Lecture 7
(compactification continued)
IR2<s2
Recall
X
IR2
,
Kleinbottle
compact
XcY
3
.
=
Y
Definition [Quotient map it is
(ii)
topological
surjective
UCY
open if
Theorem (unique
If
fix-cy
definean
equir
a
relation
②
X
a
Y--y
-
g
feal
ift
:
fab)
homeomorphism
unique
st the
hiagram
· It
y
commutes
already
X-IY
fix-cY g
than
Rock
got
surjectivity
UCE
Y->I
:
open
is
a
:
.
given special proprely(
then
are
by
have
the
both
X-c E
feat
is
preimage
quotient mup
.
g"(U)
=>
=7
at open
8"(g(us)cy
e
Got
:
flog-
cnt
bijetlon ,
defined then it has
beauotient mags
a
.
set's
way open
Lemma
Let
1) = 20 17
if
is called aquotient map
cont
-
.
.
amb
X by
on
I
o
Cu
avotient map
there exists
P
.
compet ification
is open
homeumorphism
is
IR=
(0 13
compact
not
fix-cY
.
space
&"CUX
,
compat
Y is called
(i)
,
Hasdorf ,
,
Y
5
x
·
.
taurus mobirs strip
<Totally compacti
totally
X Y
,
open
,
so
a
a
homeomorphism
.
example
x
# ~
=
y
but
.
132
.
.
27
wis
z
rewrite
partial quotient
with
=
=
10
.
13xs'
U
20
13xs'
.
construction quite clean
the
It makes
CHomotype)
2
topological spaces
ty
20
20 13x20
7
X
.
13"
=
can
y
X Y
.
**
=
z
Definition
20
harpis
are
,
g
it = H
:
:
,
X -> Y
it
you
two
think
about
continuous
X*[0 1]-cY
maps
continuous
.
it
in terms
HCX 07
sot
HCx ,
17
-
~
i
~
~
~
/
=
,
2
L
~
~
f
X
I
H : 20 13 x[0 < 17
Define
.
H(x +)
(1
=
.
.
i
.
+) +(x)
+
+
-
ycx)
(convex)
:
y
-
scales multiplication preserves continuity
Lemma
f
IR2
->
e
[0
if
.
1]->YCIR"
y
,,
42tY
,
then
ty
,
+
<1-HycX
for all
=
.
Worksheet
X [0 13 Y= IR
quotient maps
.
and
example
of
IE [0 13
.
=
fex)
gcx FxEX
.
Cpash connected) generalization
Zimme
2
Every
bg
maps
20
:
.
.
13 -Y
hounstaple
are
Definition (partn connected)
topological
I
X is called
space
continuous
a
2 20 11->x
5 C0)
Sot
:
map
it for
path connected
.
=
every
x
.
J CK)
X
yEX
=
y
Lemma (Generalization 17
originally
the
we
formula
same
H
Questin (4)
had
:
x
=
17->IR"
.
for X
holds
xxC0 13
it
me
303
y
hone
It
Hexhts
such
30
=
13
.
303
then
x
20
f(0)
-
X
=
20
.
17-230
=
.
and
Exampl
.
20 13
both
Es
open
Y=s@
⑯
13 1
or
/
=
13
H "(303) &H
empty
non
both
.
0
=
so
"
(313)
because
there is
12250
thi is 1427 30
.
.
a
=
S
f
T
03
03
y S
=
I
fis constant
f(x)
and
y
=
id
gaxs
=
(1
,
=
<
hypothest
disconnection
counter examp ↳
X
space
IR2
->
.
q(0)
no
topological
arbitraring
Yas'
xCS'
(5)
20
0) FxEs'
.
Definition (homotopic envivalence)
fix->Y
continuous
fry
Sot
XandY
Intule case
YR"
and 4
X-
vid
=
f(x)
=
yC0)
I
eair
,
Ig
it
enviratance
:
x ->X continuous
yot idy
and
-
.
emio
x=X
0
0
=
of
-idy
foy
Prove
homotopic
are
homotopic
homotophe
are
homotopy
a
is
i
I
Y
903
called
map
where
Since
idy
:
H
that
yotix
IR"X
20
H(y 07
0
=
HCX
propreties
1)
,
=
0
17->IR"
.
,
with
0
tog(x)
she
:
=
H(x +)
=
=
.
+x
+
-
Definition (contentable)
X is called
i
.
e
we
example
saw
of not
s"
:
of this
furthermore , it you have
Lemma
IR" is
that
contractible
none
and
it
is
I
contractible
can
we
points
**
19
or s'
IR")t
,
so
(303
prove
is
generally Xis
,
gal
toycx)
I
:
1141)
we
,
or
a
sphere
disconnected
,
homotopic equivalent
Think
not
contractible
about S' and
.
ID" (200 013
.
u
IR
x1)
=
got (x) idgn
:
want
.
then it's
/
/
IIII
I
Y
No
:
contractible
not
-
=
.
.
S
f(x)
point
a
.
more
are
to
envivalent
homotopic
idy's homotopic to the map fex)
sot
⑧
just
if
contractable
↳
FxC/
:
:
since
11X11=1
boy I idRY303
on any
plyX
H
=
12
"
H(X 07
In" /903
17 ->
x 20 .
X
=
,
Ac 1
Hex 1
:
y
.
<1- +x++
=
,
t)
(1
-
all
>0 v
+
Definition Cretraction map)
A
topological
Alcalled
retract it
a
Alcalled
space
a
If
,
:
X->A
deformation retract
subset
Ac
it If
H
I
:
as
is called
a
Econ
and
->
.
(X 07
formation
above
=
,
,
1
=
-
C0 13
Xx
H2x
A
feas
continuous
for all
·
not
ridy
=
fex)
front
GA
i A -X
.
X
X
=
x
X and
home
it I f It
.
HCG 1)
.
as
=
A
ara
topic equivalent
above sot
0
for
all
EEC0 1
.