Lee K.K. Lectures on Dynamical Systems, Structural Stability and Their Applications
Подождите немного. Документ загружается.
Chapter
2
Topics
in
Topology
and
Differential
Geometry
In
this
chapter,
we
assemble
most
of
the
definitions
and
theorems
about
basic
properties
of
algebra,
points
set
(also
called
general)
topology,
algebraic
and
differential
topology,
differentiable
manifolds,
and
differential
geometry,
which
are
needed
in
the
course
of
the
lectures.
our
main
purpose
is
to
establish
notation,
terminology,
concepts,
and
structures.
We
provide
definitions,
examples,
and
essential
results
without
giving
the
arguments
of
proof.
The
material
presented
is
slightly
more
than
the
essential
knowledge
for
this
lecture
series.
For
instance,
we
certainly
can
get
by
without
explicitly
talking
about
algebraic
topology,
such
as
homotopy
and
homology.
However,
we
do
utilize
the
concepts
of
orientable
manifolds
or
spaces,
so
that
everywhere
the
non-zero
volume
element
can
be
found.
We
also
discuss
the
connected
components
and
connected
sums
of
a
state
space.
Although
we
do
discuss
tubular
neighborhoods
at
the
end
of
this
chapter,
we
do
not
explicitly
use
some
of
the
results
in
discussing
dynamical
systems;
nonetheless
when
dealing
with
return
maps,
the
stability
of
orbits
and
periodic
orbits,
etc.,
we
implicitly
use
the
concept
of
tubular
neighborhoods.
Nonetheless,
these
and
many
other
concepts
and
terminology
are
frequently
used
in
research
literature.
It
is
also
our
intent
to
introduce
the
reader
to
a
more
sophisticated
mathematical
framework
so
that
when
the
reader
ventures
to
research
literature
or
further
reading,
he
will
not
feel
totally
lost
due
to
different
"culture"
or
terminology.
Furthermore,
in
global
theory,
the
state
spaces
may
be
differentiable
manifolds
with
nontrivial
topology.
In
such
cases,
concepts
from
algebraic
topology
at
times
are
essential
to
the
understanding.
In
light
of
the
above
remarks,
for
the
first
reading,
the
reader
may
want
to
skip
Section
2.3,
part
of
Section
2.4,
the
de
Rham
cohomology
part
of
Section
2.8,
and
Section
2.9.
24
There
are
several
well
written
introductory
books
on
general
topology,
algebraic
topology,
differential
topology,
and
differential
geometry,
which
can
provide
more
examples,
further
details
and
more
general
results,
e.g.,
Hicks
[1971],
Lefschetz
[1949],
Munkres
[1966],
Nach
and
Sen
[1983],
Singer
and
Thorpe
[1967],
Wallace
[1957].
For
more
advanced
readers,
the
following
books
are
highly
recommended:
Bishop
and
Crittenden
[1964],
Eilenberg
and
Steenrod
[1952],
Greenberg
[1967],
Hirsch
[1976],
Kelley
[1955],
and
Steenrod
[1951].
2.1
Getting
to
the
basics
-
algebra
Some
results
and
notations
from
group
theory
will
be
needed
later,
here
we
sketch
some
of
them.
Recall
that
a
group
is
a
set
of
elements,
denoted
by
G,
closed
under
an
operation
·
(or
+)
usually
called
multiplication
(or
addition),
satisfying
the
following
axioms:
(1)
x·(y·z)
=
(x·y)·z
for
all
x,y,z
e
G:
(2)
There
exists
a
unique
element
e e G
called
the
identity
such
that
x·e
=
e·x
= x
for
all
x e
G:
(3)
for
a
given
x e G,
there
is
a
unique
element
x-
1
e G
called
the
inverse
of
x
such
that
x·x-
1
= x-
1
·x
=
e.
A
subset
H
of
G, H c G,
is
called
a
subgroup
if
H
is
a
group
(with
the
same
operation
as
in
G).
A
subset
H c G
is
a
subgroup
of
G
iff
ab-
1
e H
for
all
a,b
e H.
Let
H,
G
be
two
groups
(with
the
same
operation
·),
a
mapping
f : G
into
....
H
is
a
homomorphism
if
f(x·y)
=
f(x)·f(y)
for
all
x,
y e G.
If
e
is
the
identity
of
H,
then
f-
1
(e)
c G
and
f-
1
(e)
is
called
the
kernel
of
f,
and
f(G)
is
the
image
of
f.
It
can
be
shown
that
f(G)
c H.
Graphically,
it
looks
like
this:
G@
f
(into)
@H
25
Two
groups
G
and
H
are
called
isomorphic
if
there
is
a
one-
to-one,
onto
map
f:G
ooto~
H
such
that
f(x·y)
=
f(x)·f(y)
for
all
x,
y E G,
and
f
is
called
an
isomorphism
and
denoted
by
G
~
H. A homomorphism
f:
G
~
H
is
an
isomorphism
iff
f
is
onto
and
its
kernel
contains
only
the
identity
element
of
G.
Graphically,
G
(-)~~f~(~1~to~1~·~o~nt~o~l-----(r)
H
If
G
is
a
group
and
S a
set
(finite
or
infinite)
of
elements
of
G
such
that
every
element
of
G
can
be
expressed
as
a
product
(here
we
use
the
multiplication
operation)
of
elements
of
S
and
their
inverses,
then
G
is
said
to
be
generated
by
S
and
elements
of
S
are
called
the
generators
of
G.
If
a
group
G
satisfies
another
axiom,
in
addition
to
(1),
(2)
and
(3)
stated
above,
specifically,
the
commutative
law,
i.e.,
x·y
=
y·x
for
all
x,
y E
G,
then
G
is
abelian.
If
the
group
operation
is
+
instead
of
·,
such
an
abelian
group
is
called
an
additive
abelian
group
and
its
identity
element
is
denoted
by
o,
and
the
inverse
of
x
is
-x,
for
all
x E
G.
If
G
is
any
additive
abelian
group
and
H c
G,
G
can
be
split
up
into
a
family
of
subsets
called
cosets
of
H,
where
any
two
elements
x,y
E G
belong
to
the
same
coset
if
x-y
E
H.
The
coset
of
H
with
the
additive
operation
form
a
group
called
the
quotient
group
of
G
with
respect
to
H,
denoted
by
G/H.
By
considering
a
sequence
of
homomorphisms,
one
can
calculate
a
group
from
other
related
groups.
This
is
the
notion
of
exact
sequence.
Let
f:
A
~
B
be
a homomorphism,
then
from
the
definition
of
the
images
and
kernel
of
f,
we
have
Im f
~
A/Ker
f.
A
sequence
A
f~
B
~
c
is
exact
at
B
iff
Im f =
Ker
g.
An
exact
sequence
has
the
following
properties:
(i)
a
sequence
id
h~
A
f~
B
is
exact
at
A
iff
f
is
1-to-1;
26
(ii)
a
sequence
A
1
~
B
~
id
is
exact
at
B
iff
f
is
onto;
(iii)
a
sequence
id
~
A
1
~
B
~
id
is
exact
(everywhere)
iff
f
is
an
isomorphism;
(iv)
a
short
exact
sequence
id
~
A
1
~
B
~
c
~
id
has
the
property
that
C
~
B/Im
f.
We
shall
utilize
these
properties
of
a
sequence
in
Chapter
3
and
thereafter.
In
the
next
section,
we
shall
discuss
some
fundamentals
of
general
topology.
2.2
Bird's
aye
view
or
general
topology
A
topological
space
consists
of
a
set
X
with
an
assignment
of
a
non-empty
family
of
subsets
of
X
to
each
element
of
X.
The
subsets
assigned
to
each
point
p e X
will
be
called
neighborhoods
of
p.
The
assignment
of
neighborhoods
to
each
point
p e X
must
satisfy:
(1)
If
U
is
a
neighborhood
of
p,
then
p e U;
(2)
Any
subset
of
X
containing
a
neighborhood
of
p
is
itself
a
neighborhood
of
p;
(3)
If
U
and
V
are
neighborhoods
of
p,
so
is
u n V;
(4)
If
U
is
a
neighborhood
of
p,
there
is
a
neighborhood
V
of
p
such
that
U
is
a
neighborhood
of
every
point
of
v.
Example:
X =
R"
, U =
(a
sphere
of
center
p}
then
the
above
conditions
are
satisfied,
so
X
is
a
topological
space.
Exercise:
Let
X
be
a
set
and
suppose
that
each
pair
of
elements
x
and
y
of
X
is
assigned
a
real
number
d(x,y)
called
the
distance
function
between
x
and
y
and
satisfying
the
following
conditions:
(1)
d(x,y)
~
0
and
d(x,y)
= 0
iff
x =
y;
(2)
d(x,y)
=
d(y,x)
for
all
x,y
e
X;
(3)
d(x,z)
S
d(x,y)
+
d(y,z)
for
any
x,
y,
z e
x.
Now
define
an
e-neighborhood
U
of
x E X
as
U
=(y
e
xl
d(x,y)
<
e}.
Then
define
a
neighborhood
of
x
eX
to
be
any
set
in
X
containing
an
e-neighborhood
of
x.
Prove
that
with
these
definitions,
X
becomes
a
topological
space.
Remark:
A
topological
space
defined
in
such
a way
is
called
a
metric
space.
Also,
prove
that
Euclidean
space
is
a
metric
space.
27
If
A
and
B
are
any
sets,
then
the
set
of
all
pairs
(a,b)
with
a e A
and
b e B
is
denoted
by
A x
B,
and
is
called
the
product
set
of
A
and
B.
Let
X
and
Y
be
two
topological
spaces.
Let
a
set
w c X x Y a
neighborhood
of
(x,y)
E X x Y
where
x e X, y e Y
if
there
is
a
neighborhood
U
of
x E X
and
a
neighborhood
V
of
y e Y
such
that
U x V c
W.
The
product
space
X x Y
becomes
a
topological
space
and
X x Y
is
called
the
topological
product
of
X
and
Y.
Clearly,
X=
Euclidean
2-plane
is
the
product
of
R x R
where
R
is
the
space
of
real
numbers.
Let
A
be
a
set
of
points
in
a
topological
space
X. A
point
p e A
is
an
interior
point
of
A
if
there
is
a
neighborhood
U
of
p
such
that
U c
A.
The
interior
of
A = A
{all
interior
points
of
A}.
Example:
X=
R'.
A=
{(x·y)
I
x'
+
y'
S
1}.
A=
{(x·y)
lx'
+
y'
<
1}.
A
set
A
in
a
topological
space
is
an
open
set
if,
for
each
p e
A,
there
is
a
neighborhood
U
of
p
such
that
U c A
(i.e.,
there
is
no
"boundary").
Example:
A=
{xl
a<
x < b
,a,b
e
R}.
Also,
any
solid
open
n-sphere
(i.e.,
LX;'
<a')
in
R"
is
an
open
set.
The
following
theorems
can
also
be
taken
as
a
set
of
axioms
to
give
an
alternative
definition
of
a
topological
space.
Theorem
2.2.1
Let
X
be
a
topological
space.
Then
(1)
the
union
of
an
arbitrary
collection
of
open
sets
in
X
is
an
open
set;
(2)
the
intersection
of
a
finite
collection
of
open
sets
in
X
is
an
open
set;
(3)
the
whole
space
X
is
an
open
set;
(4)
the
empty
set
is
an
open
set.
As
a
consequence,
we
have
the
following
theorem:
Theorem
2.2.2
If
A
is
any
set
in
a
topological
space,
then
A
is
an
open
set
and
it
is
the
largest
open
set
contained
in
A.
We
have
mentioned
prior
to
Theorem
2.2.1
that
the
properties
of
open
sets
could
be
used
to
provide
an
alternative
definition
of
a
topological
space.
Such
a
procedure
is
based
on:
28
Theorem
2.2.3
Let
X
be
an
abstract
set
and
let
0
be
a
family
of
subsets
of
X
such
that
(1)
The
union
of
any
collection
of
sets
belonging
to
0
is
a
set
belonging
to
O;
(2)
The
intersection
of
a
finite
collection
of
sets
belonging
to
0
is
a
set
belonging
to
o;
(3)
The
whole
set
X
belongs
to
O;
(4)
The
empty
set
0
belongs
to
o.
Then
there
is
one
and
only
one
way
of
making
X
into
a
topological
space
(i.e.,
by
assigning
neighborhoods
to
the
elements
of
X)
so
that
0
is
the
family
of
open
sets
of
X:
by
requiring
that
a
set
U
is
a
neighborhood
of
p e X
iff
there
is
a
set
W
beloning
to
o
such
that
p e W c U.
Intuitively,
it
is
easy
to
see
with
the
help
of
Theorem
2.2.1,
nonetheless
the
proof
is
somewhat
tedious.
The
essence
of
this
theorem
is
that
a
topological
space
can
be
defined
by
giving
the
open
sets
instead
of
assigning
neighborhoods
to
each
point.
The
original
definition
of
a
topological
space
in
terms
of
neighborhoods
is
probably
the
most
convenient
and
the
most
intuitive
one,
which
depends
on
a common
sense
notion
of
nearness.
But
there
are
advantages
from
the
viewpoint
of
open
sets.
First,
it
is
more
simple
logically
to
name
one
single
family
of
open
sets
in
X
than
to
name
a
family
of
sets
attached
to
every
point
of
X.
Moreover,
many
definitions
can
be
made
and
theorems
proven
more
easily
and
elegantly
in
terms
of
open
sets
than
in
terms
of
neighborhoods.
Let
X
be
a
topological
space
and
S a
subset.
If
p e
s,
a
subset
U
of
s
is
a
neighborhood
of
p
in
s
iff
U = S n v
for
some
neighborhood
V
of
p
in
X.
Then
the
neighborhoods
in
S
of
points
of
s
define
a
topology
on
s.
This
is
the
topology
induced
on
s
by
X,
and
s
with
this
topology
is
a
subspace
of
X.
Remark:
One
must
be
careful,
in
speaking
of
a
space
and
a
subspace,
to
distinguish
between
neighborhoods
in
the
space
and
neighborhoods
in
the
subspace;
likewise,
to
distinguish
between
open
sets
in
the
space
and
open
sets
in
29
the
subspace.
Example:
Let
X=
R
2
,
i.e.,
the
(x,y)-plane,
and
letS=
R
1
be
the
x-axis.
As
usual,
a
neighborhood
of
a
point
p e X
is
any
set
containing
a
circular
disk
with
center
p,
while
a
neighborhood
of
a
point
p e S
is
any
set
in
S
containing
an
interval
with
midpoint
p.
If
U
is
an
open
interval
in
s
(i.e.,
interval
without
endpoints),
U
is
open
inS
but
not
in
X
because
U
contains
no
circular
disk.
The
usual
topologies
on
X
and
S make S a
subspace
of
X.
This
can
be
seen
by
noting
that
if
p e
s,
the
intersection
of
s
with
a
disk
of
center
p
is
an
interval
with
midpoint
at
p.
Theorem
2.2.4
Let
X
be
a
topological
space
and
S a
subspace.
Then
a
subset
U
of
S
is
open
in
S
iff
there
is
an
open
set
V
in
X
such
that
U = V n
s.
Exercise:
Let
X
be
a
topological
space
and
S a
subset
of
X.
Define
U c S
to
be
open
in
S
iff
U = V n S
for
some
set
V
open
in
X.
Show
that
the
sets
so
defined
as
being
open
in
S
satisfy
the
conditions
of
Theorem
2.2.3
thus
defining
a
topology
on
S.
A
sequence
of
points
p
1
,
p
2
••••
in
a
topological
space
X
is
said
to
have
a
limit
p,
or
to
converge
to
p,
if
for
any
preassigned
neighborhood
U
of
p,
there
is
an
integer
N
such
that:
Pn
e U
for
all
n
~
N.
Remark:
There
are
two
warnings
to
be
made
concerning
the
limits
of
an
arbitrary
topological
space.
First,
the
cauchy
criterion
for
the
convergence
of
a
sequence
of
real
numbers
has
no
analogue
in
a
topological
space
in
general.
This
is
because
in
general
topological
space
there
is
no
uniform
standard
of
nearness
which
can
be
applied
to
a
variable
pair
of
points.
(But
cauchy
criteria
do
have
a
uniform
standard,
i.e.,
the
metric).
Second,
there
is
no
guarantee
that
the
limit
of
a
sequence
of
points,
if
it
exists,
is
unique.
It
is
desirable
to
restrict
our
attention
to
topological
spaces
satisfying
a
condition
wich
will
enable
the
uniqueness
of
the
limit
of
a
sequence
of
points
to
be
proven.
The
following
definition
will
provide
the
required
30
condition.
A
topological
space
X
is
a
Hausdorff
space,
sometimes
denoted
by
T
2
space,
if
for
every
pair
p,
q e X
with
p +
q,
there
is
a
neighborhood
U
of
p
and
a
neighborhood
v
of
q
such
that
U n V = o.
Theorem
2.2.5
Let
X
be
a
Hausdorff
space,
and
suppose
that
a
sequence
(Pn}
has
a
limit
p.
Then
this
limit
is
unique.
Example:
R
1
is
T
2
•
In
general,
R"
in
its
usual
topology
is
a T
2
space.
Let
A
be
a
set
of
points
in
a
topological
space
X.
Then
a
point
p E X
is
a
limit
point
(or
accumulation
point)
of
A
if
every
neighborhood
of
p
contains
a
point
of
A
different
from
p.
Example:
Take
X = R
1
•
Then
the
above
definition
becomes
the
one
usually
given
in
analysis.
One
can
take
a
decreasing
sequence
of
intervals
with
p
as
midpoint
of
length,
say
1,
1/2, 1/3, 1/4,
•.••
If
pis
a
limit
point
of
the
set
of
numbers
A,
each
of
these
intervals
will
contain
a
number
x
1
,
x
2
,
•••
respectively,
different
from
p
and
belonging
to
A.
If
U
is
any
neighborhood
of
the
number
p,
U
will
contain
every
interval
of
length
1/n
and
midpoint
p
for
sufficiently
large
n.
Thus
the
sequence
x
1
,
x
2
,
•••
hasp
as
its
limit.
Incidentally,
every
neighborhood
of
p
contains
infinitely
many
points
of
A.
There
are
pathological
cases
one
should
be
aware
of
(in
particular
for
non-T
2
spaces).
Corollary
2.2.6
In
a T
2
space,
only
infinite
sets
of
points
are
capable
of
having
limit
points
at
all.
Note
also
that
a
limit
point
of
a
set
A
in
a
topological
space
need
not
be
the
limit
of
any
sequence
of
points
of
A.
The
construction
of
an
example
is
a
bit
complicated.
Let
X
.
be
the
set
of
all
real
valued
functions
of
the
real
variable
x,
defined
for
all
values
of
x.
If
f
is
any
point
of
X
(i.e.,
a
function
of
X)
the
(e,
x
1
,
x
2
,
••••
)-neighborhood
of
f
will
be
defined
to
be
the
set
of
all
points
¢
of
X
such
that
l¢(x;)
-
f(x;)
I <
e,
i =
1,2,
...
n.
such
a
neighborhood
31
of
f
can
be
defined
for
every
positive
number
£,
and
every
finite
collection
of
number
x
1
,
x
2
,
•••
xn.
The
topology
of
X
can
be
defined
by
saying
that
U
is
a
neighborhood
of
f
iff
U
contains
a
(£,
x
1
,
x
2
,
•••
,xn)-
neighborhood
off
for
some£,
x
1
,
x
2
,
•••
,xn.
It
can
be
verified
that
this
assignment
of
neighborhood
satisfies
the
conditions
of
a
topological
space.
Furthermore,
one
can
show
that
this
space
is
a T
2
space.
Now
let
f
0
be
the
function
of
x
which
is
identically
zero,
and
define
a
subset
A
of
X
as
follows.
Let
x
1
,
x
2
,
•••
,xn
be
any
finite
collection
of
real
numbers
and
let
fx,x,..
..
x.,
be
the
function
of
x
defined
by
setting
fx,x
1
...
x.,(X;)
0
(i
=
1,2,
..
,n)
and
fx,x •... x..,(x)= 1
for
all
other
values
of
X.
Let
A
be
the
set
of
all
fx,x •..
x.,
for
all
possible
dinite
sets
x
1
,
x
2
••
,xn
of
real
numbers.
f
0
certainly
does
not
belong
to
A.
But
f
0
is
a
limit
point
of
A
in
the
topological
space
X.
Let
U
be
a
neighborhood
of
f
0
•
Then
U
contains
the
(£,x
1
,x
2
,
••
,xn)-neighborhood
of
f
0
for
some£
and
x
1
,
2
,
••
,xn
and
the
function
fx,
...
x,
belongs
to
this
(£,x
1
,
••
,xn)-
neighborhood,
since
I fx, .. x.,(X;) - f
0
(X;) I < £
because
both
terms
of
the
difference
are
zero
fori
=1,2,
••
,n.
Clearly
fx,
x
is
a member
of
A
and
belongs
to
the
given
neighborhood
U
;;:f
f
0
•
Thus
f
0
is
a
limit
point
of
A.
Now
we
want
to
show
that
f
0
is
not
the
limit
of
any
sequence
selected
from
A.
Suppose
f
1
,
f
2
,
••
is
a
sequence
belonging
to
A
and
having
f
0
as
the
limit.
By
the
definition
of
A,
each
fn
is
a
function
of
x
equal
to
zero
at
a
finite
set
Sn
of
values
of
x
and
equal
to
1
for
all
other
values.
The
union
of
all
the
Sn
for
n =
1,2,
...
is,
at
most,
a
denumerable
set
of
points,
and
sothere
is
a
real
number
x
0
not
belonging
to
any
of
the
sn.
Let
V
be
the
(£,x)-
neighborhood
of
f
0
for
some £ <
1.
By
the
choice
of
x
0
,
f;(X
0
)=
1
for
all
i,
and
so
none
of
the
members
of
the
sequence
f
1
,
f
2
,
•••
belongs
to
V,
it
follows
that
f
0
cannot
be
the
limit
of
this
sequence.
It
might
be
added
that
the
topological
space
constructed
32
in
the
above
example
appears
quite
naturally
and
is
used
frequently
in
analysis.
To
say
that
a
sequence
f
1
,
f
2
,
•••
of
points
of
this
space
converges
to
the
limit
f
means
exactly
that
the
functions
f
1
,
f
2
,
••
converge
pointwise
to
f,
i.e.,
for
every
fixed
value
of
x,
the
sequence
f
1
(x),
f
2
(x),
...
of
real
number
converges
to
f(x).
Just
in
passing,
suppose
a
topological
space
X
satisfies
the
following
condition:
For
each
point
p e X
there
is
a
sequence
N
1
(p),
N
2
(p),
...
of
neighborhoods
of
p
such
that
N~
1
(p)
c Nn(P)
for
each
n,
and
given
any
neighborhood
U
of
p
there
is
an
n
such
that
Nn(P) c
U.
One
can
prove
that
in
this
case
if
p
is
a
limit
point
of
a
set
A
in
X, p
is
the
limit
of
some
sequence
of
points
of
A.
The
above
condition
imposed
on
X
is
called
the
first
axiom
of
countability.
(The
second
one
is
a
condition
on
the
family
of
open
sets.
We
shall
come
to
that
later).
Exercise:
Prove
that
a
Euclidean
space
satisfies
the
first
countability
axiom.
In
fact,
one
can
show
that
the
axiom
holds
for
any
metric
space.
Let
A
be
a
set
in
a
topological
space
X.
Then
the
closure
of
A,
A,
is
defined
to
be
the
set
in
X
consisting
of
all
the
points
of
A
along
with
all
the
limit
points
of
A.
Examples:
(1)
X=
R
in
the
usual
topology,
A=
(xia
< x
<
b},
then
A=
(xia
S x S
b}.
(2)
X=
R"
and
A
is
the
set
of
points
at
distance
less
than
r
from
some
fixed
point
p,
then
A
is
the
set
of
points
whose
distance
from
p
are
less
than
or
equal
to
r.
A
set
A
in
a
topological
space
X
with
the
property
that
A = A
is
called
a
closed
set
of
X.
Theorem
2.2.7
A
set
A
in
a
topological
space
X
is
closed
iff
the
complement
of
A
in
X
is
an
open
set.
This
theorem
is
sometimes
used
to
define
a
closed
set.
one
can
also
translate
Theorem
2.2.1
into
a
theorem
about
closed
sets
by
taking
the
complements
of
all
the
open
sets
mentioned
there.
Once
again,
it
should
be
pointed
out
that
if
one
is
considering
a
subspace
S
of
a
space
X
as
well
as
S
itself,
33