

Analysis on non-Hausdorff manifolds

Smooth structures become immediatly non-trivial for non-Hausdorff manifolds, since even in one dimension, they fail to be unique.

Theorem : There exists homeomorphic $1$-dimensional manifolds with non-diffeomorphic smooth structures.

Proof : Consider the sets $A = \{ x \in \mathbb R | x \in \{ a_n \} \cup 0 \}$ and $B = \{ x \in \mathbb R | x \in \{ b_n \} \cup 0 \}$, where $a_n$ and $b_n$ are two monotone decreasing sequences with $a_n, b_n \to 0$.

One of the most important property of non-Hausdorff manifolds is that they generally lack unique limits. While some non-Hausdorff spaces can have unique limits (such as topologies where only eventually constant sequences converge to a limit, such as the cocountable topology on $\mathbb{R}$), if they are first-countable this is not the case, which is true for all manifolds.

Theorem : If two points $p, q \in M$ are adjacent, there exists a sequence $(p_n)$ such that the sequence converges to both $p$ and $q$.

Proof : Since the manifold is first-countable, there's a neighbourhood basis $(U_p)_i$ and $(U_q)_i$, and for all $n$, $U_{p_n} \cap U_{q_n} \neq \varnothing$. If we pick a sequence $p_n \in U_{p_n} \cap U_{q_n}$, this sequence will converge to both $p$ and $q$, as it is always within a neighbourhood of both.

Theorem : For any continuous function $f$ and to adjacent points $p, q \in M$, $f(p) = f(q)$

Proof : Since there always exists a sequence $p_n$ which converges to both $p$ and $q$, by continuity, this implies that the sequence $f(p_n)$ will converge both to $f(p)$ and $f(q)$. As $\mathbb{R}$ itself is Hausdorff, a sequence can only converge to a single point, implying that $f(p) = f(q)$.

Theorem : A non-Hausdorff manifold admits no partition of unity.

Proof : For a partition of unity to exist, we require a function to be $0$ outside of every open set of the manifold. Since two adjacent points $p,q$ cannot belong to the same coordinate patch, there exists open sets containing one but not the other, but on the other hand $f(p) = f(q)$, hence there cannot be a bump function on every open set of the manifold.

Definition : A bifurcate curve is a pair of curves $\gamma, \gamma'$ defined on $[0,1]$, such that there exists a $g \in (0,1)$

  • For a bifurcate curve of the first kind, $\gamma([0, g]) = \gamma'([0, g])$ and $\gamma((g, 1]) \neq \gamma'((g, 1])$
  • For a bifurcate curve of the second kind, $\gamma([0, g)) = \gamma'([0, g))$ and $\gamma([g, 1]) \neq \gamma'([g, 1])$

Fiber bundles

The construction of fiber bundles isn't terribly different in the case of non-Hausdorff manifolds, but a few peculiarities occur.

Theorem : Flow of vector fields is not unique on non-Hausdroff manifolds.

Proof : If we have the flow of a vector field $V$ passing through a point $p$ adjacent to another point $q$, consider another point $r$ on this vector flow.

Theorem : If we define a Riemannian metric $g$, two adjacent points have a distance $0$.

Proof : If two points $p, q$ are adjacent, consider the neighbourhoods defined by open balls $B_{p, \varepsilon}$ and $B_{q, \varepsilon}$. Since $p$ and $q$ are adjacent, there exists some point $r \in B_{p, \varepsilon} \cap B_{q, \varepsilon}$, by the triangle inequality we have $$d(p,q) \leq d(p,r) + d(r,q) $$ Since we're in an open ball, $d(p,r) < \varepsilon$ and $d(q,r) < \varepsilon$, $d(p, q) \leq 2 \varepsilon$ for any value of epsilon, making the Riemannian distance between those points $0$.

Theorem : A Riemannian metric tensor defines a pseudometric on the manifold.

Proof : By the same usual proofs as with manifolds, we can show that the arc length defined by the metric is symmetric and that the triangle inequality is true, but due to the distance of $0$ between adjacent points, the distance isn't positive definite and can only be a pseudometric.

Theorem : The pseudometric does not define the topology of the manifold.

Proof : A simple proof is that the manifold is a $T_1$ space and a $T_1$ space with a pseudometric that generates the topology is also a metric space, then the pseudometric cannot generate the topology. Another way to do this is that since two adjacent points have a distance of $0$, any open set of the pseudometric will contain the two points, but as seen there always exists open sets that only include one of them, meaning that those sets cannot be generated by the pseudometric.

$Y$-manifolds

From that point on, we'll restrict our consideration to $Y$-manifolds.

Definition : A $Y$-manifold is a non-Hausdorff, second-countable, n-dimensional $C^k$ manifold.

In other words, a $Y$-manifold is a "reasonable" manifold, save for the fact that it is not Hausdorff, forbidding such manifolds as variations on the long line, or adjacent sets non-countable cardinalities (as two adjacent points cannot be contained in the same chart, the cardinality of points adjacent to each other is countable at best). Since $Y$-manifolds don't admit partition of unity, it will be useful to consider the Hausdorff submanifolds of $Y$-manifolds to have the ability to use them to some degree.

For some subsets $A$, $B$ of $M$, we'll denote by $Y^B_A$ the set of points $p \in A$ such that there exists a point $q \in B$ for which $p Y q$. If $A = M$, we'll drop the index and just write $Y^B$.

Definition : An $H$-submanifold is an open submanifold of a $Y$-manifold such that the submanifold is Hausdorff and it is not the proper subset of any other open Hausdorff submanifold (it is maximal).

For instance, the branching real line admits two $H$-submanifolds, corresponding to the two coordinate patches defined earlier on it, which are just copies of $\mathbb{R}$. Similarly, the line with two origins has the same type of $H$-submanifolds. The lasso has two $H$-submanifolds, which are images of the atlases $(-1, 1) \cup (-1, +\infty)$ and $(-1, 1) \cup (-1, +\infty)$

Theorem : The set of all $H$-submanifolds $\mathscr{H}$ of a $Y$-manifolds $M$ is an open covering of $M$.

Proof : Consider the set $\Omega$ of all open Hausdorff submanifolds of $M$. Let $U,V \in \Omega$, with $U \subset V$, and a point $p \in M$ with a neighbourhood $U_p$. As the subset relation is a partial order, $\Omega$ is a partially ordered set, and we can apply Hausdorff's maximum principle : $\{ U \}$ is a totally ordered subset of $\Omega$, which means that there is a maximal totally ordered subset $\Pi \in \Omega$ that contains $U_p$ and therefore $p$. $P = \bigcup_{U_i \in \Pi} U_i$ is an open submanifold and is Hausdorff : for $x, y \in P$, there are subsets $U_x \ni x, U_y \ni y$. But since $\Pi$ is totally ordered, either $U_x \subset U_y$ or $U_y \subset U_x$, meaning that both $x$ an $y$ are in an open Hausdorff submanifold. Hence for every point $p \in M$, there is a maximal Hausdorff submanifold that contains it.

Theorem : An open submanifold $V$ of a $Y$-manifold $M$ is an $H$-submanifold if and only if $\partial{V} = \overline{Y^V}$.

Proof : If $V$ is an $H$-submanifold, for some point $p \in Y^V$, there exists a point $q \in V$ such that $p Y q$, so that $p$ is a limit point of $V$, hence $Y^V \subset \partial V$. For $p \in \partial V$, let's assume there exists a neighbourhood $U_p \ni p$ such that $U_p \cap Y^V = \varnothing$.

...

Theorem : Every connected set of adjacent points is in the boundary of the overlap of $H$-submanifolds.

Proof : Given the set of $H$-manifolds $\{ H_i \}$, $\bigcup_i H_i = M$, consider the pairwise intersection of every set, $H_i \cap H_j$, $i \neq j$. As every $H$-submanifold is maximal, $H_i \cup H_j$ is necessarily non-Hausdorff, but as $H_i \cap H_j$ is a subset of both $H_i$ and $H_j$, it is necessarily Hausdorff as well. Our adjacent points must therefore be somewhere in $H_i \Delta H_j = (H_i \cup H_j) \setminus (H_i \cap H_j)$, the symmetric difference of the two. As $H_i \cap H_j$ is open, $H_i \Delta H_j$ closed, and we have a boundary $\partial(H_i \Delta H_j)$. As $H'$ is the complement of $H_i \cap H_j$ in $H_i \cup H_j$, it can be expressed as $\overline{H_i \cap H_j}$, so that, by De Morgan's law, $H' = \overline{H}_i \cup \overline{H}_j$.

Integration

Integration on non-Hausdorff manifold cannot be done in the usual way, as the integral for pseudo-Riemannian manifolds requires a partition of unity. Any naive attempt at defining integrals is likely to results in some unforeseen results, so precautions must be taken as to the properties of integration here.

A good starting point is to consider the $H$-submanifolds. As they are Hausdorff, if they are all orientable, then integration on them is done the usual way, that is, given an $H$-manifold $H_i$ and an $n$-form $\omega$ on a coordinate chart $(U, \phi)$ of $H$, the integral is defined by

\begin{equation} \int_H \omega = \int_{\phi(U)} (\phi^{-1})^* \omega \end{equation}

or, in a more traditional way of an integral on $\mathbb{R}^n$,

\begin{equation} \int_H \omega = \int_{\phi(U)} f(x) dx^1 dx^2 \ldots dx^n \end{equation}

...

A few properties that we probably want from our integral is :

  1. The integral of a function $f$ on the splitting real line should be $$\int_Y f = \int_{-\infty}^0 f + 2 \int_0^\infty f$$
  2. Any integral over doubled points should be identical to the quotient space. For instance, $$\int_{\mathbb{R}_{\div}} f = \int_{\mathbb{R}} f$$

Fortunately, there is a definition of integration on manifolds which does not require a partition of unity.

Definition : For an oriented manifold $M$ with $\omega$ a compactly supported $n$-form on $M$, if we have a collection of open sets $\{ D_i \}$, $D_i \subset \mathbb{R}^n$, and smooth maps $F_i : \bar{D}_i \to M$, such that

  1. $F_i\bigr|_{D_i}$ is an orientation-preserving diffeomorphism $F_i\bigr|_{D_i} : D_i \to W_i \subset M$
  2. $W_i \cap W_j = \varnothing$ when $i \neq j$
  3. \text{supp} \omega \subset \bigcup_i \bar{W}_i

Then the integral of $\omega$ is defined by

\begin{equation} \int_M \omega = \sum_{i = 1}^k \int_{D_i} F^*_i \omega \end{equation}

It can be shown that, in the Hausdorff case, this is equivalent to the usual definition. For our case, let's consider a few things first.

As we have seen before, each connected part of the set of all adjacent points $Y^M_M$ is an $(n-1)$-submanifold, therefore of measure $0$. We can use those to split our manifold into non-overlapping Hausdorff manifolds with boundaries. Take the $H$-manifolds cover \{ H_i \} of $M$.

Branching spacetimes

For various reasons, the idea has been put forward that the spacetime we inhabit itself is not Hausdorff, the main one being some attempt to describe quantum mechanics, the branching of the spacetime corresponding in some way to the probabilistic nature of quantum mechanics, each possible series of measurements corresponding to some branch of spacetime.

As far as I'm aware, no such theory has ever been formulated clearly beyond the general idea (quantum mechanics having many peculiarities beyond probabilities, it would not by itself be enough to simply model point particles travelling on a non-Hausdorff manifold), but this has not stopped people from at least putting forward various ideas concerning the structure itself of such spacetimes.

While the naive splitting of spacetime would simply have it split at a time $t$, this would not be a particularly local event, and there's no need to split the entire universe for a measurement. It is generally accepted that if such a theory is to be, the splitting only occurs on a lightcone. If the spacetime is to split at a point $p \in M$, the resulting split will take place as replacing the closure of $J^+(p)$ with any number of copies of it. That is, the resulting manifold is

$$M' = (M \setminus \overline{J^+(p)}) \cup (\bigcup_{i \in I} \overline{J^+_i(p)})$$

with $I$ some indexing set.

Causal structure

The main attraction of non-Hausdorff spacetimes for those applications are the unique causal structures that they can offer, with multiple histories.

In a reasonable spacetime, the causality is usually defined by the causal space axioms, something of the form $\langle X, \leq \rangle$ such that for all $p,q,r \in X$, $p \leq p$ and if $p \leq q$ and $q \leq r$, $p \leq r$, and possibly the causality axiom $p \leq q$ and $q \leq p$ implies $p = q$.

For branching spacetimes, we'll have to first define what is meant by a history.

Definition : A history $h$ in a branching spacetime model $\langle W, \leq \rangle$ is a subset $h \subset W$ that is maximally upward-directed, that is, for every subset $g \subset W$ such that $g \not\subset h$, $g$ cannot be upward-directed, ie, it is false that $\forall e_1, e_2 \in g, $

The simplest branching spacetime is the spacetime that only has but a single branch, the $(0+1)$-dimensional spacetime $M = \mathbb{R}$. $M$ has the causal order defined by the usual order on $\mathbb{R}$. Any subset $g = (a, +\infty)$ is upward-directed, as $\forall t_1, t_2 \in g$, $|t_1| + |t_2| + 1$ will be superior to both. $M$ itself is the only set of the form $(a, \infty)$ that is maximally upward-directed, since for any other such set, $(a-1, \infty) \not\subset (a, \infty)$ is itself maximally upward-directed.

A simple example with more than one history is the branching line $\mathbb{R}_Y$, with the same order as the quotient of two copies $(\mathbb{R}, \leq)$. If we consider, say, the set defined by $(-a, 0) \cup [0_1, \infty_1) \cup [0_2, \infty_2)$, then it can be shown that this set is not upward directed, due to the fact that any event in one branch has no order relation with events in the other. So for instance, there is no common upper bound to $1_1$ and $1_2$.

Definition : $\langle W, \leq \rangle$ is a branching spacetime model, with $W$ a non-empty set and $\leq$ a partial ordering on $W$, if

  1. The ordering $\leq$ is dense ($\forall x, y \in W, x \leq y, \exists z. x \leq z \leq y$)
  2. There is no maximal element ($\nexists m. \forall x \in W, x \leq m$)
  3. Every lower bounded chain in $W$ has an infimum in $W$ ($\forall S \subset W, \exists x \in S, \forall y \in P, x \leq y, \exists m, x \leq m \leq y$)
  4. Every upper bounded chain in $W$ has a supremum in every history that contains it

Bibliography

  1. David Gauld, Non-metrisable manifolds, 2014
  2. Afshin Mardani, Topics in the General Topology of Non-metric Manifolds, 2014
  3. N. Hicks, Notes on differential geometry, 1965
  4. A. Haefliger, G. Reeb, Variétés (non séparées) à une dimension et structures feuilletées du plan, 1957
  5. P. Hajicek, Extensions of the Taub and NUT spaces and extensions of their tangent bundles
  6. P. Hajicek, Causality in non-Hausdorff space-times, 1971
  7. P. Hajicek, Bifurcate Space‐Times, 1971
  8. N. Belnap, Branching space-time, 2003
  9. L. Wroński, T. Placek, On Minkowskian Branching Structures, 2007
  10. S. Kent, R. Mimna, J. Tartir, A Note on Topological Properties of Non-Hausdorff Manifolds, 2008

Last updated : 2019-10-17 16:27:44
Tags : Topology , Differential-geometry