4
T. A . CHAPMA N
Tgfx),
\ x .
for x G C I
x
( U
2
) - U
1
g(x) = x , fo r x U ~ U
2
( J f x ) , fo r x G U
1
Certainl y g ca n be mad e smal l b y choosin g f small .
4 . Prope r Maps . A ma p f: X -• Y is proper provide d tha t f~'(C ) is compact , fo r eac h compac t
C C Y . Suc h notion s as prope r homotopy , prope r homotop y equivalence , etc. , ar e define d in
analog y wit h the correspondin g notion s fro m th e ordinar y homotop y category . A s a n exampl e
observ e tha t th e space s [0,1 ) an d (0,1 ) hav e the sam e homotop y typ e bu t not th e sam e prope r
homotop y type . (Th e reade r shoul d prov e this. ) Th e followin g basic fact s abou t prope r map s
wil l be useful . Th e reade r is agai n reminde d tha t al l space s ar e locall y compact .
4.1 . Theorem . (1) If f: X -• Y is proper, then f(X ) is closed in Y . (2) For each Y there
exists an open cover U of Y such that for any X and maps f,g : X -* Y which are U-closet
f is proper iff g is proper.
Proof . Fo r (1) it suffice s to sho w tha t f(X ) n C is compact , fo r ever y compactu m C C Y .
Certainl y f(X ) n C is close d in f(X ) an d f(X ) n C is in th e compac t subse t ff"
1
(C ) o f f(X) .
Thu s f ( X ) O C is compact .
oo
Fo r (2) writ e Y = U - C
n
, wher e th e C
n
3re compac t an d C
n
C In t (C
n +
-j). Le t
U ^ I n t f C j ) ,
U
2
= lnt(C
3
) -C^ ,
U
3
= lnt(C
4
) - C
2
,
oo
The n U-{Un} - is a n ope n cove r o f Y whic h is easil y see n to fulfil l ou r requirements .
5. A Convergenc e Criterion . In man y instance s in th e seque l we wil l nee d to kno w whe n th e
infinit e compositio n o f a sequenc e o f homeomorphism s give s us a homeomorphism . Fo r suc h
purpose s th e followin g resul t wil l be useful .
Previous Page Next Page