Non-Hausdorff Manifolds via Adjunction Spaces

There are at least two ways to glue together multiple topological spaces in a consistent way. These are:

1. 1.

   to iterate a binary construction several times over, or

2. 2.

   to glue a collection of spaces together simultaneously.

The first approach would amount to suitably modifying the standard adjunction spaces found in say [10] or [11]. Throughout this paper we will instead focus on the latter case. Formally, gluing together multiple spaces can be achieved by fixing some index set $I$ to enumerate the spaces that we would like to glue together, and by defining a triple of sets $\mathcal{F}:=(\textsf{X},\textsf{A},\textsf{f})$, where:

• •

  the set X is a collection of topological spaces $X_{i}$,

• •

  the set A is a collection of sets $A_{ij}$ such that $A_{ij}\subseteq X_{i}$ for all $j\in I$, and

• •

  the set f is a collection of continuous maps $f_{ij}:A_{ij}\rightarrow X_{j}$.

In order to yield a well-defined adjunction space, we need to impose some consistency conditions on the data contained within $\mathcal{F}$. These conditions are captured in the following definition.

Observe that the second condition above ensures that each $f_{ij}$ is a homeomorphism. Given an adjunction system $\mathcal{F}$, we can then define the adjunction space subordinate to $\mathcal{F}$, denoted $\bigcup_{\mathcal{F}}X_{i}$, as the topological space obtained from quotienting the disjoint union

$$\bigsqcup_{i}X_{i}:=\{(x,i)\ |\ x\in X_{i}\}$$

under the relation $\cong$, where $(x,i)\cong(y,j)$ iff $f_{ij}(x)=y$. The conditions of Definition 1.1 are precisely what is needed to ensure the relation $\cong$ is an equivalence relation. Points in the adjunction space $\bigcup_{\mathcal{F}}X_{i}$ can be described as equivalence classes of the form

$$[x,i]:=\{(y,j)\ |\ f_{ij}(x)=y\}.$$

By construction we have a collection of canonical maps $\phi_{i}:X_{i}\rightarrow\bigcup_{\mathcal{F}}X_{i}$ which send each $x$ in $X_{i}$ to its equivalence class in $\bigcup_{\mathcal{F}}X_{i}$. By construction these maps are continuous and injective. Moreover, these maps will commute on the relevant overlaps, i.e. the equality

$$\phi_{j}\circ f_{ij}=\phi_{i}$$

holds for all $i,j$ in $I$. Since the topology of an adjunction space is the quotient of a disjoint union, by construction we have the following useful characterisation of open sets.

In the binary version of adjunction spaces, it is well-known that the adjunction of two spaces is the pushout of the diagram below [11].

The following result shows that the adjunction space subordinate to $\mathcal{F}$ can be seen as the colimit of the diagram formed from $\mathcal{F}$.

Throughout the remainder of this paper we will consider adjunction spaces formed by gluing open sets together. In this case, we make the following useful observation.

Suppose that we have an adjunction space $\bigcup_{\mathcal{F}}X_{i}$ built from the adjunction system $\mathcal{F}$. It follows immediately from Definition 1.1 that any subset $J$ of the underlying indexing set $I$ will yield another adjunction system $\mathcal{G}$, which can be obtained by simply forgetting the parts of $\mathcal{F}$ that are not contained in $J\subset I$. As such, there is an associated adjunction space $\bigcup_{\mathcal{G}}X_{j}$, that consists of gluing only the component spaces $X_{j}$ where $j$ lies in the subset $J$. In this situation we will refer to $\mathcal{G}$ as an adjunctive subsystem of $\mathcal{F}$, and we will write $\mathcal{G}\subseteq\mathcal{F}$.

In order to distinguish objects in $\bigcup_{\mathcal{G}}X_{j}$ from objects in the full adjunction space, we will use the double-bracketed notation $\llbracket\cdot\rrbracket$ to denote objects defined in the adjunctive subspace. In particular, we will use

$$\llbracket x,j\rrbracket:=\{(y,k)\ |\ k\in J\ \wedge\ f_{jk}(x)=y\}$$

to denote the points in $\bigcup_{\mathcal{G}}X_{j}$. Furthermore, we will denote the canonical maps of the adjunctive subspace by $\chi_{j}$’s.

The universal property 1.3 yields a continuous map $g:\bigcup_{\mathcal{G}}X_{j}\rightarrow\bigcup_{\mathcal{F}}X_{i}$ which will send each equivalence class $\llbracket x,j\rrbracket$ of $\bigcup_{\mathcal{G}}X_{j}$ into the (possibly larger) equivalence class $[x,j]$ of $\bigcup_{\mathcal{F}}X_{i}$. In general there is no guarantee that the map $g$ acts as a homeomorphism between $\chi_{j}(X_{j})$ and $\phi_{j}(X_{j})$. However, this happens to be the case if we require the gluing regions $A_{ij}$ to be open.

We have seen that the adjunction of arbitrarily-many topological spaces is again a topological space. However, there is no guarantee that the gluing process will preserve any pre-existing structure. The following result is a collection of conditions that suffice to preserve topological features. We state these here without proof, since the arguments involved routinely follow from Lemma 1.4 and basic facts of topology.

We will now identify conditions under which Hausdorff violation is guaranteed in the adjunction process. In order to do so, we will first recall a binary relation that formally encodes Hausdorff violation. We will opt for the notation used in [6, 5], though it should be noted that the same relation can also be found in [12] and [13, p. 67] in essentially equivalent forms.

Let M be a locally-Euclidean second-countable space, and consider the binary relation Y, defined as $x\textsf{Y}y$ if and only if every pair of open neighbourhoods of $x$ and $y$ necessarily intersect. The relation Y is reflexive and symmetric by construction, but it need not be transitive.¹¹1Interestingly, the relation Y is weaker than the Hausdorff relation used in [14]. It is likely that their relation is the transitive closure of Y, though a proof of this is beyond the scope of our paper. It is well-known that a topological space is Hausdorff whenever convergent sequences have unique limits. Although the converse is not always true, in a first countable space one can always construct a sequence that converges to both elements of a Hausdorff-violating pair. As such, in our context the relation $x\textsf{Y}y$ asserts the existence of a sequence that converges to both $x$ and $y$ in M. Given a subset $V$ of M, we define

$$\textsf{Y}^{V}:=\{x\in\textbf{M}\ |\ \exists y(y\in V\wedge x\textsf{Y}y)\},$$

thus $\textsf{Y}^{V}$ consists of all points in M that are Hausdorff-inseparable from $V$.

In what follows, we will use the Y-relation to describe the Hausdorff violating points of the canonical subspaces $\phi_{i}(M_{i})$. In order to do so, we will make use of the following definition.

This additional requirement will allow us to describe the Hausdorff-violating points using the M-relative boundaries of the canonical subspaces $M_{i}$. Before getting to the detailed description, we will first introduce some useful notation.

According to Theorem 1.4, we may view the subspaces $M_{i}$ as embedded inside the larger space M. To simplify notation, we will identify each $M_{i}$ with its image $\phi_{i}(M_{i})$. Under this convention we may view $M_{i}$ as an honest Hausdorff (open) submanifold of M. In what follows, it will be useful to describe the boundaries of the $M_{i}$ in M. We will use the notation $\bm{\partial}$ to denote the M-relative boundaries of the $M_{i}$, that is, we define

$$\bm{\partial}M_{i}:=\partial^{\textbf{M}}(\phi_{i}(M_{i})).$$

This boldface notation will similarly be used for the M-relative closures and interiors of sets. We will also identify each $A_{ij}$ in $M_{i}$ with its images $\phi_{i}(A_{ij})$ and $\phi_{j}(A_{ji})$ in M. In general, each $A_{ij}$ may have several M-relative boundary components. We denote these as follows:

$$\bm{\partial}^{i}A_{ij}:=\left(\partial^{\textbf{M}}\phi_{i}(A_{ij})\right)\cap\phi_{i}(M_{i}).$$

We will now show that requiring an adjunction system to have homeomorphic gluing regions establishes a clear relationship between the boundary operator $\bm{\partial}$ and the binary relation Y.

The above result allows us to conclude that the Hausdorff-violating pairs of a non-Hausdorff manifold can be identified as the pairwise boundaries of open submanifolds that are glued together, provided that these boundaries exist and are homeomorphic to each other. The following result extends this idea.

It is well-known that Hausdorff manifolds are necessarily paracompact and admit partitions of unity subordinate to any open cover [15]. In this section we will explore such results in the non-Hausdorff setting.

According to our discussion thus far, we may build non-Hausdorff manifolds from adjunction spaces. Since all Hausdorff manifolds are paracompact, we may expect there to be an analogue to the results of Theorem 1.6 for paracompactness. Indeed this is true: if we consider finitely-built adjunction spaces and further restrict our attention to those adjunction spaces in which the gluing regions have homeomorphic boundaries, then we obtain the following result.

The above result confirms that certain non-Hausdorff manifolds may be paracompact. In contrast to this, in the non-Hausdorff case there may be open covers that do not admit subordinate partitions of unity. This is immediate from standard results such as those in [15], however for the sake of completeness we will include a different argument.

It follows from the above that the open cover consisting of the canonical submanifolds $M_{i}$ cannot admit a subordinate partition of unity. It is likely that Theorem 2.6 will result in some non-trivial obstructions when attempting to recreate the standard theory of differential geometry in the non-Hausdorff regime, though that is beyond the scope of this paper.

In the remainder of this section we will argue that all non-Hausdorff manifolds can be expressed as adjunction spaces. Before getting to the argument, we first need to review the notion of an $H$-submanifold. We recall the following definition, originally found in [6].

Since non-Hausdorff manifolds are locally-Euclidean, in particular they are locally-Hausdorff. We may use this observation, together with an appeal to Zorn’s Lemma to argue the following.

It is not immediately clear whether a given Hausdorff submanifold is an $H$-submanifold. Fortunately this has been resolved by Hajicek [6, Thm. 2], who provides a useful criterion for determining whether or not a given subspace is an $H$-submanifold. In our notation, Hajicek’s criterion can be stated as follows.

We can use Hajicek’s criterion together with Lemma 2.4 to conclude that any non-Hausdorff manifold M built according to 2.2 admits the canonical subspaces $M_{i}$ as $H$-submanifolds. As we will see in Section 3, it is not always the case that the $M_{i}$ are the only $H$-submanifolds of M.

We will now use Proposition 2.8 to argue that all non-Hausdorff manifolds can be described via adjunction spaces. The idea behind this proof can be found in both [4] where it is proved for vector bundles, and [3], where it is proved for manifolds in general. For completeness we will provide a proof consistent with our notation.

Take $I$ to be an indexing set of size $n$, and consider the adjunction system $\mathcal{F}$ where:

• •

  each $M_{i}$ equal to a copy of the real line equipped with the standard topology,

• •

  each $A_{ij}$ equal to the set $(-\infty,0)$ for all $i,j$ distinct, and

• •

  each $f_{ij}:A_{ij}\rightarrow M_{j}$ is the identity map.

Once we require that each $A_{ii}$ equals $M_{i}$, the data contained in $\mathcal{F}$ forms an adjunction system. The adjunction space subordinate to $\mathcal{F}$ will be a collection of $n$-many copies of the real line all glued to each other along the negative numbers, as pictured in Figure 2. According to Theorems 2.1 and 2.5 the $n$-branched real line is a paracompact non-Hausdorff manifold.

Observe that the origins remain unidentified, thus there are $n$-many distinct equivalence classes $[0,i]$ in the $n$-branched real line. These points will violate the Hausdorff property. Moreover, there are $n$-many $H$-submanifolds equalling the canonically embedded $M_{i}$.

It should be noted that there are other interesting $1$-manifolds that can be built from finitely-many copies of the real line. Indeed, consider three copies of the real line, in which $A_{12}=(-\infty,0)$, $A_{23}=(0,\infty)$, and $A_{13}=\emptyset$, and all $f$-maps equal the identity. This data defines an adjunction system, and the resulting space will be a non-Hausdorff manifold in which there are three distinct copies of the origin in which the Y-relation is not transitive.

We will now construct a non-Hausdorff manifold using countably-many copies of the real line. Suppose that the indexing set $I$ coincides with the natural numbers, ordered linearly. Consider the triple $\mathcal{F}:=(\textbf{X},\textbf{A},\textbf{f})$, where:

• •

  each $M_{i}$ equals $\mathbb{R}$ with the standard topology,

• •

  $A_{ij}=\begin{cases}(-\infty,i)&\textrm{if}\ i<j\\ (-\infty,j)&\textrm{if}\ i>j\\ \mathbb{R}&\textrm{if}\ i=j\end{cases}$, and

• •

  each $f_{ij}$ is the identity map on the appropriate domain.

The reader may verify that this collection $\mathcal{F}$ does indeed form an adjunction system. The resulting adjunction space, which we denote by T, will be a countable collection of real lines, successively splitting in two at each natural number, as pictured in Figure 3.

In this example, the copies of $\mathbb{R}$ naturally sit inside T as $H$-submanifolds. In contrast to the previous example, there is an extra $H$-submanifold, given by

$$V=\bigcup_{i,j\in I}\phi_{i}(A_{ij}).$$

It should be noted that a similar space can be found as the rigid $1$-manifold of [16].

We can form branched planes by taking the product of the $n$-branched real line with a copy of $\mathbb{R}$. The resulting spaces will still be non-Hausdorff manifolds, and can be seen as a collection of Euclidean planes that branch out from each other along $x$-axes.

Consider the $2$-branched real plane. According to Theorem 2.10 there should be a way to construct this space in terms of adjunctions. The obvious choice is to form an adjunction system using:

1. 1.

   $M_{1}=M_{2}=\mathbb{R}^{2}$,

2. 2.

   $A_{12}=\{(x,y)\in\mathbb{R}^{2}\ |\ y<0\}$ is the open half-plane, and

3. 3.

   $f:A_{12}\rightarrow M_{2}$ is the identity map.

This collection satisfies the conditions of 2.1, and the resulting adjunction space, denoted $\textbf{R}^{2}$, is a paracompact non-Hausdorff manifold. The construction is depicted in Figure 4. The Hausdorff-violating points of $\textbf{R}^{2}$ lie on the two copies of the $x$-axis. For convenience, we will denote the two $x$-axes by $X_{1}$ and $X_{2}$.

By Theorem 2.4 we see that the two copies of $\mathbb{R}^{2}$ that naturally sit inside $\textbf{R}^{2}$ will be $H$-submanifolds. However, the space $\textbf{R}^{2}$ admits many other $H$-submanifolds. We will now briefly describe one. Consider the subspace $V$ of R defined by:

$$V=\textbf{R}^{2}\backslash\big{(}\{(x,0)\in X_{1}\ |\ x\leq 0\}\cup\{(x,0)\in X_{2}\ |\ x\geq 0\}\big{)},$$

that is, we remove from $\textbf{R}^{2}$ the non-positive $x$-axis from $X_{1}$ and the non-negative $x$-axis from $X_{2}$. Observe that both copies of the origin are removed. The subset $V$ is open in $\textbf{R}^{2}$ since its complement is closed. Moreover, the Y-set of $V$ is:

$$\textsf{Y}^{V}=\{(x,0)\in X_{1}\ |\ x>0\}\cup\{(x,0)\in X_{2}\ |\ x<0\}.$$

By a simple analysis, one can see that

$$\bm{\partial}V=\textsf{Y}^{V}\cup\{[(0,0),1],[(0,0),2]\}=\textbf{Cl}(\textsf{Y}^{V}),$$

and thus Hajicek’s criterion guarantees that $V$ is an $H$-submanifold of $\textbf{R}^{2}$.

We finish this section with an example of a compact non-Hausdorff manifold. Roughly speaking, this space will be a $2$-sphere in which the equatorial copy of $S^{1}$ is wrapped around itself so that the antipodal points become Hausdorff-inseparable, as pictured in Figure 5. We will denote this non-Hausdorff sphere by $\textbf{S}^{2}$.

The original construction of $\textbf{S}^{2}$ uses a modified torus to obtain the doubled equator [7]. A more succinct construct identifies antipodes of the $2$-sphere everywhere outside the equatorial copy of $S^{1}$ [8]. Both of these constructions are not adjunction spaces, and it has even been suggested that such a colimit construction does not exist [17]. We will now remedy this by providing a construction of $\textbf{S}^{2}$ in terms of an adjunction of punctured $2$-spheres. We will start with the standard 2-sphere, embedded into $\mathbb{R}^{3}$ for convenience. Consider the equatorial copy of $S^{1}$ parameterised as

$$S^{1}=\{(x,y,0)\ |\ x^{2}+y^{2}=0\}.$$

Let $a=(1,0,0)$ and $b=(-1,0,0)$ be antipodes on $S^{1}$. Consider the following punctured spheres:

• •

  $M_{1}$ and $M_{2}$ are $S^{2}\backslash\{a\}$, and

• •

  $M_{3}$ and $M_{4}$ are $S^{2}\backslash\{b\}$.

We will now glue the $M_{i}$ in such a manner that halves of each $S^{1}$ eventually form a double-covered equator. Within each $M_{i}$ there are two arcs connecting the antipodal points $a$ and $b$. We denote these by $L^{i}_{+}$ and $L^{i}_{-}$, that is,

$$L^{i}_{+}:=\{(x,y,0)\in M_{i}\ |\ y\geq 0\}\ \textrm{and}\ L^{i}_{-}:=\{(x,y,0)\in M_{i}\ |\ y\leq 0\}.$$

Observe that in each $M_{i}$ the segments $L^{i}_{\pm}$ are closed subsets. The following table contains all of the pairwise gluing regions $A_{ij}$.

	$M_{1}$	$M_{2}$	$M_{3}$	$M_{4}$
$M_{1}$	$M_{1}$	$M_{1}\backslash S^{1}$	$M^{1}\backslash L^{1}_{-}$	$M^{1}\backslash L^{1}_{+}$
$M_{2}$	$M^{2}\backslash S^{1}$	$M_{2}$	$M^{2}\backslash L^{2}_{+}$	$M^{2}\backslash L^{2}_{-}$
$M_{3}$	$M^{3}\backslash L^{3}_{-}$	$M^{3}\backslash L^{3}_{+}$	$M_{3}$	$M^{3}\backslash S^{1}$
$M_{4}$	$M^{4}\backslash L^{4}_{+}$	$M^{4}\backslash L^{4}_{-}$	$M^{4}\backslash S^{1}$	$M_{4}$

The above data indeed defines an adjunction system. It is perhaps easiest to visualise the gluing as a successive process. Figure 6 depicts the stages in the construction.

We will restrict our attention to a basic type of non-Hausdorff manifold. Following the terminology of [5], we will refer to these as simple non-Hausdorff manifolds. The definition is as follows.

The above definition captures the idea of a non-Hausdorff manifold branching out finitely-many times from a single submanifold. Note that the $n$-branched real line and the $2$-branched plane of Section 3 are both simple, whereas the infinitely-branching real line is not.

The main benefit of considering simple non-Hausdorff manifolds is that they admit a particularly easy description of Hausdorff violation. Observe that the component spaces of a simple non-Hausdorff manifold will have homeomorphic boundaries, since we may use the identity map to define each extension $\tilde{f}_{ij}$, and thus the boundary-maps $g_{ij}$ equal $\phi_{j}\circ\phi_{i}^{-1}$ (cf. 2.2). We also make the following useful observation, which follows immediately from Definition 4.1.

It follows from the above that the Hausdorff-violating sets in a simple non-Hausdorff manifold will consist of the $I$-many (disjoint) connected components of the boundary of $A$. Thus we may conclude that the non-Hausdorff sphere of Section 3.4 is not simple, since the set $\textsf{Y}^{\textbf{S}^{2}}$ is path-connected.

We will now argue that simple non-Hausdorff manifolds admit a characterisation of their canonical subspaces $M_{i}$. There is an argument due to Müller [5] that certain simple branched Minkowski spaces admit a characterisation in terms of existent limit points. We will extend this result by passing into a broader generality, and by using a weaker condition.

Recall that Hajicek’s criterion demands that each $H$-submanifold satisfies $\bm{\partial}V=\textbf{Cl}(\textsf{Y}^{V})$. We will argue that in a simple non-Hausdorff manifold, the canonical subspaces $M_{i}$ are the unique $H$-submanifolds satisfying the stricter equality $\bm{\partial}V=\textsf{Y}^{V}$. Our argument will be via induction on the size of the indexing set underlying $\mathcal{F}$. In order to do so, we will make use of the following key lemma.

We may use the above lemma together with Theorem 2.4 to obtain the following result.