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.
- The Hausdorff property : For any two points $p, q \in M$, there exists a neighbourhood of $p$ and a neighbourhood of $q$ such that $U_p \cap U_q \neq \varnothing$.
- Paracompactness : For any open cover $\{ U_i \}_{i \in I}$, there is a refinement $\{ V_j \}_{j \in J}$, $V_j \subset U_i$, that is locally finite : for any point $p \in M$, $p$ is only in a finite number of open sets from that refinement.
- Second countability : There exists a countable basis for the topology.
If we drop the requirement of the Hausdorff property, we can get non-Hausdorff manifolds, defined by their negation of that property. As is common in mathematics, there is some ambiguity for the term : some people will define manifolds specifically as being paracompact and Hausdorff, and refer to non-Hausdorff manifolds for manifolds which are not necessarily Hausdorff. In the scope of this article, we will study manifolds which are indeed not Hausdorff (and not necessarily paracompact, but most of them will be still). Therefore, we can define them by :
In other words, a manifold is non-Hausdorff if there exists at least two points which are not well separated.
There are a few other different ways to describe non-Hausdorff manifolds, tied to possible different definitions of manifolds. For instance if we describe a manifold as a locally ringed space $(M, \mathcal{O}_M)$, with $\mathcal{O}_M$ the sheaf of local $\mathbb{R}$-algebras, then this corresponds to the existence of points which cannot be separated by any global sections $f \in C^\infty(M)$, so that for every such $f$, $f(p) = f(q)$ (we will see why this is so later).
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
- Quotient spaces of manifolds will in general not be Hausdorff
- The foliation of a manifold may form a non-Hausdorff manifold.
- Various completions of spacetimes in general relativity may fail to be Hausdorff
- The notion of spacetime itself as a non-Hausdorff manifold (so-called Branching Spacetimes) has been proposed for a variety of reasons.
1. Construction
Before looking into the properties of non-Hausdorff manifolds, let's review a few methods to generate them. Obviously as a manifold, it is possible to simply construct them by considering a collection of charts and their overlaps, and this can be a useful description for them, but there are other useful processes which can be used to generate them from other manifolds.
A lot of non-Hausdorff manifolds are built by combining other manifolds together, in particular a lot start out as the disjoint union of two manifolds. As a reminder, $M_1 \sqcup M_2$ is the disjoint union of two manifolds, defined by their union indexed by some set. If we have some indexing set $I$, then we have
\begin{equation} \bigsqcup_{i \in I} M_i = \bigcup_{i \in I} \{ M_i, i \} \end{equation}Unlike the union of set, the disjoint union will not fuse the overlap of two sets together, so that we keep identical copies of the original sets. In most cases, we will write down the indexing set as the integers $\mathbb{N}$, so that for instance $M_1 \sqcup M_2 = ( M_1, 1) \cup ( M_2, 2 ) $. Furthermore, we will keep indexing the points of the original manifolds by those, so that even if we have two copies of the same manifolds, we can keep track of which manifold they belong to at a glance. For instance if we consider $\mathbb{R} \sqcup \mathbb{R}$, we will denote by $0_1$ the zero point in the first copy, $0_1 = ( 0, 1 )$, and similarly, $0_2 = (0, 2)$.
Quotient spaces
One of the most common way of getting a non-Hausdorff manifold is via quotient spaces. If we define some equivalence relation $\sim$ on our manifold, we can define an equivalence class of points $[p]$, such that for any points $p_1, p_2 \in [p]$, $p_1 \sim p_2$. Given this, we can define the quotient space $M / \sim$ point-wise as a set where all points of $M / \sim$ correspond to equivalence classes of points of $M$. There is a surjective map
\begin{eqnarray} q : M &\to& M / \sim\\ p &\mapsto& [p] \end{eqnarray}As a topological space, the quotient topology on $M / \sim$ is defined by subsets in $M / \sim$ being open if they are composed of equivalence points corresponding to an open set in $M$. In other words, $U$ is open in $M / \sim$ if $q^{-1}(U)$ is open in $M$.
There is quite a lot of possible structures one can make using a quotient, and we can imagine for instance a retraction of $\mathbb{R}^2$ onto a cross by mapping the origin to itself and then each quadrant to its diagonal, giving us the equivalence relation we want, in which case the resulting space is not a manifold, not even a non-Hausdorff one.
There are some conditions for the resulting quotient space to itself be a manifold.
In our example of the cross space,
A simple example of a quotient space for our purpose (as we will see more in details later) is the splitting real line. As a quotient, we first consider the non-connected manifold formed by the disjoint union of two copies of the real line, $\mathbb{R} \sqcup \mathbb{R} = \{ \mathbb{R}, 1\} \cup \{ \mathbb{R}, 2 \}$. We can give it an atlas composed of identity maps on $(-\infty_i, 0_i)$, $(0_i, \infty_i)$, and an overlap patch $(-1_i, 1_i)$, for $i = 1, 2$ corresponding to each copy. Our quotient map is that for two points $x_1 \in \mathbb{R}_1$ and $x_2 \in \mathbb{R}_2$, if $x_1 = x_2$ and $x_1 < 0$, we have $x_1 \sim x_2$. For positive values and therefore the patches $(0_i, \infty_i)$, this has no effect on the open sets as the quotient map is just the identity.
The negative values are quotiented to a single line, $(-\infty, 0)$, which is also an open set as the preimage $q^{-1}((-\infty, 0)) = (-\infty_1, 0_1) \cup (-\infty_2, 0_2)$ is itself an open set
Gluing
Another 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. The gluing process is in fact a more specific kind of quotient space, except we do not have to worry about specific conditions on the quotient map.
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.
\begin{equation} \tau_{N_1, N_2} = \end{equation}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
\begin{equation} \mathbb{R}_\div = \mathbb{R} \sqcup \mathbb{R} / \sim \end{equation}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 :
$$\sim = \{ (x_1, x_2) | x_1 = x_2 \text{ and } x_1 < 0 \}$$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. There are a few variations of this manifold, but the basic one is to consider the line $\mathbb{R}$, for which every point $p$ branches in two different branches
More varied examples
From those manifolds as well as the processes we used to create them (and manifold-building processes in general), it is not too hard to create more varied examples of non-Hausdorff manifolds. We can consider the Cartesian product of any of those $1$-dimensional manifold with other manifolds (including each other)
Foliations
A field where non-Hausdorff manifolds appear a lot is in the foliation of manifolds.
A foliation (or $k$-foliation) $\mathcal{F}$ of a manifold $M$ is a partitioning into manifolds $\mathcal{L}_\alpha$ of dimensions $n - k$, the leaves of the foliation, for some indexing set $A \ni \alpha$. In other words, the leaves form a cover of the manifold and are mutually disjoint :
\begin{eqnarray} \bigcup_{\alpha \in A} \mathcal{L}_\alpha &=& M\\ \forall \alpha,\ \beta \in A, \alpha \neq \beta,\ M_\alpha \cap M_\beta = \varnothing \end{eqnarray}Definition of foliations, leaf spaces
...
An interesting case of this comes in the form of the "shape of space" for spacetimes. The idea of considering the shape of space in general relativity usually comes in play for spacetimes with an integrable spatial distribution, such as globally hyperbolic spacetimes. On the other hand, timelike distributions, being one-dimensional, are always integrable : it is always possible to foliate spacetimes with timelike lines. While the orthogonal complement of this distribution may not itself be integrable, it can be interesting to consider another definition for the "shape of space", where we simply consider the leaf space of this foliation.
Topological properties
From its definition, a non-Hausdorff manifold's most important feature will be a set of points with always overlapping neighbourhoods. As this is not a particularly common topic, there isn't really a lot of standardized terminology for this. Some papers will talk of adjacency, or of compatible apparition points. I have gone with the former here.
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 $q$.
Adjacent points are in some sense a local way in which the space fails to be locally Euclidian. Despite being arbitrarily "close" topologically, a region containing two adjacent point will not be homeomorphic to some open set of $\mathbb{R}^n$ :
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.
Although they fail to be contained in the same coordinate patch, it still remains possible for them to be in the boundary of the same one.
Theorem : If $p$ and $q$ are adjacent, there exists a coordinate neighbourhood that includes both of them in its boundary.
Proof : We can always find some charts $U_p \ni p$, $U_q \ni q$ around $p$ and $q$. By their adjacency, there exists some overlap between those two charts, $U_{pq} = U_p \cap U_q$, which can be given a coordinate chart by restricting either $\phi_{U_p}$ or $\phi_{U_{q}}$ to that neighbourhood. Since $U_p$ does not contain $q$ and $U_q$ does not contain $p$, $U_{pq}$ will not contain either. However, take the boundary $\partial U_{pq}$. If $p$ or $q$ is not in the boundary, then there exists a neighbourhood $U_p'$ or $U_q'$ such that $U_p' \cap (U_p \cap U_q) = \varnothing$, or, by associativity, $(U_p' \cap U_p) \cap U_q = \varnothing$. But since $U_p$ and $U_p'$ both contain $p$ and are open sets, their intersection is a non-empty open set containing $p$, and therefore cannot have an empty overlap with $U_q$, as the two points are adjacent. $p$ (and through a similar proof, $q$) are therefore in the boundary of this overlap.
This will be useful later on as we consider how analysis will work around adjacent points.
There is a possibly stronger sense to this depending on the type of non-Hausdorff manifold we're considering. If we take the equivalence relation $p \sim q$ if $p$ and $q$ are adjacent, and then look at the quotient space $M / \sim$, we obtain a somewhat similar space in many ways, but many of them will fail to be manifolds. The line with two origins will simply quotient to $\mathbb{R}$, but the splitting line will be a $Y$-shaped space that fails to be a manifold at $0$.
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 $(-\varepsilon_3, \varepsilon_3)$.
From the examples that we have seen, there is some intuition that the non-Hausdorff manifolds that we've seen can maybe be made into Hausdorff topological spaces. Up to their adjacent points for instance, the line with two origins is simply the line, while the splitting line is the $Y$-shaped topological space, indicating that in some cases that space may not be a manifold. This can be formalized as the Hausdorff reflection (or Hausdorffication) of our space. There are a few ways of performing this process, where given a non-Hausdorff space $X$ we obtain the Hausdorff space $HX$ via some map $h_X : X \to HX$.
An example map is to consider the equivalence class formed by adjacent points, ie two different points on the manifold will be equivalent if $p \mathrm{Y} q$, so that $\sim = \{ (p,q) | (p, q) \in X \times X,\ p \mathrm{Y} q \}$.
Analysis on non-Hausdorff manifolds
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.
As we've seen previously, adjacent points cannot both be in the same coordinate neighbourhood, but on the other hand, there exists such a neighbourhood in which they belong to the boundary. Despite our coordinates being in $\mathbb{R}^n$, a Hausdorff space, there exist
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 $a_n, b_n \to 0$, and take two lines with doubled points on the sets $A$ and $B$.
If we consider some map $h : \mathbb{R} \to \mathbb{R}$ such that $h(a_n) = b_n$, there always exists such a map that is a homeomorphism (for instance by constructing a piecewise-linear function for it), but if we consider for example the series $a_n = n^{-1}$ and $b_n = n^{-2}$, $h$ cannot be a diffeomorphism between the two [proof by showing that the derivative at $0$ does not exist].
If we make two copies of the real line quotiented by the indicating function $\mathbb{1}_{x \neq a_n}$, where every point of the series is doubled, and the same for $\mathbb{1}_{x \neq b_n}$
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.
There are two important processes to consider basic derivatives and vectors on a manifold : we can consider the derivative of a curve
This lack of unique limits has pretty important consequences on the nature of differential equations on the manifold. For instance consider two curves $\gamma_i : \mathbb{R} \to M$. We will consider some neighbourhood $U$ containing both $p$ and $q$, as such a neighbourhood always exists, and its chart $\phi : U \to O \subset \mathbb{R}^n$. such that $\gamma_i(\lambda < 0)$ coincides and $\gamma_i(\lambda = 0)$ takes each curve to a different adjacent point $p_i$ such that $p_1 \mathrm{Y} p_2$, ie $\gamma_1(0) \mathrm{Y} \gamma_2(0)$. Their respective derivatives with respect to their parameter in some chart $\phi_i$ is
\begin{equation} v_{\gamma_i, \phi_i, p_i} = \frac{d}{d\lambda} \left[ \phi_i \circ \gamma_i(\lambda) \right]|_{\lambda = 0} \end{equation}If we define our composite function $\phi_i \circ \gamma_i$ as a function $x_i : \mathbb{R} \to O_i \subset \mathbb{R}^n$, we can look into the derivative as a convergence
\begin{eqnarray} v_{\gamma_i, \phi_i, p_i} &=& \frac{d}{d\lambda} x_i(\lambda )|_{\lambda = 0}\\ &=& \lim_{h \to 0} \frac{x_i(\lambda + h) - x_i(\lambda)}{h} \end{eqnarray}Meaning that our vectors in those coordinates are defined by the following condition
\begin{equation} \forall \varepsilon > 0,\ \exists \delta > 0,\ \forall x \in \mathbb{R},\ |h - 0| < \delta \to |\frac{x_i(\lambda + h) - x_i(\lambda)}{h} - v_{\gamma_i, \phi_i, p_i}| < \varepsilon \end{equation}...
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 : The 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 :
- The integral of a function $f$ on the splitting real line should be $$\int_Y f = \int_{-\infty}^0 f + \int_{0_1}^{\infty_1} f + \int_{0_2}^{\infty_2} f$$
- Any integral over doubled points (or any other set of measure $0$) 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
- $F_i\bigr|_{D_i}$ is an orientation-preserving diffeomorphism $F_i\bigr|_{D_i} : D_i \to W_i \subset M$
- $W_i \cap W_j = \varnothing$ when $i \neq j$
- $\text{supp} \omega \subset \bigcup_i \bar{W}_i$
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 $M$.
Kripke semantics
In logics, the Kripke semantics is meant to give a model for modal logics. A Kripke frame $\langle W, \leq \rangle$ is a set $W$ of "possible worlds" and $\leq$ is a partial order relation on $W$
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
- The ordering $\leq$ is dense ($\forall x, y \in W, x \leq y, \exists z. x \leq z \leq y$)
- There is no maximal element ($\nexists m. \forall x \in W, x \leq m$)
- Every lower bounded chain in $W$ has an infimum in $W$ ($\forall S \subset W, \exists x \in S, \forall y \in P, x \leq y, \exists m, x \leq m \leq y$)
- Every upper bounded chain in $W$ has a supremum in every history that contains it
Bibliography
- David Gauld, Non-metrisable manifolds, 2014
- Afshin Mardani, Topics in the General Topology of Non-metric Manifolds, 2014
- N. Hicks, Notes on differential geometry, 1965
- A. Haefliger, G. Reeb, Variétés (non séparées) à une dimension et structures feuilletées du plan, 1957
- P. Hajicek, Extensions of the Taub and NUT spaces and extensions of their tangent bundles
- P. Hajicek, Causality in non-Hausdorff space-times, 1971
- P. Hajicek, Bifurcate Space‐Times, 1971
- N. Belnap, Branching space-time, 2003
- L. Wroński, T. Placek, On Minkowskian Branching Structures, 2007
- S. Kent, R. Mimna, J. Tartir, A Note on Topological Properties of Non-Hausdorff Manifolds, 2008
- N. Bourbaki, Variétés différentielles et analytiques : Fascicule de Résultats, 2007
Last updated : 2024-09-29 12:52:24