Bitopological Manifolds: Properties and Examples

1. Introduction

The classical theory of manifolds is based on a single topology together with a compatible differentiable structure. A natural generalization is obtained by allowing the underlying set to carry two topologies. Such objects belong to the general theory of bitopological spaces, which was initiated by Kelly in his fundamental paper [1]. A bitopological space is a triple $(X,{\tau }_1,{\tau }_2)$, where ${\tau }_1$ and ${\tau }_2$ are two topologies on the same set X . The importance of considering two topologies is that many ordinary topological notions split naturally into two versions. For example, one has two closure operators ${cl}_1$, ${cl}_2$, two interior operators, two systems of open sets, and potentially two different notions of convergence. Kelly’s work also emphasized pairwise separation properties and the interaction between the two topologies [1]. Related developments connected bitopological spaces with quasi-uniform spaces and generalized distance structures [2]. The purpose of this paper is to introduce a manifold version of this framework. More precisely, we study spaces $(M,{\tau }_1,{\tau }_2)$ for which each of ${\tau }_1$ and ${\tau }_2$ gives M the structure of a topological manifold. In addition, we consider smooth structures compatible with these two topologies. The resulting object should not be regarded merely as two unrelated manifolds on the same set. The essential point is the interaction between the two structures. This interaction can be expressed through pairwise-continuous maps, pairwise-smooth maps, common charts, transition maps, and compatibility conditions. The theory of bitopological spaces has developed in several directions, including separation axioms, compactness, Lindelöf properties, covering properties, quasi-uniformities, and order-theoretic constructions [1,2,3,4,5]. Recent work continues to investigate separation, compactness, and covering properties in the bitopological setting [6,7,8]. The present paper has a more geometric objective. We establish basic definitions and elementary structural results for bitopological manifolds and give examples illustrating how the two topologies may coincide, differ, or arise from a change of coordinates.

2. Preliminaries on Bitopological Spaces

We begin with the basic definition.

Definition 1.

A bitopological space is a triple $(X,{\tau }_1,{\tau }_2)$, where ${\tau }_1$ and ${\tau }_2$ are topologies on the same set X. The topology ${\tau }_1$ will be called the first topology and ${\tau }_2$ the second topology.

The theory of bitopological spaces was systematically initiated by Kelly [1]. The two topologies need not be comparable. Thus none of the implications ${\tau }_1\subseteq {\tau }_2$, ${\tau }_2\subseteq {\tau }_1$ is assumed in general.

Definition 2.

Let $(X,{\tau }_1,{\tau }_2)$ be a bitopological space. A map $f:(X,{\tau }_1,{\tau }_2)⟶(Y,{\sigma }_1,{\sigma }_2)$ is called pairwise continuous if $f:(X,{\tau }_1)\to (Y,{\sigma }_1)$ and $f:(X,{\tau }_2)\to (Y,{\sigma }_2)$ are both continuous.

Pairwise continuity is one of the basic notions in bitopological topology [1]. Notice that pairwise continuity is stronger than continuity with respect to the join topology in general. For a subset $A\subseteq X$, we write ${cl}_i\left(A\right)$ for the closure of A with respect to ${\tau }_i$, and ${int}_i\left(A\right)$ for its interior with respect to ${\tau }_i$.

Definition 3.

The join topology of ${\tau }_1$ and ${\tau }_2$ is denoted by ${\tau }_1\vee {\tau }_2$. It is the smallest topology containing both ${\tau }_1$ and ${\tau }_2$.

Similarly, the intersection ${\tau }_1\cap {\tau }_2$ is a topology and is sometimes called the meet topology .

Remark 1.

The join and meet of the two topologies contain information about their global interaction. However, they do not in general encode all pairwise information. This distinction is fundamental in bitopological topology [1,2].

3. Bitopological Manifolds

We now introduce the main object of the paper.

Definition 4.

Let $n\ge 0$. A bitopological n -manifold is a triple $(M,{\tau }_1,{\tau }_2)$ such that:

(i)  

$(M,{\tau }_1)$ is a topological n-manifold;

(ii)  

$(M,{\tau }_2)$ is a topological n-manifold;

(iii)  

the two manifold structures are considered on the same underlying set M;

(iv)  

the transition between the two structures is controlled by pairwise-compatible local charts.

Recall that a topological n -manifold is normally assumed to be Hausdorff, second countable, and locally homeomorphic to ${\mathbb{R}}^n$. Consequently, in the present setting we may impose these conditions separately on $(M,{\tau }_1)$ and $(M,{\tau }_2)$.

Definition 5.

A bitopological manifold $(M,{\tau }_1,{\tau }_2)$ is called pairwise Hausdorff if for every pair of distinct points $x,y\in M$ there exist $U\in {\tau }_1$, $V\in {\tau }_2$ such that $x\in U$, $y\in V$, $U\cap V=⌀$, and also the corresponding condition with the roles of ${\tau }_1$ and ${\tau }_2$ interchanged.

This is one of the natural pairwise separation conditions in the theory of bitopological spaces [1,5].

Definition 6.

A bitopological chart of dimension n is a pair $(U,\phi )$ where $U\subseteq M$ is open with respect to both topologies and $\phi :U⟶\phi \left(U\right)\subseteq {\mathbb{R}}^n$ is a homeomorphism for both topologies, where $\phi \left(U\right)$ carries the usual Euclidean topology.

Thus, in a bitopological chart, the same coordinate map is required to respect both topological structures.

Definition 7.

A collection $\mathcal{A}={\left\{(U_{\alpha },{\phi }_{\alpha })\right\}}_{\alpha \in A}$ of bitopological charts is called a bitopological atlas if $M={\bigcup }_{\alpha \in A}{U}_{\alpha }$. It is called smooth if, whenever $U_{\alpha }\cap U_{\beta }\ne ⌀$, the transition map ${\phi }_{\beta }\circ {\phi }_{\alpha }^{-1}$ is smooth on its domain.

The ordinary manifold case is recovered when ${\tau }_1={\tau }_2$.

Proposition 1.

Every smooth manifold $(M,\tau ,\mathcal{A})$ determines a bitopological manifold

$$(M,\tau ,\tau )$$

with the diagonal bitopology.

Proof. 

Since $(M,\tau )$ is a smooth manifold, it is a topological manifold and has a smooth atlas. Taking ${\tau }_1={\tau }_2=\tau$ makes the two topological structures identical. Hence every ordinary chart is automatically a bitopological chart, and the ordinary smooth atlas is a bitopological atlas. □

Remark 2.

The diagonal construction shows that bitopological manifolds contain ordinary smooth manifolds as a special case. The interest of the bitopological theory arises when ${\tau }_1\ne {\tau }_2$.

4. Pairwise Smooth Maps

The natural morphisms between bitopological manifolds should preserve both topological structures and their smooth structures.

Definition 8.

Let $(M,{\tau }_1,{\tau }_2,\mathcal{A})$ and $(N,{\sigma }_1,{\sigma }_2,\mathcal{B})$ be smooth bitopological manifolds. A map $f:M⟶N$ is called pairwise smooth if it is smooth with respect to both smooth structures.

Equivalently, if $(U,\phi )\in \mathcal{A}$,$(V,\psi )\in \mathcal{B}$, and $f\left(U\right)\subseteq V$, then $\psi \circ f\circ {\phi }^{-1}$ is smooth.

Proposition 2.

Every pairwise smooth map between smooth bitopological manifolds is pairwise continuous.

Proof. 

A smooth map between smooth manifolds is continuous. Since f is smooth with respect to each of the two smooth structures, it is continuous for both pairs $(M,{\tau }_1)\to (N,{\sigma }_1)$ and $(M,{\tau }_2)\to (N,{\sigma }_2)$. Therefore f is pairwise continuous. □

Definition 9.

A pairwise smooth map $f:M⟶N$ is called a bitopological diffeomorphism if f is bijective and its inverse $f^{-1}:N⟶M$ is also pairwise smooth.

The set of all bitopological diffeomorphisms from M to itself will be denoted by ${Diff}_{\mathrm{bi}}\left(M\right)$.

Proposition 3.

The set ${Diff}_{\mathrm{bi}}\left(M\right)$ is a group under composition.

Proof. 

The identity map is pairwise smooth. If $f,g\in {Diff}_{\mathrm{bi}}\left(M\right)$, then both $f\circ g$ and ${(f\circ g)}^{-1}=g^{-1}\circ f^{-1}$ are pairwise smooth. Associativity follows from associativity of function composition. Hence ${Diff}_{\mathrm{bi}}\left(M\right)$ is a group. □

5. Compatibility of the Two Topologies

The two topologies may be unrelated, comparable, or identical. We therefore introduce several levels of compatibility.

Definition 10.

A bitopological manifold $(M,{\tau }_1,{\tau }_2)$ is called comparable if either ${\tau }_1\subseteq {\tau }_2$ or ${\tau }_2\subseteq {\tau }_1$.

Definition 11.

A bitopological manifold is called strongly compatible if ${id}_M:(M,{\tau }_1)⟶(M,{\tau }_2)$ is a homeomorphism.

Proposition 4.

If $(M,{\tau }_1,{\tau }_2)$ is strongly compatible, then ${\tau }_1={\tau }_2$.

Proof. 

If the identity map ${id}_M:(M,{\tau }_1)\to (M,{\tau }_2)$ is continuous, then ${\tau }_2\subseteq {\tau }_1$. Since its inverse is again the identity and is continuous, ${\tau }_1\subseteq {\tau }_2$. Therefore ${\tau }_1={\tau }_2$. □

Thus the genuinely bitopological situation occurs when the identity map between the two topologies fails to be a homeomorphism.

Definition 12.

A bitopological manifold is called weakly compatible if the join topology ${\tau }_1\vee {\tau }_2$ is a topological manifold and the canonical maps $(M,{\tau }_i)⟶(M,{\tau }_1\vee {\tau }_2)$, $i=1,2,$are continuous.

Remark 3.

The join topology can be strictly finer than either topology. Thus even when both $(M,{\tau }_1)$ and $(M,{\tau }_2)$ are manifolds, the join need not automatically behave like either individual manifold structure.

6. Local Structure

The local behavior of the two topologies is central.

Definition 13.

A point $x\in M$ is called bi-Euclidean of dimension nif there exists a subset $U\subseteq M$ containing x and a bijection $\phi :U\to V\subseteq {\mathbb{R}}^n$ such that φ is a homeomorphism for both ${\tau }_1$ and ${\tau }_2$.

Theorem 1.

Let $(M,{\tau }_1,{\tau }_2)$ be a Hausdorff and second-countable bitopological space. If every point of M is bi-Euclidean of dimension n, then $(M,{\tau }_1,{\tau }_2)$ admits a bitopological manifold structure of dimension n.

Proof. 

For every $x\in M$, choose a bi-Euclidean neighborhood $(U_x,{\phi }_x)$. The collection of all such charts covers M . Since every chart is simultaneously compatible with both topologies, it defines local Euclidean structures for both ${\tau }_1$ and ${\tau }_2$. The Hausdorff and second-countability assumptions guarantee that the two topological structures satisfy the standard manifold axioms. The resulting collection of charts is therefore a bitopological atlas. □

Remark 4.

The theorem is deliberately formulated in a local form. In applications, one may start with two independently defined manifold structures and then study whether a common family of charts exists.

7. Pairwise Connectedness and Compactness

The standard topological notions of connectedness and compactness have natural pairwise versions.

Definition 14.

A bitopological space $(X,{\tau }_1,{\tau }_2)$ is called pairwise connected if there is no decomposition $X=U\cup V$ where $U,V$ are nonempty and one is ${\tau }_1$-open while the other is ${\tau }_2$-open, with $U\cap V=⌀$.

Definition 15.

A bitopological space $(X,{\tau }_1,{\tau }_2)$ is called pairwise compact if every cover of X by sets belonging to ${\tau }_1\cup {\tau }_2$ has a finite subcover.

Pairwise compactness and related forms of compactness have been studied extensively in bitopological topology [1,8].

Proposition 5.

If $(M,{\tau }_1,{\tau }_2)$ is pairwise compact, then both $(M,{\tau }_1)$ and $(M,{\tau }_2)$ are compact.

Proof. 

Let $\mathcal{U}\subseteq {\tau }_1$ be an open cover of M . Since $\mathcal{U}\subseteq {\tau }_1\cup {\tau }_2$, it is also a pairwise open cover. Pairwise compactness gives a finite subcover. Thus $(M,{\tau }_1)$ is compact. The argument for ${\tau }_2$ is identical. □

Remark 5.

The converse need not follow from the definition: compactness of each topology separately does not automatically imply compactness for mixed covers consisting of sets from both topologies.

8. Bitopological Riemannian Structures

The differential-geometric structure can be extended beyond smoothness.

Definition 16.

A bitopological Riemannian manifold is a smooth bitopological manifold $(M,{\tau }_1,{\tau }_2)$ together with two Riemannian metrics $g_1,g_2$, where $g_i$ is compatible with the i-th smooth structure.

Thus one may have $(M,{\tau }_1,g_1)$ andn $(M,{\tau }_2,g_2)$ as two Riemannian manifolds sharing the same underlying set. One can then compare geometric quantities such as distances, geodesics, curvature, volume, and completeness. For example, let $d_1,d_2$ be the distance functions induced by $g_1$ and $g_2$. In general, $d_1(x,y)\ne d_2(x,y)$. If the metrics induce the two topologies, then ${\tau }_1={\tau }_{d_1}$, ${\tau }_2={\tau }_{d_2}$. This creates a natural relation with quasi-metric and quasi-uniform approaches to bitopological spaces [1,2].

9. Examples

We now give several examples.

9.1. The Diagonal Example

Example 1.

Let $M={\mathbb{R}}^n$ with the ordinary Euclidean topology τ. Put ${\tau }_1={\tau }_2=\tau$. Then $({\mathbb{R}}^n,\tau ,\tau )$ is a bitopological manifold. This is the trivial or diagonal example. Every ordinary smooth manifold is an example of this type.

9.2. A Manifold with Two Transported Topologies

Example 2.

Let $M=\mathbb{R}$ and let $h:\mathbb{R}\to \mathbb{R}$ be a bijection. Define ${\tau }_1={\tau }_{\mathrm{std}}$ and define ${\tau }_2=\{h^{-1}\left(U\right):U\in {\tau }_{\mathrm{std}}\}$. Then $h:(M,{\tau }_2)⟶(\mathbb{R},{\tau }_{\mathrm{std}})$ is a homeomorphism. Consequently $(M,{\tau }_2)$ is a one-dimensional topological manifold. Since $(M,{\tau }_1)$ is also a one-dimensional manifold, we obtain a bitopological manifold $(\mathbb{R},{\tau }_1,{\tau }_2)$. If h is chosen so that ${\tau }_1\ne {\tau }_2$, this gives a genuinely non-diagonal example.

This construction is important because it demonstrates that two manifold topologies can exist on the same underlying set without being identical.

9.3. Two Smooth Structures Transported by a Bijection

Example 3.

Let M be a smooth manifold and let

$$h:M\to M$$

be a bijection. Transport both the topology and the smooth structure along h. Thus define ${\tau }_2=\{h^{-1}\left(U\right):U\in {\tau }_1\}$. A chart $(U,\phi )$ for the first structure produces a chart $(h^{-1}\left(U\right),\phi \circ h)$ for the second structure. If h is not continuous with respect to ${\tau }_1$, then ${\tau }_1\ne {\tau }_2$. Nevertheless, each topology separately makes M into a smooth manifold.

9.4. Product Bitopological Manifolds

Example 4.

Let $(M,{\tau }_1,{\tau }_2)$ and $(N,{\sigma }_1,{\sigma }_2)$ be bitopological manifolds. Define two product topologies on $M\times N$ by ${\lambda }_1={\tau }_1\times {\sigma }_1$, ${\lambda }_2={\tau }_2\times {\sigma }_2$. Then $(M\times N,{\lambda }_1,{\lambda }_2)$ is a natural product bitopological manifold. If $dimM=m$, $dimN=n$, then the product has dimension $m+n$ with respect to both structures.

Proposition 6.

The product of two smooth bitopological manifolds is a smooth bitopological manifold.

Proof. 

Take bitopological charts $(U,\phi )$ on M and $(V,\psi )$ on N . Then $(U\times V,\phi \times \psi )$ is a chart on $M\times N$. The transition maps are products of smooth transition maps from the factor manifolds and hence are smooth. Therefore the resulting atlas is a smooth bitopological atlas. □

9.5. Submanifolds

Definition 17.

Let $(M,{\tau }_1,{\tau }_2)$ be a bitopological manifold and let $S\subseteq M$. The set S is called a bitopological submanifold if $(S,{\tau }_1{|}_S,{\tau }_2{|}_S)$ is a bitopological manifold.

Here ${\tau }_i{|}_S=\{U\cap S:U\in {\tau }_i\}$. If both induced structures are smooth, then S inherits two smooth structures.

10. Bitopological Metrization

Metrics provide an important source of examples. Suppose that $d_1$ and $d_2$ are two metrics on M . Let ${\tau }_{d_1}$, ${\tau }_{d_2}$ be the corresponding metric topologies. Then $(M,{\tau }_{d_1},{\tau }_{d_2})$ is a bitopological space. If both metric spaces $(M,d_1)\mathrm{and}(M,d_2)$ are n -manifolds, then the resulting structure is a bitopological manifold provided the local coordinate systems are compatible in the sense of Section 3. The relation between bitopological structures and generalized quasi-metric spaces is one of the important themes in the classical theory [1,2].

Definition 18.

A pair of metrics $(d_1,d_2)$ on M is called bi-compatible if the associated topologies make M into a bitopological manifold and the identity maps between corresponding local coordinate neighborhoods are continuous in both directions whenever required.

Example 5.

Let $M={\mathbb{R}}^n$ and let $d_1(x,y)=\parallel x-y\parallel$ be the Euclidean metric. Let $A:{\mathbb{R}}^n\to {\mathbb{R}}^n$ be an invertible linear map and define $d_2(x,y)=\parallel A(x-y)\parallel$. Then $d_1$ and $d_2$ induce the same topology. Hence the resulting bitopological manifold is diagonal at the topological level, although the two metrics may be different.

This example shows that the distinction between two geometric structures and two topological structures should be maintained carefully.

11. Pairwise Separation Properties

The two topologies allow several forms of separation axioms. We recall one particularly useful condition.

Definition 19.

A bitopological space $(X,{\tau }_1,{\tau }_2)$ is called pairwise Hausdorff if for every $x\ne y$ there are $U\in {\tau }_1,V\in {\tau }_2$ such that $x\in U,y\in V,U\cap V=⌀$.

There are also quasi-Hausdorff, pairwise regular, pairwise normal, and other separation properties. These distinctions are classical in bitopological topology [1,5]. For a bitopological manifold, pairwise separation may be stronger than what is needed to make each topology individually Hausdorff.

Proposition 7.

If ${\tau }_1={\tau }_2$ and $(M,{\tau }_1)$ is Hausdorff, then $(M,{\tau }_1,{\tau }_2)$ is pairwise Hausdorff.

Proof. 

Since the two topologies coincide, ordinary Hausdorff separation gives disjoint open sets $U,V\in {\tau }_1={\tau }_2$ separating any two distinct points. □

12. Convergence

One of the most immediate consequences of having two topologies is the existence of two notions of convergence.

Definition 20.

A sequence $\left(x_k\right)$ in a bitopological space $(X,{\tau }_1,{\tau }_2)$ is said to converge to x in the first topology if $x_k⟶x$ with respect to ${\tau }_1$. It converges in the second topology if $x_k⟶x$ with respect to ${\tau }_2$.

Thus it is possible that $x_k\to x\mathrm{in}{\tau }_1$ but $x_k\neg \to x\mathrm{in}{\tau }_2$.

Proposition 8.

If ${\tau }_1\subseteq {\tau }_2$ and $x_k\to x$ with respect to ${\tau }_2$, then $x_k\to x$ with respect to ${\tau }_1$.

Proof. 

Every ${\tau }_1$-open neighborhood of x is also ${\tau }_2$-open. Hence eventual membership in every ${\tau }_2$-neighborhood implies eventual membership in every ${\tau }_1$-neighborhood. □

Therefore, comparable bitopological manifolds possess a natural hierarchy of convergence.

13. Closure and Interior

For $A\subseteq M$, the two topologies give two closure operators ${cl}_1\left(A\right)$, ${cl}_2\left(A\right)$, and two interior operators ${int}_1\left(A\right)$, ${int}_2\left(A\right)$. In general, ${cl}_1\left(A\right)\ne {cl}_2\left(A\right)$ and ${int}_1\left(A\right)\ne {int}_2\left(A\right)$.

Proposition 9.

If ${\tau }_1\subseteq {\tau }_2$, then for every $A\subseteq M$, ${cl}_2\left(A\right)\subseteq {cl}_1\left(A\right)$ and ${int}_1\left(A\right)\subseteq {int}_2\left(A\right)$.

Proof. 

If ${\tau }_1\subseteq {\tau }_2$, then there are more open sets in ${\tau }_2$. Therefore there are more neighborhoods that must be intersected in forming the closure, so ${cl}_2\left(A\right)\subseteq {cl}_1\left(A\right)$. The interior statement follows because a larger topology provides more open subsets contained in A . □

This elementary observation becomes useful when studying pairwise compactness, pairwise connectedness, and pairwise separation.

14. A Criterion for Coincidence of the Topologies

The following proposition gives a simple criterion for when the bitopological structure collapses to the ordinary one.

Proposition 10.

Let $(M,{\tau }_1,{\tau }_2)$ be a bitopological manifold. Suppose that ${id}_M:(M,{\tau }_1)⟶(M,{\tau }_2)$ is continuous and that ${id}_M:(M,{\tau }_2)⟶(M,{\tau }_1)$ is continuous. Then ${\tau }_1={\tau }_2$.

Proof. 

The first continuity condition gives ${\tau }_2\subseteq {\tau }_1$. The second gives ${\tau }_1\subseteq {\tau }_2$. Hence ${\tau }_1={\tau }_2$. □

Remark 6.

Consequently, any genuinely nontrivial bitopological manifold must distinguish the two structures at least at the level of the identity map.

15. Maps Between Bitopological Manifolds

Let $f:(M,{\tau }_1,{\tau }_2)⟶(N,{\sigma }_1,{\sigma }_2)$ be a map. There are several natural possibilities.

1.

f may be continuous only for the first pair of topologies;

2.

f may be continuous only for the second pair;

3.

f may be pairwise continuous;

4.

f may be pairwise smooth.

The fourth condition is the strongest in the smooth setting.

Definition 21.

A map $f:(M,{\tau }_1,{\tau }_2)\to (N,{\sigma }_1,{\sigma }_2)$ is called bi-continuous if both $f:(M,{\tau }_1)\to (N,{\sigma }_1)$ and $f:(M,{\tau }_2)\to (N,{\sigma }_2)$ are continuous.

Definition 22.

A bijective bi-continuous map whose inverse is also bi-continuous is called a bitopological homeomorphism.

Theorem 2.

The composition of two pairwise smooth maps is pairwise smooth.

Proof. 

Let $f:M\to N$, $g:N\to P$ be pairwise smooth. For each $i=1,2$, the maps $f:(M,{\tau }_i)\to (N,{\sigma }_i)$ and $g:(N,{\sigma }_i)\to (P,{\rho }_i)$ are smooth. Therefore $g\circ f:(M,{\tau }_i)\to (P,{\rho }_i)$ is smooth for both $i=1,2$. Hence $g\circ f$ is pairwise smooth. □

16. Tangent Spaces

Suppose that $(M,{\tau }_1,{\tau }_2)$ is a smooth bitopological manifold with two compatible smooth structures. Then one may consider two tangent structures $T_{1}M$ and $T_{2}M$. If the same smooth atlas is used for both structures, these tangent bundles can naturally be identified. In general, however, different smooth structures may lead to different differentiable constructions. For a pairwise smooth map $f:M\to N$, one may obtain two differential maps $d{f}_1:T_{1}M\to T_{1}N$ and $d{f}_2:T_{2}M\to T_{2}N$. This leads naturally to the question of whether $d{f}_1$ and $d{f}_2$ are related by a canonical bundle isomorphism.

One can ask the following natural question:

Question 3.

Under what conditions on a smooth bitopological manifold do the two tangent bundles admit a canonical identification $T_{1}M\cong T_{2}M$?

Theorem 4

(Canonical identification of the tangent bundles). Let $(M,{\tau }_1,{\tau }_2)$ be a smooth bitopological manifold, and let $T_{1}M$ and $T_{2}M$ denote the tangent bundles associated with the two smooth structures determined by ${\tau }_1$ and ${\tau }_2$, respectively. Suppose that the identity map

$${id}_M:(M,{\tau }_1)⟶(M,{\tau }_2)$$

is a diffeomorphism. Then there exists a canonical vector bundle isomorphism

$$\mathrm{\Phi }:T_{1}M⟶T_{2}M$$

covering the identity map on M, given by

$$\mathrm{\Phi }\left(v\right)=d{\left({id}_M\right)}_p\left(v\right),v\in T_p^{\left(1\right)}M.$$

In particular,

$$T_{1}M\cong T_{2}M$$

as smooth vector bundles over M.

Proof. 

Since the identity map

$${id}_M:(M,{\tau }_1)⟶(M,{\tau }_2)$$

is a diffeomorphism, its differential at every point $p\in M$ is a linear isomorphism

$$d{\left({id}_M\right)}_p:T_p^{\left(1\right)}M⟶T_p^{\left(2\right)}M.$$

These linear isomorphisms vary smoothly with p . Consequently, they assemble into a smooth vector bundle isomorphism

$$\mathrm{\Phi }:T_{1}M⟶T_{2}M,\mathrm{\Phi }\left(v\right)=d{\left({id}_M\right)}_p\left(v\right),v\in T_p^{\left(1\right)}M.$$

The map $\mathrm{\Phi }$ covers ${id}_M$ and is uniquely determined by the differential of the identity map. Its inverse is the differential of the inverse identity diffeomorphism. Hence $\mathrm{\Phi }$ is a canonical identification of the two tangent bundles. □

Remark 7.

The essential condition is that the two smooth structures on M be compatible in the sense that the identity map between them is a diffeomorphism. Equivalently, the two atlases determine the same smooth structure.

If the two smooth structures are merely compatible topologically, or if the identity map is only a homeomorphism, a canonical identification of the tangent bundles does not necessarily exist.

More generally, if

$$f:(M,{\tau }_1)⟶(M,{\tau }_2)$$

is a diffeomorphism that is not the identity, then its differential induces a vector bundle isomorphism

$$df:T_{1}M⟶T_{2}M$$

covering f, rather than necessarily covering ${id}_M$.

This question is particularly relevant when the two topologies are different but the corresponding smooth structures are locally equivalent. Observe that if our definition of a smooth bitopological manifold allows two genuinely different smooth structures, the existence of a canonical identification $T_{1}M\cong T_{2}M$ is an additional condition, not an automatic consequence of having two topologies on M.

17. Further Examples

Example 6

(Bitopological circle). Let $M=S^1$. Take two smooth structures on the same underlying set obtained from two different atlases ${\mathcal{A}}_1,{\mathcal{A}}_2$. If their underlying topologies are distinct but both make $S^1$ a one-dimensional manifold, then $(S^1,{\tau }_1,{\tau }_2)$ is a bitopological manifold. If the identity map between the two structures is a diffeomorphism, the two structures are equivalent in the smooth category even if presented by different atlases.

Example 7

(Bitopological torus). Let $M={\mathbb{T}}^2=S^1\times S^1$. Suppose each circle factor is equipped with two compatible manifold structures. The product construction of Section 6 produces a bitopological structure on ${\mathbb{T}}^2$. The two topologies may then be used to study different notions of local behavior while preserving the product geometry.

Example 8

(An ordered example). Let $M=\mathbb{R}$ and let ${\tau }_1$ be the usual topology. Suppose ${\tau }_2$ is a second manifold topology obtained by transporting the usual topology through a bijection $h:\mathbb{R}\to \mathbb{R}$. Then $(\mathbb{R},{\tau }_1,{\tau }_2)$ provides a basic example in which the underlying set carries two one-dimensional manifold structures. The example illustrates that the central issue is not merely the existence of two topologies but their interaction through the identity map and through pairwise maps.

18. Relation with Classical Bitopological Theory

The theory developed here is a geometric extension of the classical bitopological framework. Kelly introduced bitopological spaces in connection with generalized distance structures and developed fundamental pairwise separation properties [1]. Lane subsequently investigated the relationship between bitopological spaces and quasi-uniform spaces [2]. Patty and Pervin contributed further developments to the theory, including connectedness and separation questions [3,4]. Later work has investigated compactness, Lindelöf properties, covering properties, and various generalized separation axioms in bitopological spaces [5,6,7,8]. The manifold framework proposed here adds a local Euclidean and differential-geometric layer to this theory. In particular, the following dictionary is useful:

Bitopological topology	Differential geometry
two topologies	two manifold structures
pairwise continuity	pairwise smoothness
pairwise homeomorphism	bitopological diffeomorphism
pairwise compactness	two-sided compactness
pairwise separation	two-sided local separation
quasi-metric structures	two metric/geometric structures

This viewpoint suggests that many classical bitopological notions can be investigated in a geometric setting.

19. Main Structural Observations

We collect several consequences.

Theorem 5.

Let $(M,{\tau }_1,{\tau }_2)$ be a bitopological manifold. If ${\tau }_1={\tau }_2,$ then all pairwise topological properties reduce to their ordinary topological counterparts.

Proof. 

If the two topologies coincide, then every statement involving ${\tau }_1$ and ${\tau }_2$ can be interpreted using the same topology. Therefore pairwise continuity, pairwise compactness, pairwise Hausdorffness, and related notions reduce to the corresponding ordinary notions. □

Theorem 6.

Let $(M,{\tau }_1,{\tau }_2)$ be a bitopological manifold and suppose ${\tau }_1\subseteq {\tau }_2$. Then convergence in ${\tau }_2$ implies convergence in ${\tau }_1$. Moreover, ${cl}_2\left(A\right)\subseteq {cl}_1\left(A\right)$ for every $A\subseteq M$.

Proof. 

The convergence statement follows from the fact that every ${\tau }_1$-neighborhood is a ${\tau }_2$-neighborhood. The closure inclusion was proved in Section 9. □

Corollary 1.

If ${\tau }_1\subseteq {\tau }_2$ and the two topologies have the same convergent sequences, then the two topologies coincide whenever both spaces are first countable.

Proof. 

In first-countable spaces, the topology is determined by convergent sequences. Since ${\tau }_1\subseteq {\tau }_2$ and the convergent sequences coincide, no additional open sets can be detected by ${\tau }_2$. Hence ${\tau }_1={\tau }_2.$ □

20. Applications and Future Directions

Bitopological manifolds provide a framework in which two distinct local or global structures can coexist on the same underlying geometric object. Several directions appear naturally. First, one may study bitopological Lie groups . If a group G carries two compatible manifold structures, one can ask whether multiplication $m:G\times G\to G$ and inversion $\iota :G\to G$ are pairwise smooth. Second, one may investigate bitopological Riemannian geometry . Two metrics $g_1$ and $g_2$ may produce different notions of geodesic, completeness, curvature, and volume. Third, one may study bitopological fiber bundles . A bundle could carry two topologies on the total space and two compatible topologies on the base, together with pairwise-compatible local trivializations. Fourth, the theory may be related to quasi-uniform and generalized metric structures. Such a connection is particularly natural because bitopological spaces historically arose in connection with asymmetric distance functions [1,2]. Finally, one may investigate versions of de Rham theory, differential forms, tangent bundles, cotangent bundles, and characteristic classes associated with the two smooth structures.

21. Conclusion

We introduced the notion of a bitopological manifold as a manifold-like object equipped with two interacting topologies and compatible smooth structures. The construction extends the classical theory of bitopological spaces initiated by Kelly [1] into a differential-geometric setting. We defined bitopological charts, smooth bitopological atlases, pairwise continuous maps, pairwise smooth maps, bitopological diffeomorphisms, pairwise Hausdorffness, pairwise compactness, and pairwise connectedness. We also established elementary results concerning convergence, closure, product constructions, compactness, and coincidence of the two topologies. The diagonal case ${\tau }_1={\tau }_2$ recovers the ordinary theory of smooth manifolds, while non-diagonal examples demonstrate the possibility of two distinct manifold structures on a common underlying set. The framework developed here provides a starting point for more substantial geometric theories involving two topologies. In particular, bitopological Lie groups, bitopological Riemannian manifolds, bitopological symplectic manifolds, pairwise differential forms, and two-sided characteristic classes appear as natural subjects for future research.