Vector Bundles over non-Hausdorff Manifolds

In this paper we generalise the theory of real vector bundles to a certain class of non-Hausdorff manifolds. In particular, it is shown that every vector bundle fibred over these non-Hausdorff manifolds can be constructed as a colimit of standard vector bundles. We then use this description to introduce various formulas that express non-Hausdorff structures in terms of data defined on certain Hausdorff submanifolds. Finally, we use \v{C}ech cohomology to classify the real non-Hausdorff line bundles.


Introduction
The standard theory of differential geometry assumes that any pair of points in a manifold can be separated by disjoint open sets.This, known as the Hausdorff property, is typically imposed for technical convenience.Indeed, it can be shown that any Hausdorff, locally-Euclidean, second-countable topological space necessarily admits partitions of unity subordinate to any open cover.In turn, these partitions of unity can then be used to construct some familiar features of a manifold.
Conversely, non-Hausdorff manifolds will always have open covers that cannot admit partitions of unity [1].Despite this inconvenience, non-Hausdorff manifolds seem to arise within several areas of mathematics and theoretical physics.Within mathematics, one can find non-Hausdorff manifolds in the leaf spaces of foliations [2], [3], and in the spectra of certain C * algebras [4].Within theoretical physics, some general discussion of non-Hausdorff spacetimes can be found in [5]- [7], and non-Hausdorff manifolds can be found in the maximal extension of Taub-NUT spacetimes [8], in certain twistor spaces [9], and to various degrees in the study of branching spacetimes [10]- [15].Despite occurring in both mathematics and physics, a general theory of non-Hausdorff geometry is lacking.In [1], initial steps were made with a study of the topological properties of non-Hausdorff manifolds.The key idea underpinning their study were the observations of [2], [11]- [13], which suggest that non-Hausdorff manifolds can always be constructed by gluing together ordinary Hausdorff manifolds along open subspaces.This gluing operation is commonly known as an adjunction space, though it may also be seen as the topological colimit of a particular diagram.The goal of this paper is to extend the formalism of [1] to include both smooth manifolds and the vector bundles fibred over them.This paper is organised as follows.In Section 1, we start by recalling some details of [1], and by making some important restrictions on the types of non-Hausdorff manifolds that we will consider.Once this is done, we will show that our non-Hausdorff manifolds naturally inherit smooth structures from the Hausdorff submanifolds that comprise them.We will then describe the ring of smooth real-valued functions, and use a modified version of partitions of unity to extend functions from Hausdorff submanifolds into the ambient non-Hausdorff space.We will show that, due to the contravariance of the C ∞ functor, the space of functions of a non-Hausdorff manifold can be seen as the fibred product of the functions defined on each of the Hausdorff submanifolds.
In Section 2 we will begin to consider the vector bundles that one may fiber over a non-Hausdorff manifold.In a direct analogy to the theorems of [11] and [1], we will show that every vector bundle over a non-Hausdorff manifold can be realised as a colimit of bundles fibred over the Hausdorff submanifolds.We will then use this result to describe the space of sections of a non-Hausdorff vector bundle.A similar fibred-product description will emerge, this time due to the contraviance of the Γ functor.This description will then be used to construct Riemmannian metrics on our non-Hausdorff manifolds.
In Section 3 we will explore the prospect of classifying the line bundles over a non-Hausdorff manifold.We will do this by appealing to the wellknown result that real line bundles over Hausdorff manifolds can be counted by the first Čech cohomology group with Z 2 coefficients.As such, we will first provide descriptions for the Čech cohomology groups of our non-Hausdorff manifolds as Mayer-Vietoris sequences built from the cohomologies of certain Hausdorff submanifolds.We will see that, despite the Čech functor Ȟ being contravariant, in general there is no fibred product description for the cohomology groups of our non-Hausdorff manifolds.This result suggests that a general formula for the exact number of line bundles over a non-Hausdorff manifold is probably not possible.However, in the case that all of the spaces under consideration are connected, we will derive an alternating-sum formula for the number of line bundles that a non-Hausdorff manifold admits.
Throughout this paper we will assume familiarity with differential geometry up to the level of [16] or [17].In Section 3, unless otherwise stated, all results regarding the Čech cohomology are taken from [18].We will assume that all manifolds, Hausdorff or otherwise, are locally-Euclidean, secondcountable topological spaces, and as a convention we will use boldface notation to denote non-Hausdorff manifolds and their functions.

Smooth non-Hausdorff Manifolds
In this section we will introduce a formalism for smooth non-Hausdorff manifolds.We will start with the relevant topological features, and then we will use these to endow our manifolds with smooth structures.We will then discuss smooth real-valued functions on non-Hausdorff manifolds.

The Topology of non-Hausdorff Manifolds
We will now briefly review the topological properties of non-Hausdorff manifolds.All facts in this section are stated without proof, since they can already be found in [1], albeit in a slightly more general form.
We begin by generalising the adjunction spaces found in many standard texts such as [19] or [20].Our data F will consist of three key components: and a set f = {f ij } i,j∈I of continuous maps of the form f ij : M ij → M j .In order to ensure that this data induces a well-defined topological space, we will need to impose some consistency conditions.This is captured in the following definition.
Definition 1.1.The triple F = (M, A, f) is called an adjunction system if it satisfies the following conditions for all i, j ∈ I.
Given an adjunctive system F , we define the adjunction space subordinate to F to be the quotient of the disjoint union: where ∼ is the equivalence relation that identifies elements (x, i) and (y, j) of the disjoint union whenever f ij (x) = y.Observe that the conditions of Definition 1.1 are precisely the conditions needed to ensure that ∼ is a welldefined equivalence relation.Points in the adjunction space are equivalence classes of elements of each of the spaces M i .We will denote these classes by By construction there exists a collection of continuous maps φ i : M i → F M i that send each x in M i to its equivalence class [x, i] in the adjunction space.
In the case that an adjunction system F has an indexing set of size 2 we will refer to F M i as a binary adjunction space.It is well-known that these spaces can be equivalently seen as the pushout of the diagram in the category of topological spaces [20].Similarly, it can be shown that general adjunction spaces are colimits of the diagram formed from the data in F , with the maps φ i satisfying the analogous universal property.Lemma 1.2.Let F be an adjunction system and let F M i be the adjunction space subordinate to F .Suppose that ϕ i : M i → X is a collection of continuous maps such that ϕ i = ϕ j • f ij for every i, j in I. Then there is a unique continuous map α : With this in mind, we will regularly borrow language from category theory and refer to an adjunctive system and its subordinate adjunction space as a diagram and its colimit, respectively.
The canonically-induced maps φ i can be particularly well-behaved, provided that we make some extra assumptions on the data in F .The following result makes this precise.Due to the above, we will often refer to the φ i as the canonical embeddings.We will also use M, N, ... to denote adjunction spaces subordinate to any system satisfying the assumptions of Lemma 1.3.Given that each canonical embedding φ i acts as a homeomorphism, we will often simplify notation and identify each M i with its image φ i (M i ).
According to Lemma 1.3 the colimit M resulting from a diagram F of Hausdorff manifolds may be a locally-Euclidean second-countable topological space.However, there is no guarantee that any such M will be non-Hausdorff.As a matter of fact, we will need to assume that the open submanifolds M ij are proper, open submanifolds whose boundaries are pairwise homeomorphic.The following result summarises the consequences of this assumption.
Theorem 1.4.Let F be an adjunctive system that satisfies the criteria of Lemma 1.3, and let M denote the adjunction space subordinate to F .Suppose furthermore that each gluing map

The Hausdorff-violating points in M occur precisely at the M-relative
boundaries of the subspaces M i .

Each M i is a maximal Hausdorff open submanifold of M.
3. If the indexing set I is finite, then M is paracompact.

If each M i is compact and the indexing set
Throughout the remainder of this paper we will take M to be a fixed but arbitrary non-Hausdorff manifold that is built as an adjunction of finitelymany Hausdorff manifolds M i according to both Lemma 1.3 and Theorem 1.4.Consequently, M is a paracompact, locally-Euclidean second-countable space in which the manifolds M i sit inside M as maximal Hausdorff open submanifolds.

Adjunctive Subspaces and Inductive Colimits
Genreally speaking, given an adjunctive system F , we may form other diagrams by selectively deleting data pertaining to particular indices in the set I. This procedure will define what is known as an adjunctive subsystem.Given some subset J ⊂ I, and the adjunctive subsystem F ′ formed by only considering the data in J, we may use Lemma 1.2 to construct a map between the adjunction spaces subordinate to F and F ′ , given by: κ : Note that here we are using the double-bracket notation to distinguish the two types of equivalence classes.It can be shown that this map κ is an open topological embedding whenever F satisfies the criteria of Lemma 1.3 [1, §1.2].
So far we have constructed our non-Hausdorff manifold M by gluing the subspaces M i together simultaneously.However, it will also be useful to express M as a finite sequence of binary adjunction spaces.We will refer to this sequential construction as an inductive colimit.We will now justify the equivalence between these two points of view.The case for a colimit of three manifolds is shown below.Lemma 1.5.Let M be a non-Hausdorff manifold built from three manifolds M i .Then M is homeomorphic to an inductive colimit.
Proof.For readability we will only provide a sketch of the proof here -the full details can be found in the appendix.We may take the gluing region A := M 13 ∪ M 23 of M 3 , and define the map sending each element of A to M i3 along the gluing map f i3 , and then into the equivalence class f i3 , i in M 1 ∪ f 12 M 2 .Note that by the gluing condition (A3) of Definition 1.1 there is no ambiguity here.We may thus glue M 3 to M 1 ∪ f 12 M 2 along the map f 13 ∪ f 23 .The universal properties of both (M 1 ∪ f 12 M 2 ) ∪ f 13 ∪f 23 M 3 and M can then be invoked in order to create the desired homeomorphism.
As one might expect, we may generalise the previous result to all finite adjunctive systems.Theorem 1.6.Let M be a non-Hausdorff manifold built from n-many manifolds M i .Then M is homeomorphic to an inductive colimit.
Proof.We will proceed by induction on the size of the indexing set I. The case of I = 3 is already proved as the previous result.So, suppose that the hypothesis holds for all non-Hausdorff manifolds with indexing set I of size n.Let M be a non-Hausdorff manifold built from a diagram F , with indexing set I of size n + 1.Consider the subsystem F ′ formed from F by deleting all data pertaining to the space M n+1 .Then F ′ is a well-defined adjunctive system that yields a non-Hausdorff manifold N.
In analogy to Lemma 1.5, consider the gluing region of M n+1 defined by A := i≤n M (n+1)i together with the map f : A → N which sends each element x in A to its equivalence class f i(n+1) (x), i in N.This yields a binary adjunction space N ∪ f M n+1 .Again in analogy to Lemma 1.5, we may invoke the universal properties of both N ∪ f M n+1 and M to create a homeomorphism between the two spaces.The result then follows by applying the induction hypothesis to N.

Smooth Structures
In Lemma 1.3, the topological structure of each manifold M i was preserved by requiring that the gluing maps f ij act as open embeddings.The consequence was that the canonical embeddings φ i : M i → M were also open, which ensured that the local Euclidean structure of the manifolds M i can be transferred to M. Formally, this can be achieved by using a collection of atlases A i of the manifolds M i to define an atlas A on M: We will now argue that this technique defines a smooth atlas of M, provided that the gluing maps f ij are all smooth.
Lemma 1.7.Let M be a non-Hausdorff manifold built according to Lemma 1.3.If, additionally, the M i are smooth manifolds and the f ij are all smooth maps, then M admits a smooth atlas.
Proof.Consider the atlas A of M as described above.We show that the transition maps of this atlas will be smooth in the Euclidean sense.Suppose that [x, i] is a element of M, and consider two charts of M at this point.Since [x, i] is an equivalence class, in general these two charts may come from different atlases A i and A j .As such, the charts will be of the form , where (U α , ϕ α ) is a chart of M i at the point x, and (U β , ϕ β ) is a chart of M j at the point f ij (x).The transition maps in M will then be: which are smooth since each f ij is.
Since each smooth atlas has a unique maximal extension, we may consider the smooth structure induced from the atlas A described above.This allows us to effectively see our non-Hausdorff manifold M as smooth.We will now introduce some useful criteria for identifying smooth maps.The result then follows as an application of Prop.2.6 of [16].
According to the above result, we may now view the canonical embeddings φ i : M i → M as smooth open embeddings.We may also argue for a universal property as in Lemma 1.2 and thus interpret M as the colimit of the diagram F in the category of smooth locally-Euclidean spaces.The following remark makes precise the primary object of study in this paper.Remark 1.9.In addition to the conditions of Lemma 1.3 and Theorem 1.4, hereafter we will also assume that our non-Hausdorff manifold M is smooth in the sense of Theorem 1.7.Moreover, in order to study smooth objects defined on M, we will also need to assume that the gluing regions M ij have diffeomorphic boundaries, which in this context means that the closures Cl M +i (M ij ) are all smooth closed submanifolds, and the extended maps f ij of Theorem 1.4 are all smooth.

Smooth Functions
Since smoothness is a local property, we may define smooth functions on M as in the Hausdorff case.Moreover, we may still appeal to the structure of the real line to view the space C ∞ (M) as a unital associative algebra.In this section we will study C ∞ (M) in some detail.To begin with, we will discuss some techniques for constructing functions on M. We will then use these techniques to establish a relationship between C ∞ (M) and the algebras C ∞ (M i ).

The Construction of Functions on M
In the Hausdorff setting, there are two useful techniques for constructing smooth functions on a manifold.Roughly speaking, these are: 1. to glue together smooth functions defined on open subsets, and 2. to extend functions defined on a closed subset.
Will will refer to these two techniques are often known as the "Gluing Lemma" and the "Extension Lemma", respectively.For a more thorough discussion of these two constructions, the reader is encouraged to see Corollary 2.8 and Lemma 2.26 of [16].
We will now create versions of these two results for our non-Hausdorff manifold M. To begin with, we show that the smooth functions on M can be built by gluing together a collection of smooth functions that are defined on the Hausdorff submanifolds Moreover, the restriction of r to each M i equals r i , which is smooth by assumption.The result then follows from an application of Lemma 1.8.
The above result is a straightforward analogue of the Gluing Lemma.In contrast, an analogue of the Extension Lemma is more involved.The underlying complication is that the extension of a functions defined on a closed subset necessarily requires partitions of unity subordinate to any open cover.As proved in [1], in the non-Hausdorff setting we have the following obstruction to the existence of partitions of unity.

Lemma 1.11. Any open cover of M by Hausdorff sets does not admit a partition of unity subordinate to it.
Since each M i is an open, Hausdorff submanifold of M, we cannot directly use any partitions of unity subordinate to the cover {M i }.However, our restrictions on the topology of M are stringent enough so as to allow certain techniques involving partitions of unity.Indeed, the requirement that the gluing regions M ij have diffeomorphic boundaries may allow us to smoothly transfer objects between the submanifolds M i , and the requirement that M be a finite colimit may allow this transfer to be performed inductively.We now illustrate this approach with a construction of non-zero functions on M. Theorem 1.12.Any function r i on M i can be extended to a function on M.
Proof.We proceed by induction on the size of I. Suppose first that M is a binary adjunction space M 1 ∪ f 12 M 2 , and without loss generality suppose that i = 2. Let r 2 be any smooth function on M 2 .The restriction of r 2 to the closed submanifold Cl M 2 (M 12 ) is also a smooth function, and moreover the composition is a smooth function on the copy of Cl M 1 (M 12 ) that sits inside M 1 .We can now use a partition of unity argument on M 1 to extend r 2 • f 12 to a function r 1 defined on all of M 1 .We have thus created a pair of functions r i in C ∞ (M i ) that agree on M 12 .This pair of functions satisfies the antecedent of Lemma 1.10, and thus define a function r on M which restricts to r 2 on M 2 .
Suppose now the hypothesis holds for all non-Hausdorff manifolds constructed as the colimit of n-many Hausdorff manifolds M i , according to Theorem 1.4.Let M be a non-Hausdorff manifold defined as the colimit of (n + 1)-many manifolds M i .Without loss of generality, pick any smooth function r n defined of M n .According to Theorem 1.6, we may view M as the inductive colimit: where we glue along the set A := i≤n M (n+1)i .By the induction hypothesis, there exists some non-zero function r defined on the adjunction space N that extends r n .Using the fact that each f ij can be extended to a diffeomorphism of boundaries, we can extend the collective function f to a closed function f : Cl M n+1 (A) → Cl N (f (A)).The map f is well defined since the extensions f (n+1)i satisfy a cocycle condition as in Definition 1.1, and moreover f is a diffeomorphism since locally it equals f (n+1)i .
We can restrict r to Cl N (A) and then the map r • f will be a smooth function defined on Cl M n+1 (A).Using a partition of unity on M n+1 , we may extend the function r • f to some function r ′ defined on all of M n+1 .We may then use Lemma 1.10 on r and r ′ to form a globally-defined function on all of M, which by construction will restrict to r n on M n .
Usefully, smooth functions can be pulled back along smooth maps via precomposition.In the case of the canonical embeddings φ i , precomposition gives an algebra morphism As an immediate application of Theorem 1.12 we make the following observation.Corollary 1.13.For each M i , the map We saw in the previous section that we can always create smooth functions on M by gluing together functions that are defined on the component spaces M i , provided that they are compatible on the overlaps M ij .The following result expresses this principle at the level of algebras.
Theorem 1.14.The algebra C ∞ (M) is isomorphic to the fibred product Proof.Consider the map Φ * that acts on each smooth function r on M by: By the commutativity of the diagram F , we have that thus Φ * takes image in the fibred product F C ∞ (M i ).Moreover, the map Φ * is also an algebra homomorphism, since all the φ * i are.The map Φ * is injective since any pair of distinct functions r and r ′ on M must differ on one of the M i , and it is surjective by Corollary 1.13.Since every bijective algebra homomorphism is an isomorphism, this completes the proof.
We saw in the form of Lemma 1.2 that the adjunction space M is the colimit F in the category of topological spaces.Subsequent remarks in Section 1.2 confirmed that this colimit also exists in the category of smooth locally-Euclidean manifolds.Since C ∞ is a contravariant functor, in principle we may apply it to all of F to obtain a diagram in the category of unital associative algebras.The following result confirms that C ∞ (M) is the correct limit of this contravariant diagram.

Lemma 1.15. Let A be a unital associative algebra together with a collection of I-many algebra morphisms ρ
Proof.Consider the map α defined by α(a) = (ρ 1 (a), ..., ρ n (a)).This is clearly an element of the direct sum i C ∞ (M i ), and moreover the commutativity assumption of the ρ i ensures that α takes image in the fibred product.That α is an algebra morphism follows from the fact that each ρ i is.Finally, the uniqueness of α is guaranteed since the morphisms from

Vector Bundles
In this section we will construct vector bundles over our non-Hausdorff manifold M. We will start by defining a an adjunction of Hausdorff bundles E i that are fibred over each of the submanifolds M i .After this, we will argue that every vector bundle over M can be constructed in this manner.We then will generalise Theorem 1.14 by providing a description of sections of any vector bundle fibred over M, eventually finishing with a discussion of Riemannian metrics in the non-Hausdorff setting.All of the basic details of vector bundles can be found in standard texts such as [16], [17].

Colimits of Bundles
Suppose that we are given a collection of rank-k vector bundles will be an open submanifold of E i .In order to form an adjunction space from this data we need a collection of bundle morphisms F ij : E ij → E j that cover the gluing maps f ij .According to Definition 1.1, these maps also need to satisfy the cocycle condition With all of this data in hand, we may use Lemma 1.7 on the collection of bundles E i π i − → M i to form a non-Hausdorff smooth manifold E. We denote by χ i the canonical embeddings of each E i into E.In order to describe a bundle structure on E, we would first like to define a projection map π : E → M.This amounts to completing the commutative diagram simultaneously for all i, j in I.
We will denote points in E by [v, i], where v ∈ E i .Strictly speaking this is an abuse of notation, since the equivalence classes of E are different from the equivalence classes used to define points in M.However, in this notation the projection map π : E → M can be easily defined as π( Observe that π is well defined since our requirement that the bundle morphisms F ij cover the gluing map for all v in E ij .Moreover, the map π is manifestly smooth since its local expression around any point [x, i] of M will be the composition By construction the map π is surjective, and furthermore we may endow the preimages π −1 ([x, i]) with the structure of a rank-k vector space induced from the fibre π −1 i (x) of E i .In our notation, addition and scalar multiplication are given by respectively.These operations are well-defined by our assumption that the F ij are bundle morphisms, and consequently the fibres of E will indeed be k-dimensional vector spaces.
In direct analogy to the construction of smooth atlases in Section 1.2, we can describe local trivialisations of E using the bundles E i .Suppose that we have a point [x, i] in M, and fix U to be a local trivialisation of the bundle E i at the point x, with trivialising map Θ.Since φ i is an open map, we can consider the set φ i (U) as an open neighbourhood of [x, i] in M.This data can be arranged into the following diagram where the p 1 are projections onto the first factor.The trivialising map Ψ can then be defined as the composition Ψ := (φ i , id) • Θ • χ −1 i .Transition functions for local trivialisations around points in the gluing regions M ij will be: which mimic the local properties of the bundle morphisms F ij .
According to our discussion thus far, we may consider E as a rank-k vector bundle fibred over the non-Hausdorff manifold M in which the maps χ i of E are injective bundle morphisms that cover the canonical embeddings φ i .This is summarised in the following result.Theorem 2.1.Let G := (E, B, F) be a triple of sets in which: consists of the restrictions of the bundles E i to the intersections M ij , and that cover the gluing maps f ij and satisfy the condition Then the resulting adjunction space E := G E i has the structure of a non-Hausdorff rank-k vector bundle over M in which the canonical inclusions χ i : E i → E are bundle morphisms covering the canonical embeddings φ i : We will now confirm that the bundle E described satisfies a certain universal property.
Theorem 2.2.Let E be a vector bundle over M as in Theorem 2.1, and let F ρ − → M be a vector bundle.If there exist bundle morphisms ξ i : E i → F covering the canonical maps φ i satisfying ξ i = ξ j • F ij for all i, j in I. Then there exists a unique bundle morphism α : E → F such that ξ i = χ i • α for all i in I.
Proof.Since all bundles are smooth manifolds and all bundle morphisms are smooth maps, we may apply the universal property of smooth non-Hausdorff manifolds to conclude that there exists a unique smooth map α from E to F defined by Observe that since the maps χ i : E i → E and the maps ξ i : E i → F both cover the canonical embeddings φ i , we have that and thus the map α covers the identity map on M.Moreover, α acts linearly on fibres of E since α coincides with the map ξ i • χ −1 i on each E i , from which we may conclude that α is a bundle morphism.
According to the above result, we may interpret any vector bundle E constructed according to Theorem 2.1 as a colimit in the category of smooth vector bundles over locally-Euclidean, second-countable spaces.

A Reconstruction Theorem
It is well-known that all non-Hausdorff manifolds can be constructed using adjunction spaces.In essence, this result follows from the fact that maximal Hausdorff submanifolds of a given non-Hausdorff manifold form an open cover [13].It is then possible to fix a minimal open cover by Hausdorff submanifolds, and then to glue them along the identity maps defined on the pairwise intersections.The details of this result can be found in [1], [11] and [12] in different forms.
We will now argue that all vector bundles over M are colimits in the sense of Theorem 2.1.The argument is similar to the manifold case: we can always restrict a bundle down to the component spaces M i to create a collection of Hausdorff bundles that can then be re-identified.Formally, this restriction is obtained by taking the pullbacks of the bundle E along the canonical embeddings φ i .Theorem 2.3.Let E be some vector bundle over M. Then E is isomorphic to a colimit bundle of the form detailed in Theorem 2.1.
Proof.We would like to define a colimit bundle by gluing the pullback bundles φ * i E along bundle morphisms that act by identity on each fiber.Formally, this can be achieved by the data G = (E, B, G), where: • B consists of the restricted bundles This data satisfies the criteria of Theorem 2.1, thus we may conclude that F := G E i is a vector bundle over M. By construction, the pullback bundles φ * i E cover the canonical embeddings φ i via the maps p 2 which project onto the second factor of the Cartesian product.This means that pairwise we have the following diagram.
This diagram commutes since the morphisms F ij act by the identity on their second factors.According to the universal property of the colimit bundle F we may induce a (unique) bundle morphism α from F to E. Pointwise, the map α acts by [(x, v), i] → v.This map is clearly bijective, from which it follows that α is a bundle isomorphism.

Sections
In Theorem 1.14 we saw that the ring of smooth functions of M is naturally isomorphic to the fibred product F C ∞ (M i ).In categorical terms, this product can be seen as the limit of a diagram that is formed by applying the C ∞ functor to all of the data in F .We will now extend this result to sections of arbitrary bundles over M. Throughout this section we take E to be an arbitrary but fixed vector bundle over M, and we will denote by E i the restricted (Hausdorff) bundles over the subspaces M i .In analogy to Lemma 1.10, we will first show that every section of the vector bundle E can be described by gluing sections of E i that are compatible on overlaps.
Lemma 2.4.For each i in I, let s i be a section of E i .If the equality Proof.Observe first that s is well-defined, since: The map s is a right-inverse of the projection map Finally, since the restriction of s to each M i equals χ i • s i • φ −1 i , the map s is smooth by Lemma 1.8.
Following on from the approach of Section 1.3.1,we may now prove an analogue to Theorem 1.12.
Theorem 2.5.Any section s i of E i can be extended to a section s of E.
Proof.We will proceed as in Theorem 1.12, that is, by induction on the size of indexing set I. Suppose first that M is a binary adjunction space As in Theorem 1.12, we may take a non-zero section on E 2 , restrict it to the closure Cl M 2 (M 12 ), and then use the diffeomorphism f 12 to pull back s 2 to some smooth section s 12 defined on the submanifold Cl M 1 (M 12 ) of M 1 .Using the fact that F 12 is a bundle isomorphism onto its image, we may interpret s 12 as a section of the closed subbundle Cl E (E 12 ) defined over Cl M 1 (M 12 ).
In order to extend s 12 into the rest of M 1 , we will need to apply a generalisation of the Extension Lemma for sections of vector bundles (cf.Lemma 10.12 of [16]).The idea behind this generalised Extension Lemma is essentially the same as in the case of smooth functions -we may always endow the closure Cl M 1 (M 12 ) with an outward-pointing collar neighbourhood U, and then use the flow of the associated vector field to extend s 12 to all of U. Using a partition of unity subordinate to the open cover {U, M 1 \Cl M 1 (M 12 )} of M 1 , we may then create a section s 1 of E 1 .
Observe that by construction, any section s 1 created according to the above procedure will restrict to s 12 on M 12 .A global section s of E then exists by applying of Lemma 2.4 to s 1 and s 2 .The inductive case follows the same structure as Theorem 1.12, this time using the Extension Lemma for vector bundles instead.
Using the above, we can now create an argument similar to that of Theorem 1.14.
Theorem 2.6.For any vector bundle E over M, we have that Proof.The argument is the same as that of Theorem 1.14, except that this time we use Lemma 2.4 and Theorem 2.5.

Riemannian Metrics
In order to discuss Čech cohomology in the next section, we will first need to confirm that metrics exist on arbitrary vector bundles fibred over M. The precise construction of such metrics will be similar to the approach of Lemmas 1.12 and 2.4.However, in order to use this technique we first need to confirm the following.

Lemma 2.7. Let E be a vector bundle over M with colimit representation
Proof.For readability we will only provide a sketch, since the details of this argument can already be found in [12].Let us denote by F ij the bundle morphisms that are used to construct E from the E i .Since the F ij are diffeomorphisms from E ij to E ji , the differentials dF ij will be bundle isomorphisms between the tangent bundles T E ij and T E ji .Moreover, a basic property of differentials confirms that Consequently, we may glue each tangent bundle T E i along the differentials dF ij to create the bundle G T E i .The differentials dχ i : T E i → T E of the canonical embedding maps χ i : E i → E can then be used together with Theorem 2.2 to conclude that the bundle G T E i is isomorphic to T E. A similar argument can be made for the bundle T (0,2) E, except that this time we glue along the maps that pull back the (0, 2)-tensors along the diffeomorphisms F ij .
A metric tensor on E may be seen as a global non-vanishing section of the bundle T (0,2) E that is symmetric and positive-definite in its local expression.Using the above result, we may readily construct metrics on any vector bundle fibred over M. Theorem 2.8.Any vector bundle E over M admits a metric.
Proof.According to Theorem 2.3, we may view E as a colimit of bundles E i that are fibred over the Hausdorff submanifolds M i .Lemma 2.7 then allows us to express the tensor bundle T (0,2) E as a colimit of the tensor bundles T (0,2) E i .We may then apply the construction of Theorem 2.5 and use an inductive series of partitions of unity defined on each M i in order to construct a global section g of the bundle T (0,2) E. Note that we may guarantee that g is a bundle metric if we start with bundle metrics g i of E i and use the fact that bundles metrics are closed under convex combinations.
In the next section we will need to appeal to the existence of Riemannian metrics defined on M. Fortunately, these exist as an application of the previous result to T M. Corollary 2.9.M admits a Riemannian metric.

Čech Cohomology and Line Bundles
Theorems 2.1 and 2.2 tell us that any line bundle over M exists as a colimit of line bundles defined on each of the submanifolds M i .In this section we will explore how this relationship manifests in the language of Čech cohomology.To begin with, we will proceed generally and study the Čech cohomology of M in terms of the cohomologies of the M i .
Before getting to any results, we will first briefly recall the formalism of Čech cohomology.Aside from a slight change in notation, we will closely follow [18].Consider an arbitrary topological space X, with an open cover U := {U α | α ∈ A}, and let G be an Abelian group.We will use index notation to abbreviate multiple intersections of open sets in U, that is, we will write A degree-q Čech cochain f consists of a choice of constant functions from the (q + 1)-ary intersections of elements of U to the group G.In symbols: The space of Čech q-cochains, which we will denote by Čq (X, U, G), consists of all sets f of the above form.This space naturally inherits an Abelian group structure from G. We denote by δ the Čech differential, which raises the degree of each cochain by one.In additive notation, the map δ acts as follows: where here the caret notation αi denotes exclusion of that index.The Čech differential is a group homomorphism that squares to zero, and we denote the resulting cohomology groups by Ȟq (X, U, G).We may define these groups for any open cover U, and the collection of all Ȟq (X, U, G) can be made into a directed system of groups once ordered by refinement.The Čech cohomology of X is then defined as the direct limit: In what follows we will need to make use of the pullback of Čech cochains, so we recall this notion now.Suppose that ϕ : X → Y is a continuous map and U is an open cover of the topological space Y .We may define an open cover V of X by considering all sets of the form ϕ −1 (U), where U is an element of U. We may then define a map ϕ for all locally-constant functions on Y .By construction, the pullback ϕ * is a group homomorphism.

Čech Cohomology via a Mayer-Vietoris Sequence
We will now set about expressing the Čech cohomology Ȟq (M, G) in terms of the groups Ȟq (M i , G).We will obtain this relationship inductively, so throughout this section we will assume that M can be expressed as the colimit of two Hausdorff manifolds M 1 and M 2 , in accordance with Remark 1.9. 1 We will also assume that M is endowed with a fixed but arbitrary open cover U. We will not need to appeal to the particular structure of the Abelian group G, so we suppress this in our notation.
We start with a derivation of a Mayer-Vietoris sequence for M. In order to do so, we will need to make use of the pullbacks of cochains.According to our configuration, we have the following commutative diagram of pullbacks where here U i = {φ −1 i (U) | U ∈ U}, and U 12 is defined similarly.As a convention we will index these three covers using the relevant subsets of the indexing set of U.
We can combine the various pullback maps in order to create a single sequence from the above diagram.We will consider two maps: Φ * , which acts on cochains on M by concatenating the pullbacks φ * i (as in 1.14, 2.6), and the map ι * 12 − f * 12 , which pulls back a pair of cochains defined on the M i to the subset M 12 and then computes their difference.The following result confirms that this arrangement of functions forms a short exact sequence.
There is a potential ambiguity in this definition, so we must confirm that the value of h α 0 •••αq does not depend on the choice of M 1 or M 2 .So, suppose that the set U α 0 •••αq intersects both M 1 and M 2 .Then for any element in M 12 , we have two representatives of the equivalent class [x, 1], namely x and f 12 (x).We then have that as required.By construction, ȟ pulls back to ( f, ǧ) under the map Φ * .We may therefore conclude that ker (ι * 12 − f * 12 ) ⊆ Im(Φ * ), from which the equality follows.
As a consequence of the above result, we observe that the Čech cochains of M admit a fibred product structure.

Corollary 3.2. The Čech cochains on M satisfy the equality
for all q ∈ N.
Routine computations verify that the maps Φ * and ι * 12 −f * 12 commute with the Čech differential δ.Consequently, we may expand the data of Lemma 3.1 into the long exact sequence where here the maps δ * are the connecting homomorphisms induced as an application of the Snake Lemma.It should be noted that here we have tacitly passed to the cover-independent version of Čech cohomology.In fact, the relationship above is induced from a combination of refinement maps and the universal property of direct limits -the details can be found in the Appendix.
To begin with, we need to generalise the difference maps ι * 12 − f * 12 .Since there are now n-many submanifolds M i , we will have multiple pairwise intersections M ij .We may order the pairs of indices lexicographically and define a map δ : i which combines the difference maps ι * ij − f * ij in the obvious manner.Since there are now multiple intersections M i 1 •••ip , each with their own spaces of cochains Čq (M i 1 •••iq , U i 1 •••iq ), we will also have multiple pullbacks where here is the inclusion map that forgets the i th index.We can similarly define difference maps δ : In essence, the map δ plays the role of the differential that one would define for the Čech complex constructed from the open cover {M i }.In a similar manner to the binary case of Section 3.2, this map δ can be shown to square to zero and to commute with the Čech differential δ defined for Čq (M, U).
We will now set about proving the general version of Lemma 3.1, this time taking into account the multiple intersections is exact for all q in N.
Proof.We will proceed by induction on the size of the set I. The case of I = 2 is already proved as Lemma 3.1.To spare notation, we will illustrate the inductive argument for I = 3, though it should be understood that the full inductive case is near-identical.With all the data available to us, we may construct a contravariant analogue of the diagram in [18,Pg. 187] as follows, where we have suppress the open covers for readability.
The map κ * is the pullback of the inclusion κ : M 1 ∪ f 12 M 2 → M described in Section 1.1.The first row of the diagram is formed by taking kernels of the vertical maps, and the horizontal maps in the first row are defined so as to make the diagram commute.Since the columns of this diagram are all short exact sequences that split, in order to show the exactness of the center row it suffices to show that the first and third rows are exact.Observe first that the third row is a binary Mayer-Vietoris sequence, so is exact by Lemma 3.1.
Following the approach of [21], we may decompose the first row into sequences: where here this decomposition is formed by taking the kernel of the δ map.The latter sequence is a Mayer-Vietoris sequence, so is exact by Lemma 3.1.Thus our proof is complete once we argue that the former is a short exact sequence.
Observe first that the map φ * 3 is injective since any two distinct elements of ker(κ * ) will be zero on M 1 ∪ M 2 , thus can only differ somewhere on M 3 .The map −ι * is surjective by a standard extension by zero argument (cf.Lemma 3.1).Finally, we will show that Im(φ * 3 ) = ker(−ι * ).The inclusion Im(φ * 3 ) ⊆ ker(−ι * ) is guaranteed since any element in ker (κ * ) will also vanish on M 13 ∪ M 23 .For the converse inclusion, let f be some cochain on M 3 that restricts to zero on M 13 ∪ M 23 .We can extend f to a full cochain ǧ on M by defining the functions of ǧ to be The cochain ǧ will then satisfy φ * 3 ǧ = f , from which we may conclude that ker(−ι * ) ⊆ Im(φ * 3 ), whence equality.
The generalised Mayer-Vietoris long exact sequence of Theorem 3.4 can be equivalently rephrased as follows.

Corollary 3.5. The Čech cochains of M satisfy the equality
In direct analogy to the construction of the Čech-de Rham bicomplex of [18], we may arrange the cochain data of all the M i into a bicomplex as below. . . .
Theorem 3.4 ensures that the rows of this bicomplex are exact.As such, we may employ standard spectral arguments (found in, say [18]) to conclude that the cohomology of this bicomplex coincides with the Čech cohomology of M. We remark that it is possible to obtain some alternate descriptions of the Čech cohomology of M by computing the spectral sequence of the above bicomplex starting with the δ-cohomology of the columns.However, given the scope of this paper we will not expand on this observation.For our purposes, we will only use the following result.Theorem 3.6.Suppose that M is built from n-many M i in which all intersections M i 1 •••ip are connected for all p ≤ n.Then Ȟ1 (M) coincides with the fibred product F Ȟ1 (M i ).
Proof.We will proceed by induction on the size of the indexing set I. This argument revolves around an inductive form of the finite fibred product, so the reader unfamiliar with this form is invited to read the appendix first.The binary case is already proved in the form of Lemma 3.1.We will illustrate the case for I = 3, though it should be understood that the inductive argument is near-identical to the following.Let M be the colimit of the diagram formed from M 1 , M 2 and M 3 .We can use the inductive construction of M as in Lemma 1.5 to view M as an adjunction of the pair M 1 ∪ M 2 to M 3 along the union M 13 ∪ M 23 .By the Mayer-Vietoris argument of Lemma 3.1 we have the following portion of the long exact sequence: where we have used that (M 1 ∪ M 2 ) ∩ M 3 = M 13 ∪ M 23 , and ι a and ι b are the inclusions of M 13 ∪ M 23 into M 1 ∪ M 2 and M 3 , respectively.Observe that again we have passed into the cover-independent Čech cohomology by using the direct limiting process found in the Appendix.By assumption all of the Ȟ0 terms in this sequence equal G, so we may apply the same reasoning as that of Lemma 3.3 to conclude that the connecting homomorphism δ * is the zero map.Thus Ȟ1 (M) = ker(ι * a − ι * b ), which can be equivalently stated as: By assumption the triple intersection M 123 is also connected, so we apply the same reasoning to the union M 13 ∪ M 23 to conclude that Putting this all together, we have that where the final isomorphism follows from Theorem A.2.

Classifying Line Bundles
We will now discuss the prospect of classifying real line bundles over a fixed non-Hausdorff manifold M. Continuing on from the previous section, we will assume that M may be expressed as the colimit of n-many Hausdorff manifolds M i according to the requirements of Remark 1.9.
We can describe any rank-k vector bundle E over M as a choice of transition functions g αβ : U αβ → GL(k) the conditions g αα = id and g αβ g βγ g γα = id.When this is the case, we say that the open cover U = {U α } trivialises the bundle E. As in the Hausdorff case, any two sets {g αβ } and {h αβ } of trivialisations will describe the same bundle E whenever there exists a collection of maps {k α : In the case that E is a line bundle, we may use Theorem 2.8 to conclude that E admits a bundle metric.This allows us to reduce the structure group of E from GL(1) down to O(1).As such, any transition function will now take image in the Abelian group Z 2 .We may then equivalently view the construction of E from local trivialisations as a cocycle in Č1 (M, U, Z 2 ) whose cohomology class determines E up to isomorphism.Explicitly, for any open cover U of M is a bijection between the set Line R (M, U) of real line bundles trivialised by U and the U-dependent Čech cohomology Ȟ1 (M, U, Z 2 ).We will now use the direct limit construction of Ȟ1 (M, Z 2 ) to prove the following.

Theorem 3.7.
There is a bijection between the set Line R (M) of inequivalent real line bundles over M and the first Čech cohomology group Ȟ1 (M, Z 2 ).

Proof.
For each open cover U of M, denote by α U the bijection between Line R (M, U) and Ȟ1 (M, U).By construction, each group Ȟ1 (M, U, Z 2 ) maps into Ȟ1 (M, Z 2 ) via some map ψ U , and this map commutes with any refinement maps.We define a map α : Line R (M, Z 2 ) → Ȟ1 (M, Z 2 ) by sending each L to ψ U • α U (L), where U is some open cover that trivialises L. To see that map α is well-defined, suppose that L is some line bundle that is simultaneously trivialised by two open covers U and V.We may then select some open cover W that refines both U and V simultaneously.Consider two maps λ U and λ V that encode the refinements into W.By the direct limit construction of Ȟ1 (M, Z 2 ), we then have that as required.Finally, we observe that α is bijective since every α U is.
Given the results of Sections 3.1 and 3.2, it would be useful if there were formulas to express the order of Ȟ1 (M, Z 2 ) in terms of the orders of the groups Ȟ1 (M i , Z 2 ).Unfortunately, a general formula may not exist.However, we may use Theorem 3.6 together with a basic property of fibred products to yield the following.Theorem 3.8.Let M be a non-Hausdorff manifold formed from Hausdorff manifolds M i , according to 1.4.Suppose furthermore that: 1. the subspaces M i 1 •••ip are all connected, for all p ≤ n, and 2. each of the descended difference maps δ : Then the number of inequivalent line bundles on M can be expressed with the formula:

Conclusion
In this paper we have explored the prospect of fibering vector spaces over a base space which has the structure of a non-Hausdorff manifold.Using the topological criteria outlined in both Section 1.1 and [1], we saw that bundle structures can be naturally defined, reconstructed, and in some cases classified.Our initial observation was that of Theorem 1.7, which confirmed that smooth non-Hausdorff manifolds can be constructed by gluing together ordinary smooth manifolds along diffeomorphic open submanifolds.This, once combined with the contravariance of the C ∞ functor, then allowed us to express the algebra of smooth real-valued functions of a non-Hausdorff manifolds as a limit in the abelian category of unital associative algebras.
Throughout Section 2, we saw that vector bundles fibred over our non-Hausdorff manifold M can be constructed by taking a particular colimit of Hausdorff bundles.This was proved as Theorem 2.1, and it was later shown in Theorem 2.3 that every vector bundle over M can be described in this manner.Using this observation, we then showed that all sections of all bundles over M carry a description in terms of sections on each Hausdorff submanifold.In Theorem 2.8 this piecewise construction of sections was used to conclude that bundle metrics will exist for any non-Hausdorff bundle E fibred over M. In particular, Corollary 2.9 confirmed that Riemannian metrics always exist on any non-Hausdorff manifold M satisfying our particular topological criteria.
In Section 3 we studied the Čech cohomology of non-Hausdorff manifolds.In Sections 3.1 and 3.2 we related the Čech cohomology groups Ȟq (M) to the groups Ȟq (M i 1 •••ip ) in the form of (generalised) Mayer-Vietoris sequences.These sequences will be typically be non-trivial, and therefore the fibreproduct structure seen in 1.14, 2.6 and 3.5 will not present itself in Čech cohomology.Despite that, we saw in the form of Theorem 3.6 that the first Čech cohomology group Ȟ1 (M) will be a fibred product, provided that the Hausdorff submanifolds M i and all their intersections are connected.Finally, in Theorem 3.8 we identified some conditions under which the inequivalent real lines bundles over M can be expressed as an alternating sum formula.
We will finish this paper with some speculations regarding future developments.Of particular interest is the relationship between the Čech cohomology groups established here, and other theories such as de Rham cohomology.As an application of Theorem 2.6, the differential forms on M satisfy a fibred product structure, and therefore it will be interesting to compute the de Rham cohomology groups (à la [21]) and relate them to the groups Ȟq (M, R).These relationships might then be used in conjunction with a sound theory of affine connections to establish a Chern-Weil Theory for non-Hausdorff manifolds.

Lemma 1 . 3 .
Let F be an adjunction system in which the gluing regions M ij are all open submanifolds and the maps f ij are open topological embeddings.Then 1. the maps φ i are all open topological embeddings, and 2. the adjunction space subordinate to F is locally-Euclidean and secondcountable.

Lemma 1 . 8 .
Let M and N be smooth (possibly non-Hausdorff) manifolds.A map f : M → N is smooth if and only if the restrictions f | M i are smooth for all i in I. Proof.According to Lemma 1.3 the collection M i forms an open cover of M.