lectures on advanced mathematical methods for physicists pdf

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Advanced
Mathematical Methods
for Physicists
Lectures on

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Advanced
Mathematical Methods
for Physicists
Lectures on

Sunil Mukhi
Tata Institute of Fundamental Research, India

N Mukunda
formerly of Indian Institute of Science, India

~HINDUSTAN

U LQJ UBOOK AGENCY

,~World Scientific
NEW JERSEY. LONDON· SINGAPORE· BEIJING· SHANGHAI· HONG KONG· TAIPEI· CHENNAI

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Published by
World Scientific Publishing Co. Pte. Ltd.
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.

LECTURES ON ADVANCED MATHEMATICAL METHODS FOR PHYSICISTS
Copyright © 2010 Hindustan Book Agency (HBA)
Authorized edition by World Scientific Publishing Co. Pte. Ltd. for exclusive distribution worldwide
except India.
The distribution rights for print copies of the book for India remain with Hindustan Book Agency (HBA).

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.

ISBN-13 978-981-4299-73-2
ISBN-1O 981-4299-73-1

Printed in India, bookbinding made in Singapore.

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Contents
Part I: Topology and Differential Geometry

1

Introduction to Part I

3

1

2

3

4

Topology
1.1 Preliminaries
1.2 Topological Spaces
1.3 Metric spaces . . .
1.4 Basis for a topology
1.5 Closure . . . . . . .
1.6 Connected and Compact Spaces.
1.7 Continuous Functions
1.8 Homeomorphisms.

1.9 Separability

5

5
6
9
11
12
13
15

17
18

Homotopy
2.1 Loops and Homotopies . . . . . . . .
2.2 The Fundamental Group . . . . . . .
2.3 Homotopy Type and Contractibility
2.4 Higher Homotopy Groups . . . . . .

21

Differentiable Manifolds I
3.1 The Definition of a Manifold
3.2 Differentiation of Functions
3.3 Orient ability . . . . . . . . .
3.4 Calculus on Manifolds: Vector and Tensor Fields
3.5 Calculus on Manifolds: Differential Forms
3.6 Properties of Differential Forms.

3.7 More About Vectors and Forms .

41
41

Differentiable Manifolds II
4.1 Riemannian Geometry ..

65
65

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21

25
28
34

47
48
50

55
59
62


Contents

VI

4.2
4.3
4.4
4.5
4.6
4.7
4.8
5

6

Frames . . . . . . . . . . . . . . . . .
Connections, Curvature and Torsion
The Volume Form . . . . . . . .
Isometry....... . . . . . . .
Integration of Differential Forms
Stokes'Theorem . . . .
The Laplacian on Forms . .

67

69
74
76
77

80
83

Homology and Cohomology
5.1 Simplicial Homology . . . . . . . . . . . . .
5.2 De Rham Cohomology . . . . . . . . . . . .
5.3 Harmonic Forms and de Rham Cohomology

87
87
100

Fibre Bundles
6.1 The Concept of a Fibre Bundle
6.2 Tangent and Cotangent Bundles . . .
6.3 Vector Bundles and Principal Bundles

105
105
111
112

Bibliography for Part I

103

117

Part II: Group Theory and Structure and Representations of Compact Simple Lie Groups and Algebras
119
Introduction to Part II

121

7

Review of Groups and Related Structures
7.1 Definition of a Group . . . . . . . . . . . .
7.2 Conjugate Elements, Equivalence Classes .
7.3 Subgroups and Cosets . . . . . . . . . . . .
7.4 Invariant (Normal) Subgroups, the Factor Group
7.5 Abelian Groups, Commutator Subgroup . . . . .
7.6 Solvable, Nilpotent, Semi simple and Simple Groups.
7.7 Relationships Among Groups . . . . . . . . . . . . .
7.8 Ways to Combine Groups - Direct and Semidirect Products
7.9 Topological Groups, Lie Groups, Compact Lie Groups

123
123
124
124
125
126
127
129
131
132

8

Review of Group Representations
8.1 Definition of a Representation . . . . . . . . . . . .

8.2 Invariant Subspaces, Reducibility, Decomposability
8.3 Equivalence of Representations, Schur's Lemma . .
8.4 Unitary and Orthogonal Representations. . . . . .
8.5 Contragredient, Adjoint and Complex Conjugate Representations
8.6 Direct Products of Group Representations . . . . . . . . . . . . .

135
135
136
138
139
140
144

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Contents

9

Lie
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8

Vll

Groups and Lie Algebras
147
Local Coordinates in a Lie Group.
147
Analysis of Associativity . . . . . .
148
One-parameter Subgroups and Canonical Coordinates
151
Integrability Conditions and Structure Constants . . .
155
Definition of a (real) Lie Algebra: Lie Algebra of a given Lie Group157
Local Reconstruction of Lie Group from Lie Algebra
158
Comments on the G --) G Relationship . . . . . . .
160
Various Kinds of and Operations with Lie Algebras.
161

10 Linear Representations of Lie Algebras
11 Complexification and Classification of Lie
11.1 Complexification of a Real Lie Algebra. .
11.2 Solvability, Levi's Theorem, and Cartan's
(Semi) Simple Lie Algebras . . . . . . .
11.3 The Real Compact Simple Lie Algebras .

165
Algebras

171
. . . . . . . . . . . . 171
Analysis of Complex
173
. . . . . . . .
180

12 Geometry of Roots for Compact Simple Lie Algebras

183

13 Positive Roots, Simple Roots, Dynkin Diagrams
13.1 Positive Roots . . . . . . . . . ..
13.2 Simple Roots and their Properties
13.3 Dynkin Diagrams. . . . . . . . . .

189
189
189
194

14 Lie Algebras and Dynkin Diagrams for SO(2l), SO(2l+1), USp(2l),
SU(l + 1)
197
14.1 The SO(2l) Family - Dl of Cartan. . .
197
201
14.2 The SO(2l + 1) Family - Bl of Cartan
14.3 The USp(2l) Family - Gl of Cartan . .
203

207
14.4 The SU(l + 1) Family - Al of Cartan .
14.5 Coincidences for low Dimensions and Connectedness
212
15 Complete Classification of All CSLA Simple Root Systems
15.1 Series of Lemmas . . . .
15.2 The allowed Graphs r .
15.3 The Exceptional Groups

215
216
220
224

16 Representations of Compact Simple Lie Algebras
16.1 Weights and Multiplicities . . . . . . . . . . . .
16.2 Actions of En and SU(2)(a) - the Weyl Group
16.3 Dominant Weights, Highest Weight of a UlR
16.4 Fundamental UIR's, Survey of all UIR's
16.5 Fundamental UIR's for AI, B l , Gl, Dl . . . . .

227
227
228
230
233
234

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Contents

VIll

16.6 The Elementary UIR's . . . . . .
16.7 Structure of States within a UIR

240
241

17 Spinor Representations for Real Orthogonal Groups
245
17.1 The Dirac Algebra in Even Dimensions. . . . . . . . . . . . .. 246
17.2 Generators, Weights and Reducibility of U(S) - the spinor UIR's
of Dl . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
248
17.3 Conjugation Properties of Spinor UIR's of Dl . . . . . .
250
252
17.4 Remarks on Antisymmetric Tensors Under Dl = SO(2l)
17.5 The Spinor UIR's of Bl = SO(2l + 1) . . . . . .
257
17.6 Antisymmetric Tensors under Bl = SO(2l + 1) . . . . .
260
18 Spinor Representations for Real Pseudo Orthogonal Groups
18.1 Definition of SO(q,p) and Notational Matters. . . . .
18.2 Spinor Representations S(A) of SO(p, q) for p + q == 2l . . .
18.3 Representations Related to S(A) . . . . . . . . . . . . . . .
18.4 Behaviour of the Irreducible Spinor Representations S±(A)

18.5 Spinor Representations of SO(p, q) for p + q = 2l + 1
18.6 Dirac, Weyl and Majorana Spinors for SO(p, q)

261
261
262
264
265
266
267

Bibliography for Part II

273

Index

275

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Part I: Topology and
Differential Geometry
Sunil Mukhi
Department of Theoretical Physics
Tata Institute of Fundamental Research
Mumbai 400 005, India

+ J.e. Bose Fellow.

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Introduction to Part I
These notes describe the basic notions of topology and differentiable geometry in
a style that is hopefully accessible to students of physics. While in mathematics
the proof of a theorem is central to its discussion, physicists tend to think of
mathematical formalism mainly as a tool and are often willing to take a theorem
on faith. While due care must be taken to ensure that a given theorem is actually
a theorem (i.e. that it has been proved), and also that it is applicable to the
physics problem at hand, it is not necessary that physics students be able to
construct or reproduce the proofs on their own. Of course, proofs not only
provide rigour but also frequently serve to highlight the spirit of the result. I
have tried here to compensate for this loss by describing the motivation and
spirit of each theorem in simple language, and by providing some examples.
The examples provided in these notes are not usually taken from physics,
however. I have deliberately tried to avoid the tone of other mathematics-forphysicists textbooks where several physics problems are set up specifically to
illustrate each mathematical result. Instead, the attempt has been to highlight
the beauty and logical flow of the mathematics itself, starting with an abstract
set of points and adding "qualities" like a topology, a differentiable structure, a
metric and so on until one has reached all the way to fibre bundles.
Physical applications of the topics discussed here are to be found primarily
in the areas of general relativity and string theory. It is my hope that the
enterprising student interested in researching these fields will be able to use
these notes to penetrate the dense physical and mathematical formalism that

envelops (and occasionally conceals!) those profound subjects.

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Chapter 1

Topology
1.1

Preliminaries

Topology is basically the study of continuity. A topological space is an abstract
structure defined on a set, in such a way that the concept of continuous maps
makes sense. In this chapter we will first study this abstract structure and then
go on to see, in some examples, why it is a sensible approach to continuous
functions.
The physicist is familiar with ideas of continuity in the context of real
analysis, so here we will use the real line as a model of a topological space.
For a mathematician this is only one example, and by no means a typical one,
but for a physicist the real line and its direct products are sufficient to cover
essentially all cases of interest.
In subsequent chapters, we will introduce additional structures on a topological space. This will eventually lead us to manifolds and fibre bundles, the
main "stuff" of physics.
The following terms are in common use and it will be assumed that the

reader is familiar with them: set, subset, empty set, element, union, intersection,
integers, rational numbers, real numbers. The relevant symbols are illustrated
below:
subset:
union:
set of integers:

c
U

7L

empty set:
intersection:
set of rational numbers:

¢>

element:

E

n
Q

set of real numbers:

1R

We need some additional terminology from basic set theory that may also

be known to the reader, but we will explain it nevertheless.

Definition: If A c B, the complement of A in B, called A' is

A' = { x E B

I x ~A

}

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Chapter 1. Topology

6

For the reader encountering such condensed mathematical statements for
the very first time, let us express the above statement in words: A' is the set
of all elements x that are in B such that x is not in A. The reader is encouraged to similarly verbalise any mathematical statement that is not immediately
comprehensible.
We continue with our terminology.

Definition: The Cartesian product of two sets A and B is the set
A

@

B = { (a, b) I a E A, bE B }

Thus it is a collection of ordered pairs consisting of one element from each of
the original sets.
Example: The Cartesian product lRx lR is called lR 2 , the Euclidean plane. One
can interate Cartesian products any number of times, for example lRx lRx· .. x lR
is the n-dimensional Euclidean space lR n .

We continue by defining functions and their basic properties.

Definition: A function A --> B is a rule which, for each a E A, assigns a unique
bE B, called the image of a. We write b = f(a).
A function f: A --> B is surjective (onto) if every b E B is the image of at
least one a E A.
A function f: A --> B is injective (one-to-one) if every b E B is the image
of at most one a E A.
A function can be surjective, or injective, or neither, or both. If it is both,
it is called b~jective. In this case, every b E B is the image of exactly one a E A.
Armed with these basic notions, we can now introduce the concept of a
topological space.

1.2

Topological Spaces

Consider the real line lR. We are familiar with two important kinds of subsets
of lR:
Open interval: (a, b)
Closed interval: [a, bj

Ia < x < b}
= { x E lR I a :::::: x :::::: b }

={x

E

lR

(1.1 )

A closed interval contains its end-points, while an open interval does not. Let
us generalize this idea slightly.

Definition: X C lR is an open set in lR if:

x EX::::} x E (a, b)

c

X for some (a, b)

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1.2. Topological Spaces

7

In words, a subset X of IR will be called open if every point inside it can be
enclosed by an open interval (a, b) lying entirely inside it.

Examples:
(i) Every open interval (a, b) is an open set, since any point inside such an
interval can be enclosed in a smaller open interval contained within (a, b).
(ii) X = { x E IR I x > 0 } is an open set. This is not, however, an open interval.
(iii) IR itself is an open set.
(iv) The empty set cp is an open set. This may seem tricky at first, but one
merely has to check the definition, which goes "every point inside it can be ... ".
Such a sentence is always true if there are no points in the set!
(v) A closed interval is not an open set. Take X = [a, bj. Then a E X and b E X
cannot be enclosed by any open interval lying entirely in [a, bj. All the remaining
points in the closed interval can actually be enclosed in an open interval inside
[a, bj, but the desired property does not hold for all points, and therefore the
set fails to be open.
(vi) A single point is not an open set. To see this, check the definition.
Next, we define a closed set in IR.

Definition: A closed set in IR is the complement in IR of any open set.
Examples:
(i) A closed interval [a, bj is a closed set. It is the complement of the open set
X = { x E IR I x > b } U { x E IR I x < a }.
(ii) A single point {a} is a closed set. It is the complement in IR of the open set
X = { x E IR I x > a } U { x E IR I x < a }.
(iii) The full set IR and the empty set cp are both closed sets, since they are both
open sets and are the complements of each other.
We see that a set can be open, or closed, or neither, or both. For example,
[a, b) is neither closed or open (this is the set that contains the end-point a but
not the end-point b). Since a cannot be enclosed by an open set, [a, b) is not
open. In its complement, b cannot be enclosed by any open set. So [a, b) is not
closed either. In IR, one can check that the only sets which are both open and
closed according to our definition are IR and cp.

It is important to emphasize that so far, we are only talking about open
and closed sets in the real line IR. The idea is to extract some key properties
of these sets and use those to define open sets and closed sets in an abstract
setting. Continuing in this direction, we note some interesting properties of
open and closed sets in IR:
(a) The union of any number of open sets in IR is open. This follows from the
definition, and should be checked carefully.
(b) The intersection of a finite number of open sets in IR is open.
Why did we have to specify a finite number? The answer is that by taking
the intersection of an infinite number of open sets in IR, we can actually manu-

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Chapter 1. Topology

8

facture a set which is not open. As an example, consider the following infinite
collection of open intervals:

{ (-~,~) i n =1,2,3, ...

}

(1.2)

°

As n becomes very large, the sets keep becoming smaller. But the point is

contained in each set, whatever the value of n. Hence it is also contained in
their intersection. However any other chosen point a E JR will lie outside the
set (- ~, ~) once n becomes larger than I!I' Thus the infinite intersection over
all n contains only the single point 0, which as we have seen is not an open set.
This shows that only finite intersections of open sets are guaranteed to be open.
Having discussed open and closed sets on JR, we are now in a position to
extend this to a more abstract setting, which is that of a topological space. This
will allow us to formulate the concepts of continuity, limit points, compactness,
connectedness etc. in a context far more general than real analysis.
Definition: A topological space is a set S together with a collection U of subsets
of S, called open sets, satisfying the following conditions:
(i) ¢ E U, S E U.
(ii) The union of arbitrarily many Ui E U is again in U (thus the union of any
number of open sets is open).
(iii) The intersection of finitely many subsets Ui E U is again in U (thus the
intersection of finitely many open sets is open).
The pair (S, U) is called a topological space. The collection U of subsets of S is
called a topology on S.

The reader will notice that if the set S happens to coincide with the real
line JR, then the familiar collection of open sets on JR satisfies the axioms above,
and so JR together with its usual collection of open sets provides an example of
a topological space. But as we will soon see, we can put different topologies on
the same set of points, by choosing a different collection of subsets as open sets.
This will lead to more general (and strange!) examples of topological spaces.
Having defined open sets, it is natural to define closed sets as follows:
Definition: In a given topological space, a closed set is a subset of S which is
the complement of an open set. Thus, if U E U, then U' == { xES I x rf:.U } is
closed.
Examples:

(i) Let S be any set whatsoever. Choose U to be the collection of two sets
{ ¢, S}. This is the smallest collection we can take consistent with the axioms! Clearly all the axioms are trivially satisfied. This is called the trivial or
sometimes indiscrete topology on S.
(ii) Let S again be any set. Choose U to be the collection of all subsets of S.
This is clearly the largest collection we can possibly take. In this topology, all
subsets are open. But they are all closed as well (check this!). This is another
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9

1.3. Metric spaces

trivial example of a topology, called the discrete topology, on S. We see, from
this example and the one above, that it is possible to define more than one
topology on the same set. We can also guess that if the topology has too few
or too many open sets, it is liable to be trivial and quite uninteresting.
(iii)] Let S be the real line JR. U is the collection of all subsets X of JR such
that
(1.3)
x EX=} x E (a, b) eX.
This is our old definition of "open set in JR". We realise now that it was not the
unique choice of topology, but it was certainly the most natural and familiar.
Accordingly, this topology is called the usual topology on JR.
(iv) S is a finite set, consisting of, say, six elements. We write
S = {a, b, c, d, e, f}

(1.4)

U= {¢,S,{a,b},{b},{b,c},{a,b,c}}

(1.5)

Choose
This defines a topology. Some of the closed sets (besides ¢ and S) are {d, e, f}
and {a, c, d, e, fl. This example shows that we can very well define a topology
on a finite set.
Exercise: If we leave out {a, b} in U, do we still get a topology? What if we
leave out {b}? What if we add {d}?

1.3

Metric spaces

A topological space carries no intrinsic notion of metric, or distance between
points. We may choose to define this notion on a given topological space if
we like. We will find that among other things, introducing a metric helps to
generate many more examples of topological spaces.

Definition: A metric space is a set S along with a map which assigns a real
number:::: 0 to each pair of points in S, satisfying the conditions below. If
xES, yES then d(x, y) should be thought of as the distance between x and
y. The conditions on this map are:
(i) d(x, y) = 0 if and only if x = y.
(ii) d(x, y) = d(y, x)
(iii) d(x, z) :::; d(x, y) + d(y , z) (triangle inequality).
The map d: S x S - 7 JR+ is called a metric on S .
We see from the list of axioms above, that we are generalising to abstract
topological spaces the familiar notion of distance on Euclidean space. Later
on we will see that Euclidean space is really a "differentiable manifold with a

metric", which involves a lot more structure than we have encountered so far.
It is useful to keep in mind that a metric can make sense without any of that
other structure - all we need is a set, not even a topological space. In fact we
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Chapter 1. Topology

10

will see in a moment that defining a metric on a set actually helps us define a
topological space.
Examples: On lR we may choose the usual metric d(x, y) = Ix - yl. On lR2 we
can similarly choose d(x, y) = Ix - 111- The reader should check that the axioms
are satisfied.
Exercise: Does d(x, y)

= (x -

yf

define a metric on lR?

Given any metric on a set S, we can define a particular topology on S,
called the metric topology.
Definition: With respect to a given metric on a set S, the open disc on S at
the point xES with radius a > 0 is the set

Dx(a) = { yES I d(x, y) < a }
The open disc is just the set of points strictly within a distance a of the

chosen point. However, we must remember that this distance is defined with
respect to an abstract metric that may not necessarily be the one familiar to
us.
Having defined open discs via the metric, we can now define a topology
(called the m etric topology) on S as follows. A subset XeS will be called an
open set if every x E X can be enclosed by some open disc contained entirely
in X:
X is open if x EX=} x E Dx(a) eX

(1.6)

for some a. This gives us our collection of open sets X, and together with the
set S, we claim this defines a topological space.
Exercise: Check that this satisfies the axioms for a topological space.
We realise that the usual topology on lR is just the metric topology with
d(x, y) = Ix - YI. The metric topology also gives the usual topology on ndimensional Euclidean space IRn. The metric there is:
n

d(x\ yi)

=

2)xi _ yi)2

(1.7)

i=l

where Xi, yi , i = 1,2, ... ,n are the familiar coordinates of points in lRn.
The open discs are familiar too. For lR 2 they are the familiar discs on the

plane (not including their boundary), while for lR3 they are the interior of a
solid ball about the point. In higher dimensions they are generalisations of this,
so we will use the term "open ball" to describe open discs in any lRn.
Note that for a given set of points we can define many inequivalent metrics. On lR, for example, we could use the rather unorthodox (for physicists)
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11

1.4. Basis for a topology

definition:
d(x,y)

= 1, x =f.
= 0, x =

y
y.

Thus every pair of distinct points is the same (unit) distance apart.
Exercise: Check that this satisfies the axioms for a metric. What do the open
discs look like in this metric? Show that the associated metric topology is one
that we have already introduced above.

1.4

Basis for a topology

Defining a topological space requires listing all the open sets in the collection

U. This can be quite tedious. If there are infinitely many open sets it might
be impossible to list all of them. Therefore, for convenience we introduce the
concept of a basis.
For this we return to the familiar case - the usual topology on IR. Here, the
open intervals (a, b) form a distinguished subclass of the open sets. But they
are not all the open sets (for example the union of several open intervals is an
open set, but is not itself an open interval. If this point was not clear then it is
time to go back and carefully review the definition of open sets on IR!).
The open intervals on IR have the following properties:
(i) The union of all open intervals is the whole set IR:
U

(ai, bi ) = IR

(1.8)

(ii) The intersection of two open intervals can be expressed as the union of other
open intervals. For example, if al < a2 < b1 < b2 then

(1.9)
(iii) cp is an open interval: (a, a) = cp.
These properties can be abstracted to define a basis for an arbitrary topological
space.
Definition: In an arbitrary topological space (S, U), any collection of open sets
(a subset of the full collection U) satisfying the above three conditions is called
a basis for the topology.
A basis contains only a preferred collection of open sets that "generates"
(via arbitrary unions and finite intersections) the complete collection U. And
there can be many different bases for the same topology.
Example: In any metric space, the open discs provide a basis for the metric

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12

Chapter 1. Topology

topology. The reader should check this statement.
Exercise: In the usual topology on IR 2 , give some examples of open sets that
are not open discs. Also find a basis for the same topology consisting of open
sets that are different from open discs.

1.5

Closure

In many familiar cases, we have seen that the distinction between open and
closed sets is that the former do not contain their "end points" while the latter
do. In trying to make this more precise, we are led to the concept of closure of
a set.
In IR2 , for example, the set Xl = { x c IR2 I i!2 < 1 } defines an open
set, the open unit disc. It is open because every point in it can be enclosed in
a small open disc which lies inside the bigger one (see Fig. l.1). Points on the
unit circle i!2 = 1 are not in the unit open disc, so we don't have to worry about
enclosing them.

..

(~"'.

....... ,,,
,,

~ ~"' ...
'"

......... ,

Figure l.1: The open unit disc. Every point in it can be enclosed in an open
disc.

:s

On the other hand consider the set X 2 = { xC IR2 I i!2
1 }. In addition
to points within the unit circle, this set contains all points on the unit circle.
But clearly X 2 is not an open set, since points on the boundary circle cannot
be enclosed by open discs in X 2 . In fact X 2 is a closed set, as one can check by
going back to the axioms.
So it seems that we can add some points to an open set and make it a
closed set. Let us make this precise via some definitions.
Definition: Let (S, U) be a topological space. Let A C S. A point s E S is
called a limit point of A if, whenever s is contained in an open set u E U, we
have

(u-{s})nA#¢
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1.6. Connected and Compact Spaces

13

In other words, S is a limit point of A if every open neighbourhood of S has a
non-empty intersection with A.
Exercise: Show that all points on the boundary of an open disc are limit points
of the open disc. They do not, however, belong to the open disc. This shows
that in general, a limit point of a set A need not be contained in A.

Definition: The closure of a set A

A=

A

c

Sis:

u { limit points of A }

In other words, if we add to a set all its limit points, we get the closure of the
set. This is so named because of the following result:

Theorem: The closure of any set A c S is a closed set.
Exercise: Prove this theorem. As always, it helps to go back to the definition.
What you have to show is that the complement of the closure A is an open set.
Given a topological space (S, U), we can define a topology on any subset
A c S. Simply choose UA to be the collection of sets ui n A, Ui C U. This
topology is called the relative topology on A. Note that sets which are open in

A C S in the relative topology need not themselves be open in S.
Exercise: Consider subsets of IR and find an example to illustrate this point.

1.6

Connected and Compact Spaces

Consider the real line IR with the usual topology. If we delete one point, say
{O}, then IR - {O} falls into two disjoint pieces. It is natural to say that IR is
connected but IR - {O} is disconnected. To make this precise and general, we
need to give a definition in terms of a given topology on a set.

Definition: A topological space (S, U) is connected if it cannot be expressed
as the union of two disjoint open sets in its topology. If on the other hand we
can express it as the union of disjoint open sets, in other words if we can find
open sets Ul, U2 E U such that Ul n U2 = ¢, Ul U U2 = S, then the space is said
to be disconnected.
Theorem: In a connected topological space (S, U), the only sets which are
both closed and open are Sand ¢.
Exercise: Prove this theorem.
This definition tells us in particular that with the usual topology, IRn is
connected for all n. IR - {O} is disconnected, but IRn - {O} is connect~d for
n 2': 2. On the other hand, the space ]R.2 - { (x, 0) I x E IR}, which is IR2
with a line removed, is disconnected. Similarly, IR2 - { (x, y) Ix 2 + y2 = I},
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14

Chapter 1. Topology

which is IR2 with the unit circle removed, is disconnected. IR3 minus a plane is
disconnected, and so on. In these examples it is sufficient to rely on our intuition
about connectedness, but in more abstract cases one needs to carefully follow
the definition above.
Note that connectivity depends in an essential way on not just the set, but
also the collection of open sets U, in other words on the topology. For example,
IR with the discrete topology is disconnected:

IR=Ua{a}, aEIR

(1.10)

Recall that each {a} is an open set, disjoint from every other, in the discrete
topology. We can also conclude that IR is disconnected in the discrete topology
from a theorem stated earlier. In this topology, IR and ¢ are not the only sets
which are both closed and open, since each {a} is also both closed and open.
We now turn to the study of closed, bounded 3ubsets of IR which will turn
out to be rather special.

Definition: A cover of a set X is a family of sets {Fa} = F such that their
union contains X, thus
Xc UaFa

If, (S, U) is a topological space and XeS, then a cover { Fa} is said to be
an open cover if Fa E U for all n, namely, if Fa are all open sets.
Now there is a famous theorem:

Heine-Borel Theorem: Any open cover of a closed bounded subset of IR n (in
the usual topology) admits a finite subcover.

Let us see what this means for IR. An example of a closed bounded subset is an open interval [a, bj. The theorem says that if we have any (possibly
infinite) collection of open sets {Fa} which cover [a, bj then a finite subset of
this collection also exists which covers [a, bj. The reader should try to convince
herself of this with an example.
An open interval (a, b) is bounded but not closed, while the semi-infinite
interval [0,00) = { x E IR I x > 0 } is closed but not bounded. So the HeineBorel theorem does not apply in these cases. And indeed, here is an example of
an open cover of (a, b) with no finite sub cover. Take (a, b) = (-1, 1). Let

Fn=

(-1+~,1-~)
n
n

n=2,3,4, ...

(1.11)

With a little thought, one sees that U~=l Fn = (-1,1). Therefore Fn provides
an open cover of (-1,1). But no finite subset of the collection {Fn} is a cover
of (-1,1).
Exercise: Find an open cover of [0, 00) which has no finite subcover.
Closed bounded subsets of IRn have several good properties. They are
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15

1.7. Continuous Functions

known as compact sets on lRn. Now we need to generalize this notion to arbitrary
topological spaces. In that case we cannot always give a meaning to "bounded" ,
so we proceed using the equivalent notions provided by the Heine-Borel theorem.

Definition: Given a topological space (3, U), a set X
if every open cover admits a finite sub cover.

c

3 is said to be compact

Theorem: Let 3 be a compact topological space. Then every infinite subset
of 3 has a limit point. This is one example of the special properties of compact
sets.
Exercise: Show by an example that this is not true for non-compact sets. It
is worth looking up the proof of the above theorem.

Theorem: Every closed subset ofa compact space is compact in the relative
topology. Thus, compactness is preserved on passing to a topological subspace.

1.7

Continuous Functions

Using the general concept of topological spaces developed above, we now turn
to the definition of continuity. Suppose (3, U) and (T, V) are topological spaces,
and f : 3 ----t T is a function. Since f is not in general injective (one to one), it
does not have an inverse in the sense of a function f- 1 : T ----t 3. But we can
define a more general notion of inverse, which takes any subset of T to some
subset of 3. This will provide the key to understanding continuity.

Definition: If T'

c

T, then the inverse

f- 1 (T') c

f-l(V) = { s E 3

I f(s)

3 is defined by:

E T' }

Note that the inverse is defined on all subsets T' of T, including the individual elements {t} C T treated as special, single-element subsets. However it
does not necessarily map the latter to individual elements s E 3, but rather to
subsets of 3 as above. The inverse evaluated on a particular subset of T may
of course be the empty set ¢ C 3.
For the special case of bijective (one-to-one and onto) functions, singleelement sets {t} c T will be mapped to single-element sets f- 1 ( {t}) c 3. In
this case we can treat the inverse as a function on T, and the definition above
coincides with the usual inverse function.
Consider an example in lR with the usual topology:

Example: f : lR ----t lR+ is defined by f(x) = x 2 . Then, f- 1 : {y} C lR+ ----t
{Vfj, -Vfj} c lR (Fig. 1.2). Now take f- 1 on an open set of lR+, say (1,4).
Clearly
(1.12)

(1,4) C lR+ ----t {(I, 2), (-1, -2)} C lR.

rl:

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16

Chapter 1. Topology

x
-2

2

-I

Figure 1.2: f(x) = x 2 , an example of a continuous function.

Thus the inverse maps open sets of lR.+ to open sets of lR..
One can convince oneself that this is true for any continuous function lR. ---+
lR. +. Moreover, it is false for discontinuous functions, as the following example
shows (Fig. 1.3):

f(x)

+ 1,
= x + 2,
=

x

x <0
x

~

0

f(x)

x

-\

112

Figure 1.3: An example of a discontinuous function.
In this example, the open set (~, ~) c lR. is mapped by f- 1 to the set [O,~)
which is not open. In fact it can be shown that on lR. with the usual topology,
the continuous functions are those for whom the inverse always takes open sets
to open sets. For discontinuous functions (i.e. functions having a "break" in
their graph) there will be at least one open set which is mapped by the inverse

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