MATH535 lecture 3838.1
Last time : . defined degf for ft (s ', S' ) and used it to prove that any complex
polynomial pay of positive degree has a root.
. Observed that for any 3 sets X , Y , Z there is a bijectionHornsea (Z ,
Y"
)→ Hourset ( Z xx , Y ) , (h : z → Y
" ) t th : ZxXii ) where I text -- th th) ta)Here Hom
sa (A , B) = set of functions from A to Be BA.
Note we have the evaluation map evy : YXx X - Y,er, If, a) e- f- Cx)
So Hornsea ( 7,4×1 - Monger ( ZXX, Y) is That -- Wx ( HH ,a )
or, equivalently , I = ego ( hxidx)
Recalls the compaet-op.ee topology on co (XM) is generated by the sets
M (Kil) =L f :X- Y I f- (Kl EUI ,where KEX is compact, U E Y open .
We want to show : suppose AZ are locally compact Hausdorff , Y Hausdorff .
Then t he co ( Z , Hill) , I = Wx o'
(hxidx ) E C'
( Z xx, Y ) and-
: (z ,co ( X, Yl) - co ( Z xx , Y ) is a homeomorphism .
First step :
Lemma38 Suppose X is LCH ( hoc . compact Hausdorff ) . Then ev : CYXIY) xx - Y,If, a) he fix)
is continuous.
Proofs Let Me Y be open .We want to show : W
- ' (UI is open in (X, Y ) xx.Let Hin ) E W
-
YU) .Then fail -- ev (fault U.
Since f is continuous, f- ' (UI is
an open nbd of n in X.Since X is locally compact , A compact nbd K of a
so that K e f- '
(U) . But then f CK) EU. This implies that f e M (Kil) .
tf g e Mlk , U) t y e- K, gas) = err ( g , y) c- U. ⇒ er ( Mlk, U) x K ) E U.
⇒ Neck , h) x K E WY U) .•
'o ei
'
f UI is open .
Hence ev : co (X,Y ) x Y - Y is continuous . D
Grokary3 Suppose X is LCH. Then
the ft , O ( X, Y ) ) , Tn 't exo (hxidx ) e- Co ( Zx Y ,Y )
.
38.2
Proofs Since hxidx : Zx X - C ' IX.Y) x X and evx : CoA ,4) x X e Y are continuous,so is
T : = eve, o (hxidxl. D
Remand Suppose k : 2- xx - Y is continuous.Then b- z e Z
, Ftz) : X - Y , ICH la) : - k Hix)is continuous. This is because I #I = ko iz where iz :X- 2-xx is the inclusion
iz ca) z CE ,r ) txt X . We thus have a map
-
: ( 2- xx , Y ) - Homsee. ft , COCKY ) ) , k 1- I
.
The next lemma shows that the image of-
lands in co ( Z, co Hill) E House. (Z , COCKY) ) .
↳mma38c3_ Let 11,4, 't be three spaces . Then the (ZXX , Y) the map
K : Z - co ( X, Y ) , z te Te Lz) = k Cz , . ) is continuous , hence we have
a map-
: co (Z xx , Y ) - of -2 , CTX , Y) ) .
Proof Enough to show : t KEX compact ,they open , k
- '
(Mlk , U)) is open in -2.
Let t t k-
I MLK , U )) . Then kfhzsxk ) = ICH ( K) EU.⇒ aah x k e k
- '
(U) .Since his continuous
,k ' (UI is open, in 2-xx
. By Tube Lemma, 7 open nbdW of t sit Wx K E k
- ' ( U) . ⇒ t w c- IN U2 kdcwsxk) - ( Flw)) 1kt .⇒ a wt W KIWI c- Mlk , U ) .⇒ W E (Te)
"
( MLK , U )) . ⇒ I ' (Mlk ,U) ) is open in Z . D
Recaps If X is LCH,Y,Z are spaces we have maps
① -
i. Co ( Z ,
C-
(X, Y l) - co (z xx, Y ) , h teh,tile, x) = ditz ) ) ( x ) thx) EZXX
② - i ( 2- XX, Y ) - Colt , 01441) , kn I, KH - k tail ,htt EZ
.
It's easy to check that the two maps are inverses of each other.
He'd like to prove :
theorems Suppose X , Z are LCH, Y Hausdorff. Then the bijectionco ( Zxx, Y ) → Colt , (X, Y ) )
is a homeomorphism .
To prove 38.4 we need :
38.3
Lemma38 Let P be an LCH space , Q a space , B a sub basis for a topology on Q ,K -
- E KE P compact ) K is a nbd of some at PS .Then
I - l M IK ,U ) I KEK , U C- BS
is a sub basis for the compact - open topology on CYP,Q )
.
Proof Enough to show : tf k e P compact, t U '- Q open ,t f E M ( K ,U) , 7 Ui . . - Un EB
Ki , . - kn E K sit f E,§,
M (Ki , Ui) E M (K ,U) .
For each seek I Use c- B with text Uae EU and a compact nbd Ka of se with Ka e- f''
fun ) .( Note that f- E M (ka
,Ua) ta C- KI
.
Since K is compact 3 a , - - an EP sit . KE Ka,u - - o Kaen . ⇒ f t M (Kai , Uni ) .
Since age II,
M (Kai, Uni ) , g (K) e- GCU Kai ) -- U g ( Kai) EU Uai E U,
in,
M (Kai , Uno ) E Mlk, U) . D
proofofss.ly Consider
K = LK , x Kat 2- XX / K , is a compact- nbd of some # c- Z,Ke is a compact nbd of some at Xl
.
By 38.5, I M I Kixkz , U ) I Kixkz C-K,he 4in open4 is a subbasis of the compact - open
topology on co ( 2- xx, Y ) .
By 38.5 , l M ( K , , M (Ka, U)) I klxkz C-K,he th opens is a subbasis for the compact
open topology on co (Z , COCKY ) ) .Now he Mlk , xka , U)⇒ k CK, xkz) EU ⇐s CI ( ki )) (ka ) EU ⇐
ti (ki) E M (K2 ,U) ⇐ I E M (Ki , M ( Kz , U ) ) .
Hence the bijection- t.co/ZxX, Y ) - C ° (Z , co (X, Y ) ) takes a sub basis of the
topology to a sub basis.⇒-
is a homeomorphism . D
Compact -open topology and pullbacks ( restrictions .
Lemma3# Let 4 : X'- X be a continuous map and Y a space . t f e (X, Y)
,
44 Cfl : -- foul E (X', Y ) . The map44 : co (X,Y ) - (X' , Y ) is continuous (w.at . the compact - open topologies)
38.4
Proof f k EX' compact and HUEY open ,
④*5' ( Mlk ,U) ) - If e- COCKY) I 44ft e- Mlk , uh = If I Hot ) (K) e US
= If I f ( 41kt) E UG - M ( 4 (Kl , U) .
( Note that since 4 is continuous and K is compact , 4( K ) is compact ) .
⇒ Yt is continuous . D
Note that if X' EX is a subspace , then the map COCKY)→ cold, Y ), f 1-flyis continuous since the inclusion map i :X'as X is continuous and f 1×1 -
- fo i .