4 T. A . CHAPMA N Tgfx), \ x . for x G C I x ( U 2 ) - U 1 g(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 .
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