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 .