# Unauthorized Drone Detection: Experiments and Prototypes ††thanks: This
publication was made possible by the NPRP award NPRP12S-0313-190348 from the
Qatar National Research Fund (a member of The Qatar Foundation). The
statements made herein are solely the responsibility of the authors. Qatar
National Library funded the publications of this article.

Muhammad Asif Khan1, Hamid Menouar1, Osama Muhammad Khalid1 and Adnan Abu-
Dayya1  Qatar Mobility Innovations Center (QMIC), Qatar University, Doha,
Qatar.1
Email: {muhammada, hamidm, osamam<EMAIL_ADDRESS>
###### Abstract

The increase in the number of unmanned aerial vehicles a.k.a. drones pose
several threats to public privacy, critical infrastructure and cyber security.
Hence, detecting unauthorized drones is a significant problem which received
attention in the last few years. In this paper, we present our experimental
work on three drone detection methods (i.e., acoustic detection, radio
frequency (RF) detection, and visual detection) to evaluate their efficacy in
both indoor and outdoor environments. Owing to the limitations of these
schemes, we present a novel encryption-based drone detection scheme that uses
a two-stage verification of the drone’s received signal strength indicator
(RSSI) and the encryption key generated from the drone’s position coordinates
to reliably detect an unauthorized drone in the presence of authorized drones.

###### Index Terms:

Acoustic, drone detection, radar, radio frequency, visual

©2022 IEEE. This paper has been accepted for presentation in 23rd IEEE
International Conference on Industrial Technology (ICIT22), 22 - 25 August,
2022, Shanghai, China. Personal use of this material is permitted. Permission
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works. DOI: ¡DOI No.¿

##  I Introduction

Unmanned aerial vehicles (UAVs) a.k.a. drones are becoming a commonplace in
many commercial applications such as aerial photography, geological mapping
and surveying, traffic and crowd monitoring, search and rescue, emergency
response and disaster relief, shipping, crop monitoring and spraying, and
emergency communication networking. These widespread applications of drones
will lead to an unprecedented growth in the number of commercial drones.
According to Federal Aviation Authority (FAA), the number of civilian drones
in United States (US) alone is expected to reach 1.66 million by the end of
2023 [1]. The presence of such a huge number of civilian drones in the low-
altitude airspace is associated with severe threats such as public privacy
invasion, terrorism, and cyber attacks. Thus, to protect people and
infrastructure from the illegitimate uses of commercial drones, an efficient
stand-alone drone detection system is urgently needed.

Some countries and drone manufacturers have implemented regulatory measures to
restrict the illegitimate uses of drones. For instance, some Da-Jiang
Innovations (DJI) uses special override commands to restrict drones’ entry
into no-fly zones. However, these measures do not solve the problem of
preventing unauthorized drones operations in non-restricted areas.
Furthermore, these restrictions in drones’ software can be altered and thus
are not sufficient to fully protect NFZs from unauthorized drones’ entry.

Recently, several drone detection solutions have been proposed by researchers
from academia and industry. These systems are typically categorized into four
categories based on the underlying detection technology. These include (i)
acoustic (sound-based) detection [2], (ii) radio frequency (RF) detection [3],
(iii) radar detection [4], and (iv) visual (camera-based) detection [5, 6, 7].
Acoustic detection uses sound sensors (i.e., microphones) to detect the sound
produced by a drone’s engine or other auxiliary components such as its
propeller blades. The signal is detected by highly sensitive microphones
(single or an array of multiple microphones) and processed to extract
significant features. The features are then mapped with a database of pre-
saved drones’ signatures (features indexed by drone’s ID) or fed to a machine
learning (ML) model to detect and identify the drone. Sound-based detection
has a short detection range ($\sim 200$ meter) and is highly susceptible to
background noise. However, its advantage is that it does not require visual
line-of-sight (LOS) to the target drone. In RF-based detection, an RF sensor
is employed to listen to the control signal between a drone and its ground
controller. The RF signals are then processed and fed to a machine learning
model to classify the drone. RF-detection has long detection range (up to
several kilometers) and also does not require LOS. However, it fails to detect
autonomous drones. Radar-detection is a relatively more accurate and robust
method with proven accuracy in aircraft detection. Radars transmit a high
frequency signal and receive the reflected version of it. The doppler shift
between the transmitted and reflected signal is used to identify a moving
drone. However, radars fail to detect hovering drones (stationery) and can not
differentiate between a drone and a moving bird. Visual detection uses visual
sensors (i.e., daylight and infrared cameras) to monitor the airspace. The
video frames are processed using deep learning methods such as convolution
neural networks (CNN) to detect the presence of drones. It is relatively more
accurate and efficient, but requires a clear LOS to the target. Furthermore,
the accuracy deteriorate in harsh weather such as heavy fog, rain, and dust
etc.

The theoretical limitations of drone detection approaches motivates
experimental evaluation of these systems. Hence, several recent works dedicate
to evaluate the performance of these schemes [8]. For instance, [9, 10, 2]
show how acoustic detection can be improved by using different number of
microphones. Similarly, the works in [11, 12, 13] explore feature engineering
to process sensors’ outputs to improve the detection accuracy. Whereas, [14,
2, 15] investigate different machine learning techniques to develop robust
system. For RF detection, the research efforts are mainly focused on the
advancement and sophistication in the machine learning models to classify the
RF signals. Some researchers propose to use simpler ML techniques such as
support vector machines (SVM), k-nearest neighbors (kNN), and neural networks
(NN) [16], while others propose deep learning techniques including convolution
neural networks (CNN) [3, 17], Deep Recurrent Neural Network (DRNN) [18].
Radar-based detection relatively received little attention and is typically
focused on Micro-Doppler radars to enhance detection of smaller objects [19].
Visual detection [20, 21] is gaining popularity due to the growing computer
vision community. The recent research in this area advocates CNN-based
techniques [22] for higher detection accuracy. While most of the research
works attempt to improve the detection accuracy, the problem of unauthorized
drone detection in the presence of authorized drones is yet to be
investigated.

In this work, we first present our experimental analysis of the three
approaches for drone detection i.e., sound-based, RF-based, and camera-based
detection. We then propose our prototype system for encryption-based
unauthorized drone detection. To the best of our knowledge, the proposed
prototype system is novel and is highly effective to detect unauthorised
drones in the presence of authorized drones.

##  II Experiments and Prototypes

In this section, we provide details of our experiments and prototypes using
three drone detection schemes i.e., acoustic, RF-based, and visual detection.
Radar-based detection could not be studied due to regulatory requirements.
Additionally, we also provide a novel encryption-based drone detection scheme
to cope with the shortcomings of the basic RF detection.

###  II-A Acoustic Detection System

The prototype for acoustic detection system is depicted in Fig. 1. It mainly
consists of (i) a sound sensors (i.e., a 4-mic array board, ReSpeaker) to
detect drone’s sound (ii) a Raspberry Pi processor to process sensor data, and
(iii) an Open EmbeddeD Audition System (ODAS) user interface.

Figure 1: Prototype of acoustic drone detection system.

The microphones array captures the sound waves and passes it to the Raspberry
Pi processor. At Raspberry Pi, the sound waves are processed to detect
statistical features. The features are then searched in an existing database
of drones’ sound signatures to detect the drone’s presence and its location
information (i.e., source azimuth). These information is then displayed on the
ODAS GUI.

###  II-B RF Detection System

Our prototype for RF-based detection using passive detection consists of a
universal software radio peripheral (USRP) E312 which is a portable stand-
alone software-defined radio (SDR) developed by Ettus research™), an embedded
field-programmable gate array (FPGA) along with a dual-core CPU with ARM
architecture for accelerating computation, and a GPS device. The USRP comes
with a $2\times 2$ multiple input-multiple output (MIMO) transceiver with an
instantaneous bandwidth of 56 MHz, and a frequency range that varies from 70
MHz up to 6 GHz.

(a)

(b)

Figure 2: RF-based detection prototype.

In our prototype, the USRP has been tuned around 2.4GHz since the sub-band of
the controller signals varies from 2.4GHz to 2.47GHz. Then the baseband signal
is sampled with a sampling frequency of 20MHz. The signals were acquired for
100ms and sent through an Ethernet cable to the host computer in the form of
frames where each frame is 4k sample long. The sampled signal is then
processed at the host computer using a custom-written MATLAB script, where the
original signal and its Fourier transform is reconstructed from the IQ complex
signal. Then the spectrum of the reconstructed signal is observed in the
present and absents of controller signals under different channel condition
such as in the present/absent of Wi-Fi signals, Bluetooth signals other types
of EM environmental interference that use the same band to establish the
communication.

###  II-C Visual Detection System

The prototype consists of a PTZ (Pan-Tilt-Zoom) camera (model: SD6CE445XA-HNR
WizSense series), an NVIDIA Jetson Xavier board to run the CNN model (YOLOv4
[23]), and a web interface. The PTZ camera is installed on a box facing upward
to cover the sky view. The camera is programmed to continuously scan for the
presence of drone operating in two operating modes i.e., (i) user-defined, and
(ii) step scanning. In user-defined mode, the user set the camera zoom,
starting angle, and final angle of the camera. The camera then continuously
scan the specified field of view (FOV). In the step mode, horizontal FOV is
divided into steps instead of a continuous range. The camera changes its FOV
by changing its FOV by one step, stops for a second, capture a frame, and move
to the next step. The experimental set up in depicted in Fig. 2 showing
multiple scenarios i.e., in an underground parking garage, inside building
corridors, and outdoor environment.

###  II-D Encryption-based Detection System

The encryption-based method is proposed to detect an unauthorized drone in a
swarm of authorized drones. Our prototype system consists of two drones and a
ground controller (Holybro Pixhawk Mini). The drones are equipped with
Raspberry Pi 4 Model B processors, while the controller is equipped with
NVIDIA Jetson AGX Xavier Developer Kit and is integrated with the web-portal
user interface. The functional diagram for our encryption-based detection
system is exhibited in Fig. 3. The proposed scheme makes use of the RF-based
detection with additional encryption method to increase the reliability of the
detection.

(a) The BLE Broadcast module. (b) The BLE Subsystem module.

Figure 3: The proposed encryption-based detection scheme.

Each drone runs two scripts i.e., BLE broadcast and BLE sub-system. The BLE
broadcast is responsible for acquiring all drone’s information from the
controller. It consists of three threads. The first thread continuously reads
the drone’s data, generates the verification key, and sends to the BLE
subsystem. The second thread receives the drone’s flight data, the encryption
key, and RSSI for each drone from the BLE subsystem. The key and RSSI are both
verified. If any of the two tests for key and RSSI matching fails, the drone’s
list is updated by flagging the drone as unauthorized. The updated list is
shared with the third thread in the BLE broadcast. The list updates every 0.1
seconds. The third thread reads the updated list and transmit it to the
controller via Wi-Fi link. The encryption key is generated from drone’s ID and
its GPS coordinates as follow.

$KEY=\\{(5\times ID)+(4\times LAT)+(3\times LON)+(2\times ALT)\\}\%256$

The modulus 256 is used to generate a key of size 8 bits which can be easily
packed in the BLE advertisement packet. The BLE subsystem is responsible to
broadcast flight data from all drones along with their RSSI values and the
encryption key over standard Bluetooth advertisements. It also consists of
three threads. The first thread receive extracted data from drone’s flight
controller via the BLE Broadcast script locally over TCP. The second thread
receives the flight data from the first thread and formats the data into BLE
packets. The third thread receives the BLE advertisement packets, unpacks, and
transmits them locally to the BLE Broadcast script.

##  III Results

We evaluated the performance of each detection scheme. In acoustic detection,
sound signatures of each drone were analyzed using a spectogram. A sample
sound signature and its spectogram is presented in Fig. 4.

Figure 4: Acoustic signature of the flying drone.

We used a direction of arrival (DOA) extraction algorithm which outputs up to
4 source angels. Fig. 5 shows the location estimates from acoustic signatures.
We observed random errors in the drone’s location due to significant
background noise in outdoor environment. This affirms that the use of larger
arrays of highly sensitive microphones to improve the performance of acoustic
detection [10].

Figure 5: A snapshot of ODAS GUI showing drone’s estimated locations.

To evaluate RF-based detection system, several tests are conducted in multiple
scenarios as depicted in Fig. 2. Raw data ($\sim 1000$ samples) was collected
to form a balanced binary labeled dataset and then to construct the receiver
operating characteristic curve (ROC) as depicted in Fig. 6.

(a) ROC Curve.

(b) Precision-Recall Curve.

Figure 6: RF-based detection performance analysis.

It can be observed in Fig. 6 that the area under the ROC curve (AUC) reaches
to almost $0.98$. The optimal operating point is acquired from the ROC curve
which corresponds to the threshold value of $74.02$, precision $0.985$ and
recall $0.94$.

TABLE I: System performance under different threshold values. Threshold |  Accuracy |  Precision |  Recall
---|---|---|---
60 |  0.9281 |  0.8927 |  0.9673
65 |  0.9459 |  0.9376 |  0.9510
70 |  0.9596 |  0.9687 |  0.9469
75 |  0.9656 |  0.9851 |  0.9429
80 |  0.9587 |  0.9912 |  0.9224
85 |  0.9537 |  0.9978 |  0.9061

The results in Table I presents the the recall, precision, and accuracy of the
proposed system under different threshold values. These values clearly show
the reliability and the robustness of the developed system. However, we expect
that the performance of this system can potentially degrade when adopted in
large-scaled problems and other more dynamic environments. One possible way to
reduce false alarms is to introduce two-samples, one-alarm policy, in which a
drone will only be declared in case of two consecutive positive detections.
This also eliminates the need to adjust the system threshold. The downside is
that it compromises the real-time performance of the system due to doubling of
the processing.

In visual detection, we used You Only Look Once (YOLOv4) deep learning model
to train a model on a two-class dataset containing $944$ drone images and
$270$ helicopter images collected from several public datasets. We also
acquired more drone images using the PTZ camera and labelled these new images.
We further trained the YOLOv4 model using transfer learning on the server. The
system achieved $90\%$ prediction accuracy on the test data (unseen). The
trained model is then converted into TensorRT model and deployed over NVIDIA
Jetson Xavier. When a drone is detected in a video frame captured by the
camera, the system automatically estimates the location of the drone based on
the direction of the camera. We developed an algorithm to automatically adjust
the camera direction to track a moving drone. The algorithm uses the detection
box dimensions, and updates the camera direction based on the box dimensions
relative to the frame. The web-interface showing a detected drone in red color
whereas the camera’s FOV as a green beam is depicted in Fig. 7. The width of
the beam indicates the current zoom level of the camera i.e., beam width is
automatically adjusted as inversely proportional to the zoom level.

Figure 7: Graphical interface of camera-based detection prototype system: An
unauthorized drone is detected. The drone is shown as red circle while the
camera’s FOV is displayed as green beam. The beam provides the estimated
location and direction of the detected drone.

It is worthy to note that the use of a single camera can provide limited
coverage of the region of interest at a time. To monitor $360^{\circ}{}$ area,
multiple cameras configured in an array at the same or different locations can
be used. Camera-based detection also has a limited range and can’t detect
drone that are too far due to limited optical zoom capability. Visual
detection also requires LOS to the target, we suggest to integrate acoustic or
RF detection as a first stage to provide an estimated location of the drone to
the camera to rotate towards the target drone.

Lastly, we evaluated our encryption-based scheme on real drones for accuracy.
The proposed scheme produced highly reliable results in different set ups.
Fig. 8 shows the intruder drone detected using the proposed prototype.

Figure 8: Graphical interface of encryption-based detection prototype system:
An intruder drone (red) located close to a a non-intruder drone (green) is
accurately detected and localized on the map.

##  IV Conclusion and Future Work

In this paper, we present our preliminary experiments and prototypes for three
drone detection systems i.e., acoustic detection, RF-based detection, and
visual detection respectively. We present only candidate results for each
method (due to space limitation). However, our detailed investigation shows
serious limitations of each method in different scenarios. We propose a novel
encryption-based unauthorized drone detection in the presence of other
authorized drones. The prototype is experimentally validated which accurately
detects an unauthorized drone in the presence of an authorized drone. As
future work, we suggest other encryption-based schemes (e.g., [24]) as well as
sensor fusion to overcome the limitations of individual drone detection
system. This may include a two-stage hybrid system using sensor fusion e.g.,
hybrid RF and visual sensors system. In such a system, the RF sensor provides
an early detection of the presence of a drone at a larger distance and
provides an estimated location /direction of the remote drone to the camera
which detects, identify, and tracks the drone more accurately (second stage).
Furthermore, the visual detection approach rely on the availability of large
datasets of drone’s images, whereas the existence dataset only contains images
of few drone models. The existing visual detection schemes can be further
improved and tested on large datasets of containing signatures of a variety of
drone models with different prospective and scale captures in diverse
environmental conditions.

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∎

11institutetext: Qianqian Xia 22institutetext: Nanjing University of
Information Science & Technology, Nanjing, 210044, China
22email<EMAIL_ADDRESS>
# Smooth distributions on subcartesian spaces are globally finitely generated

Qianqian Xia

(Received: date / Accepted: date)

###### Abstract

We prove that a connected subcartesian space admits embedding in a Euclidean
space. The Whitney Embedding Theorem is then stated as a corollary of our
result. Based on the above result together with the theory of distribution on
smooth manifolds, we show that smooth generalized distributions on connected
subcartesian spaces are globally finitely generated. We also show that smooth
generalized subbundles of vector bundles on connected subcartesian spaces are
globally finitely generated.

###### Keywords:

Subcartesian spaceEmbedding and Generalized distribution

##  1 Introduction

Distribution theory has its application in many fields such as geometric
mechanics and control 11 ; 1 . Given a smooth manifold $M$, by assigning to
each point $x\in M$ a subspace of $T_{x}M$, we get a generalized distribution
$D$. Sussmann and Stefan considered a generalized distribution $D$ to be
smooth if for each $x\in M$, there are locally defined smooth vector fields
that are sections of $D$ whose values at $x$ span $D_{x}$. If $E$ is a vector
bundle over $M$, a generalized subbunlde $F$ of $E$ is an assignment
$x\rightarrow F_{x}$ of a subspace of $F_{x}$ of the fiber $E_{x}$ of $E$ over
$x$, for each $x\in M$. $F$ is said to be smooth if for each $x\in M$, there
are locally defined smooth sections of $F$ whose value at $x$ span $F_{x}$.

In 1 , Lewis and Bullo give an interesting discussion of generalized
distributions and generalized subbundles. A question whether a generalized
subbundle has a finite set of global generators is proposed there.

In 3 , Drager et.al. show that any smooth generalized distribution on a smooth
manifold is globally finitely generated, that is, there are finitely many
globally defined sections whose values span the generalized distribution at
each point. The result in 3  provides a complete answer to the conjecture
which was first proposed in 1 . A more recently detailed discussion on this
topic can be found in 2 .

The existing results on distribution theory are restricted to smooth
manifolds, since differential geometry was traditionally regarded as the study
of smooth manifolds. However, there has long been perceived the need for an
extension of the framework of smooth manifolds in differential geometry, which
is too restrictive and does not admit certain basic geometric intuitions.
Sikorski¡¯s theory of differential spaces 6 ; 7  studies the differential
geometry of a large class of singular spaces which both contains the theory of
manifolds and allows the investigation of singularities. It is the
investigation of geometry in terms of differentiable functions. A similar
expansion occurred in the realm of algebra geometry, which is the
investigation of geometry in terms of polynomials. The difference between the
two theories is in the choice of the space of functions. A differential space
$S$ is said to be subcartesian 4 ; 5  if every point of $S$ has a
neighbourhood diffeomorphic to a subset of some Cartesian space
$\mathbb{R}^{n}$. It has been shown in 5  that the orbit space of a proper
action of a connected Lie group on a smooth manifold is a subcartesian space.
The theory of subcartesian spaces has been developed by Śniatycki et.al. in
recent few years 12 ; 13 ; 14 ; 15 ; 16 . See 5  for a systematic treatment on
this topic.

Given a differential space $S$, the tangent space $T_{x}S$ of $S$ is defined
as the set of all derivations of $C^{\infty}(S)$ at $x\in S$ and the tangent
bundle $TS$ is the union of all tangent spaces to $S$ at all points in $S$.
$TS$ is a differential space with the differential structure $C^{\infty}(TS)$
inherited from $C^{\infty}(S)$.

Analogous to the terminology in 1 ; 3 ; 2 , a generalized distribution on a
subcartesian space $S$ is a subset $D\subseteq TS$ such that for each $x\in
S$, the intersection $D_{x}=D\cap T_{x}S$ is a vector subspace of $T_{x}S$. A
generalized distribution on $S$ is smooth, if for any $v_{x}\in T_{x}S$, there
exist a neighborhood $U$ of $x$ and a local smooth section $X\in\Gamma(TS;U)$
such that $X(x)=v_{x}$ and $X(y)\in D(y)$ for each $y\in U$. We say that a
generalized distribution $D$ on $S$ is globally finitely generated, if there
exist derivations $X_{1},\cdots,X_{l}$ of $C^{\infty}(S)$ such that
$X_{1}(x),\cdots,X_{l}(x)$ span $D_{x}$ for each $x\in S$.

Analogous to the definition of vector bundle on smooth manifolds, a vector
bundle $E$ on a subcartesian space $S$ is a smooth map between subcartesian
spaces which has the local product property with respect to $\mathbb{R}^{k}$.
Besides, for each $x\in S$, fiber $E_{x}$ is a $k$-dimensional real vector
space. A generalized subbundle $F$ of a vector bundle $E$ on a subcartesian
space $S$ is a subset of $E$, such that $F_{x}=E_{x}\cap F$ is a subspace of
$E$. A generalized subbundle $F$ of $E$ is smooth, if for any $e_{x}\in
F_{x}$, there exist a neighborhood $U$ of $x$ and a local smooth section
$\sigma:U\rightarrow E$ such that $\sigma(x)=e_{x}$ and $\sigma(y)\in F_{y}$
for each $y\in U$. We say that a smooth generalized subbundle $F$ on $S$ is
globally finitely generated, if there exist global smooth sections
$\sigma_{1},\cdots,\sigma_{l}$ of $F$ such that
$\sigma_{1}(x),\cdots,\sigma_{l}(x)$ span $F_{x}$ for each $x\in S$.

In this paper, we study distribution theory on subcartesian space. We first
present embedding theorem for subcartesian space. We show that every connected
subcartesian space admits a smooth embedding into a Euclidean space, where the
smoothness is within the category of differential spaces. We provide a
complete proof which has not been found in the existing literature. We then
present the Whitney Embedding Theorem as a corollary of our embedding theorem
here. Based on this, together with Theorem 3.1.6 in 5  and the main theorem in
3 , we show that every smooth generalized distribution on a connected
subcartesian space is globally finitely generated. We also show that every
smooth generalized subbundle on a connected subcartesian space is globally
finitely generated.

##  2 Preliminaries

###### Definition 1

A differential structure on a topological space $S$ is a family
$C^{\infty}(S)$ of real-valued functions on $S$ satisfying the following
conditions:

1\. The family

$\\{f^{-1}(I)|f\in C^{\infty}(S)\text{and}\,I\,\text{is an open interval
in}\,\mathbb{R}\\}$

is a subbasis for the topology of $S$.

2\. If $f_{1},\cdots,f_{n}\in C^{\infty}(S)$ and $F\in
C^{\infty}(\mathbb{R}^{n})$, then $F(f_{1},\cdots,f_{n})\in C^{\infty}(S)$.

3\. If $f:S\rightarrow\mathbb{R}$ is a function such that, for every $x\in S$,
there exist an open neighborhood $U$ of $x$, and a function $f_{x}\in
C^{\infty}(S)$ satisfying

$f_{x}|U=f|U,$

then $f\in C^{\infty}(S)$. Here, the subscript vertical bar $|$ denotes a
restriction.

###### Proposition 1

Let $S$ be a differential space. Then for any open set $U\subseteq S$, for
each $x\in U$, there exists an open subset $V\subseteq U$ that contains $x$
such that $cl(V)\subseteq U$. Here $cl(V)$ denotes the closure of the subset
$V$ in $S$.

###### Proof

According to condition 1 in Definition 1, for any open set $U\subseteq S$ and
any $x\in U$, there exist $f_{1},\cdots,f_{k}\in C^{\infty}(S)$ and intervals
$(a_{1},b_{1}),\cdots,(a_{k},b_{k})$ in $\mathbb{R}$, such that
$f_{1}^{-1}(a_{1},b_{1})\cap\cdots\cap f_{k}^{-1}(a_{k},b_{k})\subseteq U$
contains $x$. Let
$[c_{1},d_{1}]\subseteq(a_{1},b_{1}),\cdots,[c_{k},d_{k}]\subseteq(a_{k},b_{k})$
be such that $f_{1}(x)\in(c_{1},d_{1}),\cdots,f_{k}(x)\in(c_{k},d_{k})$.
Consider the subset $V=f_{1}^{-1}(c_{1},d_{1})\cap\cdots\cap
f_{k}^{-1}(c_{k},d_{k})$. Then $x\in V$. It follows from condition 1 in
Definition 1 that $V$ is open. Beside,
$cl(V)=f_{1}^{-1}[c_{1},d_{1}]\cap\cdots\cap f_{k}^{-1}[c_{k},d_{k}]\subseteq
f_{1}^{-1}(a_{1},b_{1})\cap\cdots\cap f_{k}^{-1}(a_{k},b_{k})\subseteq U$.
Hence the result follows immediately.

###### Definition 2

A map $\phi:R\rightarrow S$ is $C^{\infty}$ if $\phi^{*}f=f\circ\phi\in
C^{\infty}(R)$ for every $f\in C^{\infty}(S)$. A $C^{\infty}$ map $\phi$
between differential spaces is a $C^{\infty}$ diffeomorphism if it is
invertible and its inverse is $C^{\infty}$.

###### Definition 3

A differential space $S$ is subcartesian if it is Hausdorff and every point
$x\in S$ has a neighbourhood $U$ diffeomorphic to a subset $V$ of
$\mathbb{R}^{n}$ by the map $\phi:U\rightarrow\mathbb{R}^{n}$.
$(U,\mathbb{R}^{n},\phi)$ is said to be a local chart of $S$. The subcartesian
space is said to be locally connected if for every point $x\in S$, there
exists a connected neighborhood $U$ such that $(U,\mathbb{R}^{n},\phi)$ is a
local chart of $S$.

###### Definition 4

Let $S$ be a subcartesian space. The structural dimension of $S$ at a point
$x\in S$ is the smallest integer $n_{x}$ such that for some open neighborhood
$U\subseteq S$ of $x$, there is a diffeomorphism of $U$ onto a subset
$V\subseteq\mathbb{R}^{n_{x}}$. The structural dimension of $S$ is the
smallest integer $n$ such that for every point $x\in S$, the structural
dimension $n_{x}$ of $S$ at $x$ satisfies that $n_{x}\leq n$.

###### Definition 5

Let $f\in C^{\infty}(S)$. For each $x\in S$, the smooth function $g$ on
$\mathbb{R}^{n}$ is said to be a smooth extension of $f$ at $x\in S$, if there
exists a local chart $(U,\mathbb{R}^{n},\phi)$ of $S$ with $x\in U$ such that
$g\circ\phi|U=f|U$.

Given any smooth function on $S$ and any $x\in S$, the existence of smooth
extension is ensured by the above definition of subcartesian space.

###### Lemma 1

5  For every open subset $U$ of a differential space $S$ and every $x\in U$,
there exists $f\in C^{\infty}(S)$ satisfying $f|V=1$ for some neighbourhood
$V$ of $x$ contained in $U$, and $f|W=0$ for some open subset $W$ of $S$ such
that $U\cup W=S$.

###### Lemma 2

5  Let $S$ be locally compact, Hausdorff and second countable. Then every open
cover $\\{U_{\alpha}\\}$ of $S$ has a countable, locally finite refinement
consisting of open sets with compact closures.

###### Definition 6

A countable partition of unity on a differential space $S$ is a countable
family of functions $\\{f_{i}\\}\in C^{\infty}$ such that:
(a) The collection of their supports is locally finite.
(b) $f_{i}(x)\geq 0$ for each $i$ and each $x\in S$.
(c) $\sum_{i=1}^{\infty}f_{i}(x)=1$ for each $x\in S$.

###### Theorem 2.1

5  Let $S$ be a differential space with differential structure
$C^{\infty}(S)$, and let $\\{U_{\alpha}\\}$ be an open cover of $S$. If $S$ is
Hausdorff, locally compact and second countable, then there exists a countable
partition of unity $\\{f_{i}\\}\in C^{\infty}(S)$, subordinate to
$\\{U_{\alpha}\\}$ and such that the support of each $f_{i}$ is compact.

###### Definition 7

A derivation of $C^{\infty}(S)$ is a linear map

$X:C^{\infty}(S)\rightarrow C^{\infty}(S):f\rightarrow X(f)$

satisfying Leibniz¡®s rule

$X(f_{1}f_{2})=X(f_{1})f_{2}+f_{1}X(f_{2})$

for every $f_{1},f_{2}\in C^{\infty}(S)$.

We denote by $\text{Der}C^{\infty}(S)$ the space of derivations of
$C^{\infty}(S)$.

###### Definition 8

A derivation of $C^{\infty}(S)$ at $x\in S$ is a linear map
$v:C^{\infty}(S)\rightarrow\mathbb{R}$ such that

$v(f_{1}f_{2})=v(f_{1})f_{2}(x)+f_{1}(x)v(f_{2})$

for every $f_{1},f_{2}\in C^{\infty}(S)$.

We interpret derivations of $C^{\infty}(S)$ at $x\in S$ as tangent vectors to
$S$ at $x$. The set of all derivations of $C^{\infty}(S)$ at $x$ is denoted by
$T_{x}S$ and is called the tangent space to $S$ at $x$.

###### Lemma 3

5  If $f\in C^{\infty}(S)$ vanishes identically in an open set $U\subseteq S$,
then $X(f)|U=0$ for all $X\in\text{Der}C^{\infty}(S)$.

###### Definition 9

The tangent bundle of a differential space $S$ is the union $TS$ of all
tangent spaces to $S$ at all points in $S$. In other words, $TS=\cup_{x\in
S}T_{x}S$. The tangent bundle projection is the map $\tau:TS\rightarrow S$
that assigns to each $v\in TS$ the point $x\in S$ such that $v\in T_{x}S$.

###### Definition 10

The differential structure $C^{\infty}(TS)$ of $TS$ is generated by the family
of functions $\\{f\circ\tau,df|f\in C^{\infty}(S)\\}$.

###### Definition 11

A smooth section of $\tau:TS\rightarrow S$ is a smooth map
$\sigma:S\rightarrow TS$ such that $\tau\circ\sigma=\text{identity}$. A local
smooth section of $\tau:TS\rightarrow S$ is a smooth map
$\gamma:U\rightarrow\tau^{-1}(U)$ such that $\tau\circ\gamma=\text{identity}$.

###### Definition 12

A generalized distribution on a subcartesian space $S$ is a subset $D\subseteq
TS$ such that for each $x\in S$, the intersection $D_{x}=D\cap T_{x}S$ is a
vector subspace of $T_{x}S$. $D$ is smooth, if for any $v_{x}\in D_{x}$, there
exists a neighborhood $U$ of $x$ and a local smooth section
$X:U\rightarrow\tau^{-1}(U)$ such that $X(x)=v_{x},X(y)\in D_{y}$, for each
$y\in U$.

Let $\pi:E\rightarrow S$ be a smooth map between subcartesian spaces. The map
$\pi$ will be said to have the local product property with respect to
$\mathbb{R}^{k}$, if there is an open covering $\\{U_{\alpha}\\}$ of $S$ and a
family $\\{\psi_{\alpha}\\}$ of diffeomorphisms

$\psi_{\alpha}:U_{\alpha}\times\mathbb{R}^{k}\rightarrow\pi^{-1}(U_{\alpha})$

such that $\pi\circ\psi_{\alpha}(x,y)=x$, for each $x\in
U_{\alpha},y\in\mathbb{R}^{k}$. The system $\\{(U_{\alpha},\psi_{\alpha})\\}$
will be called a local decomposition of $\pi$.

###### Definition 13

A smooth vector bundle is a four-tuple $(E,\pi,S,\mathbb{R}^{k})$ where
$\pi:E\rightarrow S$ is a smooth map between subcartesian spaces which has the
local product property with respect to $\mathbb{R}^{k}$. Besides, for each
$x\in S$, $\pi^{-1}(x)$ is a $k$-dimensional real vector space, and
$\psi_{\alpha}^{-1}|\pi^{-1}(x):\pi^{-1}(x)\rightarrow\\{x\\}\times\mathbb{R}^{k}$
is an linear isomorphism, for each $x\in U_{\alpha}$. A local decomposition
for $\pi$ is called a coordinate representation for the vector bundle.

###### Definition 14

A smooth section of $(E,\pi,S,\mathbb{R}^{k})$ is a smooth map
$\sigma:S\rightarrow E$ such that $\pi\circ\sigma=\text{identity}$. A local
smooth section of $(E,\pi,S,\mathbb{R}^{k})$ is a smooth map
$\gamma:U\rightarrow\pi^{-1}(U)$ such that $\pi\circ\gamma=\text{identity}|U$,
where $U$ is an open subset on $S$.

###### Definition 15

A generalized subbundle of a smooth vector bundle $(E,\pi,S,\mathbb{R}^{k})$
is a subset $F\subseteq E$ such that for each $x\in S$, $F_{x}=\pi^{-1}(x)\cap
F$ is a subspace of $\pi^{-1}(x)$. $F$ is smooth, if for any $e_{x}\in F_{x}$,
there exist a neighborhood $U$ of $x$ and a local smooth section
$\sigma:U\rightarrow\pi^{-1}(U)$ such that $\sigma(x)=e_{x}$, and
$\sigma(y)\in F_{y}$, for each $y\in U$.

###### Definition 16

A smooth Riemannian metric in $(E,\pi,S,\mathbb{R}^{k})$ is a symmetric
positive definite bilinear form $g(x)$ in $\pi^{-1}(x)$, for each $x\in S$,
such that for each smooth section $\sigma$ of $(E,\pi,S,\mathbb{R}^{k})$, the
function $g(x)(\sigma(x),\sigma(x))\in C^{\infty}(S)$.

###### Proposition 2

Each smooth vector bundle admits a smooth Riemannian metric.

###### Proof

Consider the trivial bundle $S\times\mathbb{R}^{k}$. Let $<,>$ be a Euclidean
metric in $\mathbb{R}^{k}$, then

$g(x)(y_{1},y_{2})=<y_{1},y_{2}>,x\in S,y_{1},y_{2}\in\mathbb{R}^{k}$

defines a smooth Riemannian metric in $S\times\mathbb{R}^{k}$.

Now consider any smooth vector bundle $(E,\pi,S,\mathbb{R}^{k})$. Let
$\\{(U_{\alpha},\psi_{\alpha})\\}$ be a local decomposition of $\pi$. It
follows from Theorem 2.1 that there exists a subordinate partition of unity
$\\{p_{\alpha}\\}$.

Since the restriction of $(E,\pi,S,\mathbb{R}^{k})$ to $U_{\alpha}$ is
trivial, there is a Riemannian metric $g_{\alpha}$. Define $g$ by
$\Sigma_{\alpha}p_{\alpha}g_{\alpha}$. Then $g(x)$ is a Euclidean metric in
$\pi^{-1}(x)$; hence $g$ is a smooth Riemannian metric in
$(E,\pi,S,\mathbb{R}^{k})$.

In the following we restrict our attention to Hausdorff, locally compact,
second countable subcartesian spaces. Hence, we shall be able to rely on the
existence of partitions of unity. Note that a subcartesian space is Hausdorff;
see Definition 3. Moreover, Definition 3 and condition 1 of Definition 1 imply
that a subcartesian space is locally compact. Thus, we make the following
assumption.

Assumption. Throughout the paper, all subcartesian spaces are second countable
and connected.

##  3 Dimension Theory

###### Definition 17

9  A space $A$ has covering dimension at most $m$ if every open covering of
$A$ has a refinement of order at most $m$ (which means that no point of $A$
lies in more than $m+1$ elements of the refinement). If $A$ is a closed subset
of the topological space $M$, this is equivalent to the statement that every
open covering of $M$ has a refinement whose restriction to $A$ has order at
most $m$. If $A^{\prime}$ is a closed subset of $A$ and $A$ has covering
dimension at most $m$, so does $A^{\prime}$.

###### Lemma 4

8  Let $M$ be a second countable locally compact Hausdorff topological space.
Then there exist countable many open sets $G_{1},G_{2},\cdots,G_{k},\cdots$
satisfying

(1) $cl(G_{j})$ is compact, $j=1,2,\cdots$;

(2) $cl(G_{j})\subseteq G_{j+1},j=1,2,\cdots$;

(3) $\cup G_{j}=\cup cl(G_{j})=M,$
where $cl(G_{j})$ denotes the closure of $G_{j},j=1,2,\cdots$.

###### Lemma 5

Let $S$ be a subcartesian space and $\Phi\in C^{\infty}(S)$. Then there exist
locally finite open covers
$(U_{j})_{j\in\mathbb{Z}_{>0}},(V_{j})_{j\in\mathbb{Z}_{>0}},(W_{j})_{j\in\mathbb{Z}_{>0}}$
such that $cl(U_{j})\subseteq V_{j},cl(V_{j})\subseteq W_{j}$, and $cl(W_{j})$
is compact, for each $j>0$, where $(W_{j},\mathbb{R}^{n_{j}},\phi_{j})$ is a
local chart of $S$, such that there exists a smooth extension $f_{j}$ of
$\Phi$ on $W_{j}$, that is, $f_{j}\circ\phi_{j}|W_{j}=\Phi|W_{j}$.

###### Proof

From Lemma 4 we know that there exist countable open sets
$G_{1},\cdots,G_{k},\cdots$ satisfying conditions (1), (2) and (3) in Lemma 4.
It follows that $cl(G_{h})/{G_{h-1}}$ is compact, $G_{h+1}/{cl(G_{h-2})}$ is
open and $cl(G_{h})/{G_{h-1}}\subseteq G_{h+1}/{cl(G_{h-2})}$. Since $S$ is
locally compact, for any $p\in cl({G}_{h})/{G_{h-1}}$, there exists a compact
set $C$ in $S$ containing an open neighborhood $U$ of $p$. On the other hand
we know the local charts of $S$ form an open cover of $S$. Then for $p\in
cl({G}_{h})/{G_{h-1}}$, there exist a local chart $(\mathscr{V},\phi)$ and an
open set $W\subset\mathscr{V}$ such that

(1) There exists a smooth function $f$ on $\mathbb{R}^{n}$ satisfying
$f\circ\phi|W=\Phi|W$;

(2) $p\in W\subseteq(G_{h+1}/{cl({G}_{h-2})})\cap\mathscr{V}$;

(3) $\phi(p)=0$ and $\phi(W)=\phi(\mathscr{W})\subseteq\phi(\mathscr{V})$;
Here $\mathscr{W}$ is an open subset containing $p$ such that
$cl(\mathscr{W})\subseteq(G_{h+1}/{cl({G}_{h-2})})\cap\mathscr{V}\cap U$. The
existence of $\mathscr{W}$ is ensured by Proposition 1. Since $cl(W)\subseteq
U\subseteq C$ and $C$ is compact, it follows that $cl(W)$ is compact.

Let $\mathscr{W}_{1}$ be an open set containing $p$ such that
$cl(\mathscr{W}_{1})\subseteq W$. Denote by $V=\mathscr{W}_{1}$. And let
$\mathscr{W}_{2}$ be an open set containing $p$ such that
$cl(\mathscr{W}_{2})\subseteq V$. Denote by $U=\mathscr{W}_{2}$. Then we have
$cl(U)\subseteq V$ and $cl(V)\subseteq W$.

Since ${cl({G}_{h}})/{G_{h-1}}$ is compact, there exist finitely many points
$p_{h,1},p_{h,2},\cdots$, $p_{h,k_{h}}\in cl({G}_{h})/{G_{h-1}}$, such that
the corresponding open sets $U_{h,1},U_{h,2},\cdots,U_{h,k_{h}}$ form an open
cover of $cl(G_{h})/G_{h-1}$. We claim that the corresponding open sets

$(U_{1,1},U_{1,2},\cdots,U_{1,k_{1}};U_{2,1},U_{2,2},\cdots,U_{2,k_{2}};\cdots),$
$(V_{1,1},V_{1,2},\cdots,V_{1,k_{1}};V_{2,1},V_{2,2},\cdots,V_{2,k_{2}};\cdots),$
$(W_{1,1},W_{1,2},\cdots,W_{1,k_{1}};W_{2,1},W_{2,2},\cdots,W_{2,k_{2}};\cdots),$

satisfy the conditions in the lemma. We only need to prove the local
finiteness of $(W_{i,j})$. Given $p\in S$, assume that $p\in G_{r}$, then it
follows from the above construction that there exist finite many $W_{i,j}$
that intersect $G_{r}$. In fact,

$W_{i,j}\cap G_{r}=\emptyset,i\geq r+2,1\leq j\leq k_{i}.$

This completes the proof of the lemma.

###### Lemma 6

9  Let $A_{n}$ be a closed subset of $M$, such that
$A_{n}\subseteq\text{Int}A_{n+1}$ and $\cup A_{n}=M$. If $A_{l}$ and each set
$cl(A_{n+1}-A_{n})$ have covering dimension at most $m$, so does $M$.

###### Theorem 3.1

Every subcartesian space $S$ has covering dimension at most $n$, where $n$ is
the structural dimension of $S$.

###### Proof

Cover $S$ with the sets $\\{cl(V_{j})\\}_{j\in\mathbb{Z}_{>0}}$ as given in
Lemma 5. Let $A_{1}=cl(V_{1})$. In general, given
$A_{l}=cl(V_{1})\cup\cdots\cup cl(V_{q})$, since $cl(V_{1})\cup\cdots\cup
cl(V_{q})$ is compact, we can choose an integer $p>q$ such that
$A_{l}\subseteq\text{Int}(cl(V_{1})\cup\cdots\cup cl(V_{p}))$, and let
$A_{l+1}=cl(V_{1})\cup\cdots\cup cl(V_{p})$. Since $\phi_{j}(cl(V_{j}))$ is
compact in $\mathbb{R}^{n_{j}}$, it follows from Theorem 2.15 in 9  that
$\phi_{j}(cl(V_{j}))$ has covering dimension at most $n_{j}$, which yields
that $cl(V_{j})$ has covering dimension at most $n_{j}$ since $\phi_{j}$ is a
diffeomorphism. Hence according to Corollary 2.14 in 9 , each $A_{l}$ has
covering dimension at most $n$. Then the result follows immediately from Lemma
6.

###### Lemma 7

Let $S$ be a subcartesian space with structural dimension $n$. Let
$\mathcal{O}$ be an open covering of $S$. There is a locally finite open
refinement of $\mathcal{O}$ which is the union of $n+1$ collections
$\mathcal{O}_{0},\cdots,\mathcal{O}_{m}$ such that the sets belonging to any
one $\mathcal{O}_{i}$ are pairwise disjoint.

###### Proof

According to Lemma 2, let $(U_{j})_{j\in\mathbb{Z}>0}$ be a countable locally
finite open refinement of $\mathcal{O}$. It follows from Theorem 2.1 that
there exists a partition of unity $(\phi_{j}){{}_{j\in\mathbb{Z}>0}}$ such
that the support of $\phi_{j}$ is contained in $U_{j}$, for each $j$. Then by
using the same arguments as given in the proof of Lemma 2.7 in 9 , the result
follows.

###### Proposition 3

A subcartesian space $S$ admits a finite atlas.

###### Proof

Cover $S$ with the atlas $\\{(U_{j},\phi_{j})_{j\in\mathbb{Z}_{>0}}\\}$ as
given in Lemma 5. Then we can choose the mappings
$\psi_{j}:\mathbb{R}^{n_{j}}\rightarrow\mathbb{R}^{n},j\in\mathbb{Z}_{>0}$
such that the sets $\psi_{j}\circ\phi_{j}(U_{j})$ are disjoint. Let
$\\{V_{ij}|i\leq n+1,j\in\mathbb{N}\\}$ be the refinement of Lemma 7. Choose
$\alpha(i,j)$ such that $V_{ij}\subseteq U_{\alpha(i,j)}$. Now let
$V_{i}=\cup_{j\in\mathbb{N}}V_{ij}$ and define
$\upsilon_{i}:V_{i}\rightarrow\mathbb{R}^{n}$ by
$\upsilon_{i}(x)=\phi_{\alpha(i,j)}(x)$, for $x\in V_{ij}$. Since $V_{ij}$ are
disjoint for fixed $i$, and the sets $\psi_{j}\circ\phi_{j}(U_{j})$ are
disjoint, it follows immediately that $\upsilon_{i}$ is a diffeomorphism
between $V_{i}$ and a subset of $\mathbb{R}^{n}$, for $i=1,\cdots,n+1$. Hence
$\\{(V_{i},\upsilon_{i}),i=1,\cdots,n+1\\}$ is a finite atlas for $S$.

###### Proposition 4

Every smooth vector bundle $(E,\pi,S,\mathbb{R}^{k})$ has a finite coordinate
representation.

###### Proof

Let $\\{(U_{\alpha},\psi_{\alpha})\\}$ be any coordinate representation for
$(E,\pi,S,\mathbb{R}^{k})$. It follows from Lemma 7 that there exists a
refinement $\\{V_{ij}|i=1,\cdots,m,j\in\mathbb{N}\\}$ of $\\{U_{\alpha}\\}$
such that $V_{ij}\cap V_{ik}=\emptyset$ for $j\neq k$. Let
$V_{i}=\cup_{j}V_{ij}$ and define
$\psi_{i}:V_{i}\times\mathbb{R}^{k}\rightarrow\pi^{-1}(V_{i})$ by
$\psi_{i}(x,y)=\psi_{ij}(x,y)$ if $x\in V_{ij},y\in\mathbb{R}^{k}$, where
$\psi_{ij}$ is the restriction of some $\psi_{\alpha}$. It’s easy to see that
$\psi_{i}$ is a diffeomorphism. Hence $\\{(V_{i},\psi_{i})|i=1,\cdots,m\\}$ is
a coordinate representation for the vector bundle.

##  4 Embedding theorem

###### Lemma 8

8  If $\Phi:U\rightarrow\mathbb{R}^{m}$ is a smooth mapping from an open
subset $U\subseteq\mathbb{R}^{n}$ with $m\geq 2n$, then for
$\epsilon\in\mathbb{R}_{>0}$, there exists $A\in\mathbb{R}^{m\times n}$ such
that (i) $|A_{j}^{i}|<\epsilon$ for
$i\in\\{1,\cdots,m\\},j\in\\{1,\cdots,n\\}$. (ii) the mapping
$x\rightarrow\Phi(x)+Ax$ is an immersion on $U$.

Let $U$ be an open set in $\mathbb{R}^{n}$ and let $K\subseteq U$ be compact
in $\mathbb{R}^{n}$. Consider the smooth mapping
$\Phi:U\rightarrow\mathbb{R}^{m}$. For $x=(x_{1},\cdots,x_{n})\in U$,
$\Phi(x)=(\Phi_{1}(x),\cdots,\Phi_{m}(x))=(\Phi_{1}(x_{1},\cdots,x_{n}),\cdots,\Phi_{m}(x_{1},\cdots,x_{n}))$.
Denote by

$|\Phi|_{K}^{(0)}=\max_{1\leq i\leq m}\sup_{x\in K}\\{|\Phi_{i}(x)|\\},$
$|\Phi|_{K}^{(1)}=\max\\{|\Phi|_{K}^{(0)},|\frac{\partial\Phi}{\partial
x_{1}}|_{K}^{(0)},\cdots,|\frac{\partial\Phi}{\partial x_{n}}|_{K}^{(0)}\\}.$

###### Lemma 9

8  If $\Phi:U\rightarrow\mathbb{R}^{m}$ is an immersion on the compact set
$K\subseteq U$, where $U$ is an open subset of $\mathbb{R}^{n}$, then there
exists $\delta\in\mathbb{R}_{>0}$, such that for any smooth mapping
$\Psi:U\rightarrow\mathbb{R}^{m}$ satisfying that
$|\Psi-\Phi|_{K}^{(1)}<\delta$, $\Psi$ is an immersion on $K$.

###### Lemma 10

Let $(U,\phi,\mathbb{R}^{k})$ and $(V,\psi,\mathbb{R}^{m})$ be two local
charts of the subcartesian space $S$ such that $U\cap V\neq\emptyset$. Then
for each $x\in U\cap V$, there exists a smooth mapping
$F:\mathbb{R}^{k}\rightarrow\mathbb{R}^{m}$ such that $F\circ\phi|W=\psi|W$,
where $W\subseteq U\cap V$ is a neighborhood of $x$.

###### Proof

Consider the mapping $\psi\circ\phi^{-1}:\phi(U\cap
V)\rightarrow\mathbb{R}^{m}$. According to Definition 3, it’s a smooth mapping
from the subcartesian space $\phi(U\cap V)$ to $\mathbb{R}^{m}$, where the
differential structure of $\phi(U\cap V)$ is generated by restrictions of
smooth functions on $\mathbb{R}^{k}$. It follows that for each
$\phi(x)\in\phi(U\cap V)$, there exist a smooth mapping
$F:\mathbb{R}^{k}\rightarrow\mathbb{R}^{m}$ and a neighborhood $\tilde{W}$ of
$\phi(x)$ in $\phi(U\cap V)$, such that
$F|\tilde{W}=\psi\circ\phi^{-1}|\tilde{W}$. Let $W=\phi^{-1}(\tilde{W})$. Then
the result follows immediately.

###### Lemma 11

Let $S$ be a subcartesian space with structural dimension $n$. Let
$\Phi:S\rightarrow\mathbb{R}^{m}$ be a smooth mapping with $m\geq 2n$. Then
for any $\delta>0$, there exists a smooth mapping
$\Psi:S\rightarrow\mathbb{R}^{m}$ satisfying $\|\Phi(x)-\Psi(x)\|<\delta$ for
every $x\in S$, where $\|.\|$ denotes the Euclidean norm. Besides, for every
$x\in S$, there exist some local chart $(U,\mathbb{R}^{k},\phi)$ where $x\in
U$, and a smooth extension $\Psi_{x}:\mathbb{R}^{k}\rightarrow\mathbb{R}^{m}$
of $\Phi$ at $x$, such that $\Psi_{x}$ is an immersion on a neighborhood $V$
of $\phi(x)$ in $\mathbb{R}^{k}$.

###### Proof

Let
$(U_{j})_{j\in\mathbb{Z}_{>0}},(V_{j})_{j\in\mathbb{Z}_{>0}},(W_{j})_{j\in\mathbb{Z}_{>0}}$
be the open covers satisfying the conditions in Lemma 5. For each
$j\in\mathbb{Z}_{>0}$, we claim that there exist smooth function
$\rho_{j}:S\rightarrow[0,1]$ such that $\rho_{j}(x)=1$ for $x\in cl(U_{j})$
and $\rho_{j}(x)=0$ for $x\in S/cl(V_{j})$. Consider the open cover
$\\{V_{j},S/cl(U_{j})\\}$. Let $\\{(\mathscr{V}_{i},g_{i})\\}$ be a
subordinate partition on unity and $f=\sum g_{i}$ where the sum is over those
$i$ for which $\mathscr{V}_{i}\cap cl(U_{j})\neq\emptyset$. Then $f$ is
smooth, is one on $cl(U_{j})$, and zero on $S/cl(V_{j})$.

We inductively define a sequence of maps $\Phi_{k}:S\rightarrow\mathbb{R}^{m}$
such that $\Phi_{k}-\Phi$ is smooth and has support contained in
$K_{k}=\cup_{j=1}^{k}cl(W_{j})$ for each $k\in\mathbb{Z}_{>0}$. For $k=1$, let
$A_{1}\in\mathbb{R}^{m\times n}$ and define

$\Phi_{1}(x)=\Phi(x)+\rho_{1}(x)A_{1}\phi_{1}(x).$

Since $\rho_{1}(x)=1$ for $x\in cl(U_{1})$. We have
$\Phi_{1}(x)=\Phi(x)+A_{1}\phi_{1}(x)$ for $x\in U_{1}$. Let $f_{1}$ be the
smooth extension of $\Phi$ on $W_{1}$ according to Lemma 5. Then it follows
from Lemma 8 that we can choose $A_{1}$ such that (1) the smooth extension
$F(y)=f_{1}(y)+A_{1}y$ of $\Phi_{1}$ on $U_{1}$ is an immersion on
$\mathbb{R}^{n_{1}}$. Hence for each $x\in cl(U_{1})\subseteq V_{1}$, there
exists a smooth extension of $\Phi_{1}$ at $x$ which is also an immersion; (2)
$\|\Phi_{1}(x)-\Phi(x)\|<\frac{\delta}{2}$ for $x\in S$.

Now suppose that we have defined $\Phi_{1},\cdots,\Phi_{k}$ such that, for
each $j\in\\{1,\cdots,k\\},\Phi_{j}-\Phi$ is smooth, has support in $K_{j}$,
and satisfies $\|\Phi_{j}(x)-\Phi_{j-1}(x)\|<\frac{\delta}{2^{j}}$ for $x\in
S$. Besides, for each $x\in\cup_{i=1}^{k}cl(U_{i})$, suppose that there exists
a smooth extension of $\Phi_{k}$ at $x$ which is also an immersion.

For $A_{k+1}\in\mathbb{R}^{m\times n}$ denote

$\Phi_{k+1}(x)=\Phi_{k}(x)+\rho_{k+1}(x)A_{k+1}\phi_{k+1}(x).$

We claim that we can choose $A_{k+1}$ such that (1) there exists a smooth
extension of $\Phi_{k+1}$ at $x$ which is also an immersion, for each
$x\in\cup_{i=1}^{k+1}cl(U_{i})$; (2)
$\|\Phi_{k+1}(x)-\Phi_{k}(x)\|<\frac{\delta}{2^{k+1}}$ for $x\in S$.

To prove the above claim. Let $x\in(\cup_{i=1}^{k}cl(U_{i}))\cap cl(W_{k+1})$.
Suppose that $x\in cl(U_{i})$ for some $i$ satisfying $1\leq i\leq k$. Let
$(W_{i},\mathbb{R}^{n_{i}},\phi_{i})$ and
$(W_{k+1},\mathbb{R}^{n_{k+1}},\phi_{k+1})$ be two local charts of $S$
containing $x$. It follows from Lemma 10 that there exist a neighborhood
$W_{x}$ of $x$ and a smooth mapping
$F_{i{k+1}}:\mathbb{R}^{n_{i}}\rightarrow\mathbb{R}^{n_{k+1}}$ such that
$F_{i{k+1}}\circ\phi_{i}|W_{x}=\phi_{k+1}|W_{x}$. Hence we have

$\Phi_{k+1}(x)=\Phi_{k}(x)+\rho_{k+1}(x)A_{k+1}(F_{i{k+1}}\circ\phi_{i})(x),$

for $x\in W_{x}$. Then by shrinking $W_{x}$ if necessary, it follows from
Lemma 9 that we can choose $A_{k+1}$ sufficiently small such that there exists
a smooth extension of $\Phi_{k+1}$ on $W_{x}$ which is an immersion on a local
neighborhood of $\phi_{i}(x)$ in $\mathbb{R}^{n_{i}}$ containing
$\phi(W_{x})$. Since $(\cup_{i=1}^{k}cl(U_{i}))\cap cl(W_{k+1})\subseteq
cl(W_{k+1})$ and $cl(W_{k+1})$ is compact, it follows that
$(\cup_{i=1}^{k}cl(U_{i}))\cap cl(W_{k+1})$ is compact. Hence it can be
covered by finitely many $W_{x}$, for $x\in(\cup_{i=1}^{k}cl(U_{i}))\cap
cl(W_{k+1})$. This means that we can choose $A_{k+1}$ such that for each
$x\in(\cup_{i=1}^{k}cl(U_{i}))\cap cl(W_{k+1})$, there exist a local chart
$(W,\mathbb{R}^{k},\phi)$ and smooth extension of $\Phi_{k+1}$ at $x$, which
is an immersion in a neighborhood of $\phi(x)$ in $\mathbb{R}^{k}$. For $x\in
S/cl(W_{k+1})$, we have $\Phi_{k+1}(x)=\Phi_{k}(x)$. It follows that for each
$x\in(\cup_{i=1}^{k}cl(U_{i}))/cl(W_{k+1})$, there exists a smooth extension
of $\Phi_{k+1}$ at $x$, which is an immersion. For $x\in cl(U_{k+1})$, the
existence of smooth extension of $\Phi_{k+1}$ at $x$ that is an immersion is
ensured by Lemma 8. Finally, we can choose $A_{k+1}$ sufficiently small such
that $\|\Phi_{k+1}(x)-\Phi_{k}(x)\|<\frac{\delta}{2^{k+1}}$ for $x\in S$. This
completes the proof for the above claim.

Let $\Psi(x)=\lim_{k\rightarrow\infty}\Phi_{k}(x)$, this limit existing and
$\Psi$ being smooth since, for each $x\in S$, according to Lemma 5, there
exists $N\in\mathbb{Z}_{>0}$ such that $y\notin W_{N+1},W_{N+2},\cdots$, for
each $y$ in some neighbourhood of $x$. Hence $\Phi_{N}(y)=\Psi(y)$, for $y$ in
some neighbourhood of $x$. Besides, since $U_{i}\subseteq
W_{i},i=N+1,N+2,\cdots$, it follows that $x\in S/\cup_{i=N+1}^{\infty}U_{i}$,
which yields that $x\in\cup_{i=1}^{N}U_{i}$. It follows from the above
construction of $\Phi_{N}$ that there exists a smooth extension of $\Psi$ at
$x$ which is an immersion. And $\|\Phi-\Psi\|<\delta$ for every $x\in S$. This
completes the proof of the theorem.

###### Lemma 12

Let $S$ be a subcartesian space with structural dimension $n$. Let
$\Phi:S\rightarrow\mathbb{R}^{m}$ be a smooth mapping with $m\geq 2n+1$. Then
for any $\delta>0$, there exists a smooth injective mapping
$\Psi:S\rightarrow\mathbb{R}^{m}$ satisfying $\|\Phi(x)-\Psi(x)\|<\delta$ for
every $s\in S$. Besides, for every $x\in S$, there exist some local chart
$(U,\mathbb{R}^{l},\phi)$ where $x\in U$, and a smooth extension
$\Psi_{x}:\mathbb{R}^{l}\rightarrow\mathbb{R}^{m}$ of $\Phi$ at $x$, such that
$\Psi_{x}$ is an immersion on a neighborhood of $\phi(x)$ in $\mathbb{R}^{l}$.

###### Proof

According to Lemma 11, let $\Upsilon:S\rightarrow\mathbb{R}^{m}$ be a smooth
mapping such that $\|\Upsilon(x)-\Phi(x)\|<\delta$ for every $x\in S$.
Besides, for every $x\in S$, there exist some local chart
$(U,\mathbb{R}^{l},\phi)$ where $x\in U$, and a smooth extension
$\Upsilon_{x}:\mathbb{R}^{l}\rightarrow\mathbb{R}^{m}$ of $\Phi$ at $x$, such
that $\Upsilon_{x}$ is an immersion on a neighborhood of $\phi(x)$ in
$\mathbb{R}^{l}$. It follows that for each $x\in S$ there exists a
neighbourhood $U_{x}$ of $x$ such that $\Upsilon|U_{x}$ is injective. By using
similar constructions as given in Lemma 5, we can get a locally finite open
cover $(U_{j})_{j\in\mathbb{Z}_{>0}}$ of $S$ by relatively compact open sets
such that $\Upsilon|U_{j}$ is injective.

For each $j\in\mathbb{Z}_{>0}$, let $\rho_{j}$ be a smooth function on $S$
taking values in $[0,1]$ and satisfying $\rho_{j}(x)=0$ for $x\in S/U_{j}$,
and suppose that $\sum_{j=1}^{\infty}\rho_{j}(x)=1$ for every $x\in S$, i.e.,
$(\rho_{j})_{j\in\mathbb{Z}_{>0}}$ is a partition of unity. For any sequence
$(y_{j})_{j\in\mathbb{Z}_{>0}}$ in $\mathbb{R}^{m}$ satisfying
$\|y_{j}\|<\frac{\delta}{2^{j+1}}$ for $x\in U_{j}$. for $k\in\mathbb{Z}_{>0}$
define

$\Psi_{k}(x)=\Upsilon(x)+\sum_{j=1}^{k}\rho_{j}(x)y_{j}.$

By Lemma 9 above the sequence $(y_{j})_{j\in\mathbb{Z}_{>0}}$ can be chosen
such that for every $x\in S$, there exist some local chart
$(U,\mathbb{R}^{l},\phi)$ and a smooth extension
$\Psi_{k_{x}}:\mathbb{R}^{l}\rightarrow\mathbb{R}^{m}$ of $\Psi_{k}$ at $x$,
satisfying that $\Psi_{k_{x}}$ is an immersion on a neighborhood of $\phi(x)$
in $\mathbb{R}^{l}$, for each $k\in\mathbb{Z}_{>0}$.

For each $k\in\mathbb{Z}_{>0}$, let

$D_{k}=\\{(p,q)\in S\times S|\rho_{k}(p)\neq\rho_{k}(q)\\}$

and define the mapping $\theta_{k}:D_{k}\rightarrow\mathbb{R}^{m}$ by

$\theta_{k}(x,y)=\frac{\Psi_{k-1}(x)-\Psi_{k-1}(y)}{\rho_{k}(x)-\rho_{k}(y)}$

Since ${\theta_{k}(D_{k})}$ has measure zero ($m\geq 2n+1$), the sequence
$(y_{j})_{j\in\mathbb{Z}_{>0}}$ can be chosen by additionally asking that
$y_{k}\notin\theta_{k}(D_{k})$. We then define
$\Psi(x)=\lim_{k\rightarrow\infty}\Psi_{k}(x)$. It follows that $\Psi$ is a
smooth mapping satisfying that $\|\Psi(x)-\Phi(x)\|<\delta(x)$. Besides, for
every $x\in S$, there exists a smooth extension
$\Psi_{x}:\mathbb{R}^{l}\rightarrow\mathbb{R}^{m}$ of $\Psi$ at $x$, with
local charts $(U,\mathbb{R}^{l},\phi)$, satisfying that $\Psi_{x}$ is an
immersion on a neighborhood of $\phi(x)$ in $\mathbb{R}^{l}$.

It remains to show that $\Psi$ is injective. Suppose that $\Psi(x)=\Psi(y)$.
Let $N\in\mathbb{Z}_{>0}$ be sufficiently large that
$\rho_{j+1}(x)=\rho_{j+1}(y)$ and $\Psi_{j}(x)=\Psi_{j}(y)=\Psi(x)=\Psi(y)$
for every $j\geq N$. Since $y_{N-1}\notin\theta_{N-1}(D_{N-1})$ we have
$\rho_{N-1}(x)=\rho_{N-1}(y)$. Otherwise, since
$\Psi_{N-1}(x)+\rho_{N}(x)y_{N}=\Psi_{N}(x)=\Psi_{N}(y)=\Psi_{N-1}(y)+\rho_{N}(y)y_{N}$,
we have
$y_{N}=\frac{\Psi_{N-1}(x)-\Psi_{N-1}(y)}{\rho_{N}(x)-\rho_{N}(y)}\in\theta_{N}(D_{N})$.
This makes contradiction.

Consequently, we have $\Psi_{N-1}(x)=\Psi_{N-1}(y)$. We can proceed
inductively backwards to conclude that $\rho_{j}(x)=\rho_{j}(y)$ and
$\Psi_{j}(x)=\Psi_{j}(y)$ for every $j\in\mathbb{Z}_{>0}$. Consequently,
$\Upsilon(x)=\Upsilon(y)$. Since $\rho_{j}(x)\neq 0$ for at least one
$j\in\mathbb{Z}_{>0}$, it follows that $x,y\in U_{j}$ for some
$j\in\mathbb{Z}_{>0}$. However, this is in contradiction with the fact that
$\Upsilon|U_{j}$ is injective. This completes the proof of the above theorem.

###### Definition 18

Let $S$ be a topological space. A function $u:S\rightarrow[-\infty,\infty)$ is
an exhaustion function for $S$ if the sublevel set $u^{-1}([-\infty,a))$ is a
relatively compact subset of $S$ for every $a\in\mathbb{R}$.

###### Lemma 13

A continuous map $u:S\rightarrow[-\infty,\infty)$ is an exhaustion function
for a topological space $S$ if and only if, for any $a\in\mathbb{R}$, there
exists a compact set $K\subseteq S$ such that $u(x)>a$ for every $x\in S/K$.

###### Proof

First suppose that $u$ is an exhaustion function and let
$a\in\mathbb{R}_{>0}$. Since $u$ is an exhaustion function,
$u^{-1}([-\infty,a])$ is compact. Moreover, $u(x)>a$ for every $x\in
S/u^{-1}([-\infty,a])$. For the converse, suppose that, for any
$a\in\mathbb{R}_{>0}$, there exists a compact set $K\subseteq S$ such that
$u(x)>a$ for every $x\in S/K$. Let $a\in\mathbb{R}$ and note that
$u^{-1}([-\infty,a])$ is compact, being a closed subset of a compact set,
showing that $u$ is an exhaustion function.

###### Lemma 14

Let $S$ be a subcartesian space. Then there exists a smooth exhaustion
function on $S$.

###### Proof

Since $S$ is locally compact, Hausdorff and second countable, it possesses a
countable open cover by relatively compact open sets. According to Theorem
2.1, there exists a partition of unity $\\{\rho_{i}\\}\in C^{\infty}(S)$
subordinate to such an open cover. Consider the function
$f=\sum_{j=1}^{\infty}j\rho_{j}$. It’s easy to see that $f\in C^{\infty}(S)$.
We claim that $f$ is an exhaustion function. Indeed, let $a\in\mathbb{R}$ and
let $N\in\mathbb{Z}_{>0}$ be such that $N>a$. Let $K$ be a compact set
containing the supports of $\rho_{1},\cdots,\rho_{N}$. We claim that
$f^{-1}((-\infty,a])\subseteq K$. Indeed, suppose that $x\notin K$, then we
have $x\notin\cup_{j=1}^{N}\text{supp}(\rho_{j})$. It follows that
$\sum_{j=N+1}^{\infty}\rho_{j}(x)=\sum_{j=1}^{\infty}\rho_{j}(x)=1$. Hence
$f(x)=\sum_{j=1}^{\infty}j\rho_{j}(x)=\sum_{j=N+1}^{\infty}j\rho_{j}(x)>N\sum_{j=N+1}^{\infty}\rho_{j}(x)=N>a$.
By Lemma 13 we know that $f$ is an exhaustion function on $S$. Hence the
result follows immediately.

###### Definition 19

For topological space $S$ and $T$, a continuous mapping $f:S\rightarrow T$ is
proper if $f^{-1}(K)$ is compact for every compact subset $K\subseteq T$.

###### Lemma 15

8  Let $S$ be a Hausdorff topological space and $T$ be a locally compact
Hausdorff topological space. Let $f:S\rightarrow T$ be a proper injective
mapping. Then $f:S\rightarrow f(S)$ is a homeomorphism provided that $f(M)$ is
equipped with the subspace topology inherited from $T$.

The following result indicates embedding theorem for subcartesian space.

###### Theorem 4.1

Let $S$ be a subcartesian space with structural dimension $n$. Then $S$ is
diffeomorphic to a subset of $\mathbb{R}^{m}$, where $m\geq 2n+1$.

###### Proof

According to Lemma 14, let $u$ be a smooth exhaustion function on $S$. Now
define $\Phi:S\rightarrow\mathbb{R}^{m}$ by

$\Phi(x)=(u(x),0,\cdots,0).$

By Lemma 12, let $\Psi:S\rightarrow\mathbb{R}^{m}$ be a smooth injective
mapping satisfying that $\|\Psi(x)-\Phi(x)\|<1$ for every $x\in S$. Besides,
for every $x\in S$, there exists a smooth extension
$\Psi_{x}:\mathbb{R}^{l}\rightarrow\mathbb{R}^{m}$ of $\Psi$ at $x$, with
local charts $(U,\mathbb{R}^{l},\phi)$, satisfying that $\Psi_{x}$ is an
immersion on a neighborhood of $\phi(x)$ in $\mathbb{R}^{l}$.

We claim that $\Psi$ is proper. Let $K\subseteq\mathbb{R}^{m}$ be compact. If
$x\in\Psi^{-1}(K)$ then

$\|\Psi(x)\|\leq\text{sup}\\{\|x\||x\in K\\}=C_{k},$

and so $\|\Phi(x)\|\leq C_{K}+1$. Since $\|\Phi(x)\|=u(x)$, we have

$\Psi^{-1}(K)\subseteq u^{-1}([-\infty,C_{K}+1]).$

Since $u$ is an exhaustion function we conclude that $\Psi^{-1}(K)$ is
compact, being a closed subset of the compact set $u^{-1}([-\infty,C_{K}+1])$.
It follows from Lemma 15 that $\Psi:S\rightarrow\Psi(S)$ is a homeomorphism.

It remains to show that: (1) for any $f\in C^{\infty}(\mathbb{R}^{m})$,
$\Psi^{*}f\in C^{\infty}(S)$; (2) for any $g\in C^{\infty}(S)$,
$(\Psi^{-1})^{*}g\in C^{\infty}(\Psi(S))$, with the differential structure
$C^{\infty}(\Psi(S))$ generated by restrictions of smooth differential
structure on $\mathbb{R}^{m}$.

To prove (1), let $\Psi=(\Psi_{1},\cdots,\Psi_{m})$. Since $\Psi$ is smooth,
it follows that $\Psi_{i}\in C^{\infty}(S)$, for $i=1,\cdots,m$. Since $f\in
C^{\infty}(\mathbb{R}^{m})$, it follows from condition 2 in Definition 1 that
$\Psi^{*}f=f(\Psi_{1},\cdots,\Psi_{m})\in C^{\infty}(S)$. To prove (2), it
follows from Lemma 12 that for each $x\in S$, there exists a smooth extension
$\Psi_{x}:\mathbb{R}^{l}\rightarrow\mathbb{R}^{m}$ of $\Psi$ at $x$, with
local chart $(U,\mathbb{R}^{l},\phi)$, such that $\Psi_{x}$ is an immersion on
a neighborhood $V$ of $\phi(x)$ in $\mathbb{R}^{l}$. By shrinking $V$ if
necessary, there exists a smooth function $g_{x}$ on $\mathbb{R}^{l}$, such
that $g|V\cap\phi(U)=g_{x}\circ\phi|\phi^{-1}(V)\cap U$. Due to the local
representatives for the immersion $\Psi_{x}$, there exist a neighborhood $W$
of $\Psi(x)$ in $\mathbb{R}^{m}$, and $h\in C^{\infty}(W)$, such that
$h|W\cap\Psi(S)=(\Psi^{-1})^{*}g|W\cap\Psi(S)$. By using bump function, we
know that for each $y\in\Psi(S)$, there exists a function $h_{y}\in
C^{\infty}(\mathbb{R}^{m})$, such that $(\Psi^{-1})^{*}g|W_{y}=h_{y}|W_{y}$,
where $W_{y}$ is a neighborhood of $y$ in $\Psi(S)$. It follows that
$(\Psi^{-1})^{*}g\in C^{\infty}(\Psi(S))$. This completes the proof of (2).
Hence the result of the theorem follows immediately.

###### Corollary 1

(The Whitney Embedding Theorem) If $M$ is a smooth, paracompact, Hausdorff and
connected manifold of dimension $n$, then there exists a proper smooth
embedding of $M$ in $\mathbb{R}^{2n+1}$.

###### Proof

Since a smooth manifold is a subcartesian space with differential structure
given by all the smooth functions on $M$, it follows from Theorem 4.1 that
there exists a proper injective mapping
$\Psi=(\Psi_{1},\cdots,\Psi_{2n+1}):M\rightarrow\mathbb{R}^{2n+1}$ that is
smooth in the sense of differential spaces. Let $(x_{1},\cdots,x_{2n+1})$ be
coordinates for $\mathbb{R}^{2n+1}$, it follows that $\Psi_{i}=\Psi^{*}x_{i}$
is a smooth function on $M$. That is, $\Psi$ is smooth in the sense of
manifolds. From the proof of Lemma 11 and Lemma 12 it can also be ensured that
$\Psi$ is an immersion. Lemma 15 ensures that $\Psi:M\rightarrow\Psi(M)$ is a
homeomorphism. It follows that $\Psi$ is an embedding. Hence the result
follows immediately.

##  5 Smooth generalized distributions on subcartesian spaces

###### Theorem 5.1

5  Let $S$ be a differential subspace of $\mathbb{R}^{n}$, and let $X$ be a
derivation of $C^{\infty}(S)$. For each $x\in S\subseteq\mathbb{R}^{n}$, there
exist a neighborhood $U$ of $x$ in $\mathbb{R}^{n}$ and a vector field $Y$ on
$\mathbb{R}^{n}$ such that

$X(F|S)|U\cap S=(Y(F))|U\cap S$ (1)

for every $F\in C^{\infty}(\mathbb{R}^{n})$.

###### Theorem 5.2

3  Let $M$ be a smooth connected manifold and let $E$ be a smooth vector
bundle over $M$. Let $F$ be a smooth generalized subbundle of $E$. Then $F$ is
globally finitely generated.

###### Theorem 5.3

Let $S$ be a subcartesian space. Let $D$ be a smooth generalized distribution
on $S$. Then $D$ is globally finitely generated.

###### Proof

According to Theorem 4.1, $S$ is diffeomorphic to a subset $\bar{S}$ of
$\mathbb{R}^{m}$. It remains to show that for any smooth generalized
distribution $D$ on a subset $\bar{S}$ of $\mathbb{R}^{m}$, $D$ is globally
finitely generated. We first construct a neighborhood of $\bar{S}$ in
$\mathbb{R}^{m}$ with a smooth generalized distribution on it as follows:
according to Definition 12, for each $x\in\bar{S}$ and $v_{x}\in D_{x}$, by
using bump function, there exist a derivation $X_{x}$ of
$C^{\infty}(\bar{S})$, such that $X_{x}$ is a smooth section of $D$ and
$X_{x}(x)=v_{x}$. It follows from Theorem 5.1 that there exist a neighborhood
$U_{v_{x}}$ of $x$ and a vector field $Y_{v_{x}}$ on $\mathbb{R}^{n}$ such
that

$X_{x}(F|\bar{S})|U_{v_{x}}\cap\bar{S}=(Y_{v_{x}}(F))|U_{v_{x}}\cap\bar{S},$
(2)

for every $F\in C^{\infty}(\mathbb{R}^{n})$. Now consider the open subset
$\cup_{x\in\bar{S}}\cup_{v_{x}\in D_{x}}U_{v_{x}}$ of $\mathbb{R}^{m}$. For
each $y\in\cup_{x\in\bar{S}}\cup_{v_{x}\in D_{x}}U_{v_{x}}$, let $\Lambda$ be
the set of all points in $T\bar{S}$ such that $U_{v}$ contains $y$ for every
$v\in\Lambda$. We define a generalized distribution $\hat{D}$ on
$\cup_{x\in\bar{S}}\cup_{v_{x}\in D_{x}}U_{v_{x}}$ by

$\hat{D}_{y}=\text{span}_{\mathbb{R}}\\{Y_{v}(y)|v\in\Lambda\\}.$ (3)

It’s easy to see that $\hat{D}$ is smooth on the open submanifold
$\cup_{x\in\bar{S}}\cup_{v_{x}\in D_{x}}U_{v_{x}}$ of $\mathbb{R}^{m}$. Then
according to Theorem 5.2, there exist smooth sections $Y_{1},\cdots,Y_{l}$ of
$\hat{D}$ such that $\hat{D}_{y}$ is spanned by $Y_{1}(y),\cdots,Y_{l}(y)$ for
every $y\in\cup_{x\in\bar{S}}\cup_{v_{x}\in D_{x}}U_{v_{x}}$. We claim that
$Y_{1},\cdots,Y_{l}$ yield derivations $X_{1},\cdots,X_{l}$ of
$C^{\infty}(\bar{S})$ that span $D_{x}$ for each $x\in\bar{S}$.

For each $f\in C^{\infty}(\bar{S})$, for each $x\in\bar{S}$, there exist a
neighborhood $U$ of $x$ in $\cup_{x\in\bar{S}}\cup_{v_{x}\in D_{x}}U_{v_{x}}$,
and a function $F_{x}\in C^{\infty}(\cup_{x\in\bar{S}}\cup_{v_{x}\in
D_{x}}U_{v_{x}})$, such that $f|U\cap\bar{S}=F_{x}|U\cap\bar{S}$. Set
$X_{i}(f)|U\cap\bar{S}=Y_{i}(F_{x})|U\cap\bar{S}$. We claim that $X_{i}(f)\in
C^{\infty}(\bar{S})$ is well-defined. Let $V$ be another neighborhood of $x$
in $\cup_{x\in\bar{S}}\cup_{v_{x}\in D_{x}}U_{v_{x}}$, and let $H_{x}\in
C^{\infty}(\cup_{x\in\bar{S}}\cup_{v_{x}\in D_{x}}U_{v_{x}})$ be a function
such that $f|V\cap\bar{S}=H_{x}|V\cap\bar{S}$. We have that $F_{x}|U\cap
V\cap\bar{S}=H_{x}|U\cap V\cap\bar{S}$. Then it follows from Lemma 3, (2) and
(3) that $Y_{i}(F_{x}-H_{x})|U\cap V\cap\bar{S}=0$. This yields that
$X_{i}(f)$ is a well-defined smooth function on $\bar{S}$. Hence we get a
derivation $X_{i}$ of $C^{\infty}(\bar{S})$. From (2) and (3) we can also see
that $X_{i}(x)\in D_{x}$ for each $x\in\bar{S}$.

It remains to show that $X_{1}(x),\cdots,X_{l}(x)$ span $D_{x}$ at each
$x\in\bar{S}$. Let $w_{x}\in D_{x}$. Then according to the above construction
of $\hat{D}$, we know that there exist a derivation $X$ of
$C^{\infty}(\bar{S})$ such that $X(x)=w_{x}$, a neighborhood $U_{w_{x}}$ of
$x$ and a vector field $Y_{w_{x}}$ on $\mathbb{R}^{m}$ such that
$X(F|\bar{S})|U_{w_{x}}\cap\bar{S}=(Y_{w_{x}}(F))|U_{w_{x}}\cap\bar{S}$, for
every $F\in C^{\infty}(\mathbb{R}^{m})$, where $Y_{w_{x}}(x)\in\bar{D}_{x}$.
So there exist constants $c_{1},\cdots,c_{l}$ such that
$Y_{w_{x}}(x)=\sum_{i=1}^{l}c_{i}Y_{i}(x)$. Then it follows from the above
definition of $X_{i}$ that $\sum_{i=1}^{l}c_{i}X_{i}(x)=w_{x}$. This completes
the proof for the above claim. Hence the result of the theorem follows
immediately.

##  6 Smooth generalized subbundles on subcartesian spaces

###### Lemma 16

Let $S$ be a subcartesian space. Let $(E,\pi,S,\mathbb{R}^{k})$ be a smooth
vector bundle. Then $E$ is globally finitely generated.

###### Proof

It follows from Proposition 4 that $(E,\pi,S,\mathbb{R}^{k})$ admits a finite
coordinate representation $\\{(U_{i},\psi_{i})|i=1,\cdots,m\\}$. Then on each
$U_{i}$, there exist local sections $\gamma_{i1},\cdots,\gamma_{ik}$ of
$(E,\pi,S,\mathbb{R}^{k})$ such that
$\pi^{-1}(x)=\text{span}_{\mathbb{R}}\\{\gamma_{i1}(x),\cdots,\gamma_{ik}(x)\\}$
for each $x\in U_{i},i=1,\cdots,m$. Let $\\{\rho_{j}|j=1,2,\cdots\\}$ be the
partition of unity for $S$ subordinate to the covering
$\\{U_{i}|i=1,\cdots,m\\}$. For $i\in\\{1,2,\cdots,m\\}$, let

$I_{i}=\\{l\in\\{1,2,\cdots\\}|\text{the support of}\,\rho_{l}\,\text{is
contained in}\,U_{i}\\}.$

Define global sections $\sigma_{ij}$ by

$\sigma_{ij}=\sum_{l\in I_{i}}\rho_{l}\gamma_{ij},$

for $i=1,\cdots,m$ and $j=1,\cdots,k$. Then for each $e_{x}\in\pi^{-1}(x)$,
there exists some $\rho_{i}$ such that $\rho_{i}(x)\neq 0$ and the support of
$\rho_{i}$ is contained in some $U_{j}$. It follows that there exist constants
$c_{1},\cdots,c_{k}$ such that $e_{x}=\sum_{l=1}^{k}c_{l}\sigma_{jl}(x)$.
Hence the result follows.

###### Proposition 5

For every smooth vector bundle $(E,\pi,S,\mathbb{R}^{k})$ there exist a
trivial bundle $S\times\mathbb{R}^{l}$, for some $l\in\mathbb{N}$ and a smooth
injective bundle map $\varphi:E\rightarrow S\times\mathbb{R}^{l}$.

###### Proof

According to Lemma 16, we know that there exist finite global generators
$\sigma_{1},\cdots,\sigma_{l}$ for $E$. Consider the trivial bundle
$S\times\mathbb{R}^{l}$ and the bundle map
$\psi:S\times\mathbb{R}^{l}\rightarrow E$ given by

$\psi(x,\sum_{i=1}^{l}\lambda_{i}e_{i})=\sum_{i=1}^{l}\lambda_{i}\sigma_{i}(x),x\in
S,$

where $e_{1},\cdots,e_{l}$ is a basis for $\mathbb{R}^{l}$. Then each linear
map $\psi_{x}:\mathbb{R}^{l}\rightarrow\pi^{-1}(x)$ is surjective. From
Proposition 2, there exist smooth Riemannian metrics on
$(E,\pi,S,\mathbb{R}^{k})$ and $(S\times\mathbb{R}^{l},\pi,S,\mathbb{R}^{l})$.
Define $\varphi:E\rightarrow S\times\mathbb{R}^{l}$ by
$\varphi_{x}=\psi_{x}^{*}(\psi_{x}\circ\psi_{x}^{*})^{-1},x\in S$, where
$\psi_{x}^{*}$ is the adjoint map of $\psi_{x}$ associated with the smooth
Riemannian metrics on $(E,\pi,S,\mathbb{R}^{k})$ and
$(S\times\mathbb{R}^{l},\pi,S,\mathbb{R}^{l})$. $\varphi$ is a smooth bundle
map satisfying that $\psi\circ\varphi=\text{identity}$. Hence $\varphi$ is
injective.

###### Theorem 6.1

Let $S$ be a subcartesian space. Let $(E,\pi,S,\mathbb{R}^{k})$ be a smooth
vector bundle. Let $F$ be a smooth generalized subbundle of $E$. Then $F$ is
globally finitely generated.

###### Proof

According to Proposition 5, it suffices to show that every smooth generalized
subbundle of the trivial bundle $S\times\mathbb{R}^{l}$ is globally finitely
generated. We identify each subspace $F(\pi^{-1}(x))$ with a subspace of
$\mathbb{R}^{l}$ and identify sections of $S\times\mathbb{R}^{l}$, and hence
sections of $F$, with $R^{l}$-valued functions. Further, it follows from
Theorem 4.1 that $S$ is diffeomorphic to a subset of $\mathbb{R}^{m}$. Hence
for each $e_{x}\in F(\pi^{-1}(x))$, since $F$ is smooth, there exist a
neighborhood $U_{e_{x}}$ of $x$ in $\mathbb{R}^{m}$ and a smooth section
$\xi_{U_{e_{x}}}$ satisfying that $\xi_{U_{e_{x}}}(x)=e_{x}$ and
$\xi_{U_{e_{x}}}|U_{e_{x}}\cap S$ is a local smooth section of $F$. Now
consider the open subset $U=\cup_{x\in S}\cup_{e_{x}\in
F(\pi^{-1}(x))}U_{e_{x}}$ and the generalized subbundle $\hat{F}$ of
$U\times\mathbb{R}^{l}$ defined by

$\hat{F}(y)=\text{span}_{\mathbb{R}}\\{\xi_{U_{e_{x}}}(y)|y\in U_{e_{x}}\\}.$

It’s obvious that $\hat{F}$ is smooth. It follows from Theorem 5.2 that there
exist smooth global generators $\xi_{1},\cdots,\xi_{s}$ of $\hat{F}$. We claim
that $\xi_{1}|S,\cdots,\xi_{s}|S$ are smooth global generators of $F$. It
follows from the definition of $\xi_{U_{e_{x}}}$ and $\hat{F}$ that
$\xi_{1}|S,\cdots,\xi_{s}|S$ are smooth sections of $F$. Besides, since
$e_{x}\in\hat{F}(x)$ for each $x\in S$ and $e_{x}\in F(x)$, it follows
immediately that $\xi_{1}|S,\cdots,\xi_{s}|S$ are global generators of $F$.
This completes the proof of the theorem.

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