Exponential law for uniformly continous proper maps

Publicacions Ma emá iques, Vol 32 (1988), 123-127 . Abs ac EXPONENTIAL LAW FOR UNIFORMLY CONTINUOUS PROPER MAPS R . AYALA, E . DOMINGUEZ, A . QUINTERO The pu pose o his no e is o p o e he exponen ial law o uni o mly con inuous p ope maps . Le X be a egula space and Y a locally compac egula space . I is well known ha he spaces o con inuousmaps C(X x Y, Z) and C(X, C(Y Z)) a e homeomo phic conside ing he compac -open opology . This p ope y has impo an consequences in he s udy o he pa h-componen s o he unc ion spaces and in Homo opy Theo y : The exponen ial law o he uni o mly con- inuous p ope maps has simila consequences in some pa icula cases . All he spaces we conside , unless o he wise men ioned, a e me ic spaces . A p ope map will . b e a con inuous map : X -+ Y such ha o e e y compac subspace K o Y, -1(K) is á compac in X . To abb e ia e,- we will say ha is a p-map . A u-map is a uni o mly,cgn inuous map, and a up-map will be a uni o mly con inuous p-map . A up-isomo phism l a homeomo phism such ha and -1 a e u-maps ., By C(X, Y),  C p (X, Y) and C  p (X, Y) we will deno e he se s o con inuous maps, p-maps and up-maps be ween X and Y, espec i ely . Wi h Cú p (X, Y) we will ep esen he space o up-maps wi h he opology o uni o m con e gen e . In his no e we p o e ha he up-maps ollow he exponen ial law i X is compac ; ha is, he unc o s X x (-) and Cú p (X, -) a e adjoin . We also p o e ha i X is no compac hese unc o s a e no gene ally adjoin . A up-homo opy (p-homo opy) be ween up-maps (p-maps is a homo opy which is a up-map (p-map . Wi h [-, -], [-, -] P , and [-, -]  p we will ep- esen he se s o homo opy, p-homo opy and up-homo opy espec i ely . Also, he co esponding homo opy classes will be deno ed by [ ], [ ] p and [ ]up . R" will s and o he n-dimensional Euclidean space, and I o he uni in e al [0,1] wi h ha dis an e . The Euclidean no m will be ep esen ed by  - ~, and he dis an e o a me ic space by d(-, -) . This wo k has been suppo ed in pa by CAICYT g an 0812-84 12 4  R . AYALA, E . DOMINGUEZ, A . QUINTERO Theo em . Le X, Y, Z me ic spaces . We can de ine an injec i e map ID : Cup (X x Y, Z) -) Cup(X, Cü p (Y, Z)) as 1( ) (x) (y) = (x, y) . I X is compac , hen 1 is on o . P oo : I is easy o see ha : (a) Fo each x E X he map oD( )(x) is a up-map, because i is he composi- ion o wo up-maps . (b) oD ( ) is a u-map, because is a u-map . (c) Le us see ha 4>( ) is a p-map . Gi en a compac subspace K C Cú p (Y,Z), we only ha e o p o e ha any sequence {xn} in 1( ) -1 (K) has a clus e poin . Le {z,,} be a subsequence o {xj such ha {-D( ) (z n )} is con e gen ; le B E K be he limi poin . Then, o each yo E Y he sequence { (zn, yo)} con e ges o B(yo) . Consequen ly, H = { (zn, yo) ; n E N}U{e(yo)} is a compac subspace o Z . This implies ha he sequence {x,,} has a clus e poin . Hence -P is well de ined and i is an injec i e map . Since X is compac , each con inuous map de ined on X is also uni o mly con inuous . Then, gi en a con inuous map g : X -) Cú p (Y, Z) i is enough o show ha : X x Y-> Z de ined as (x, y) = g (x) (y) is a up-map . To p o e ha is a u-map, le ,, : Y -+ Z and , : X --> Z he maps de ined by = (y) = (x, y) = (x) o each couple (x, y) . Acco ding o [1, X .2 .1 .2] i su ices o show ha he se s H={ ., ;xEX} and C={ , ;yEY} a e uni o mly equicon inuous . Bu H = g (X) is a compac subse o Cú p (Y, Z), hence i is uni o mly equicon inuous by he heo em o Ascoli (see [1, .X .2 .5 .2]) . Since g is a up-map, i can be easily shown ha C is uni o mly equicon inuous se . I emains o show ha is a p-map . Le K be a compac subse o Z . I M = U{g(x)-'(K) ; xE X}, i is easy o check ha -1 (K) C X x M . I su ices o p o e ha M is a compac subse o Y . Gi en a sequence {y,,} in M, he e is a sequence {x,,} C X such ha g (x ,, ) (y,,) E K, o each n E N . Because X and K a e compac , we can assume ha {xn} and {g(xn)(yn)} con e ge o x o E X and zo E K espec i ely . Then, {g(xn)} con e ges o g(x o ), and i is ob ious ha o each e > 0 he e exis s n o such ha d(zo,g(xo)(yn)) < e i n >_ no . The e o e, K = {g(xo)(yn) ; n E N} U {zo} UNIFORMILY CONTINUOUS PROPER MAPS  12 5 is compac . Since g(x0) is p ope , g(xo) -1 (K) is also compac and {y,,} C g(xo) -1 (K) implies ha {y,,} has a clus e poin . We conclude ha M is compac . The ollowing s a emen is easily p o ed : Co olla y . Le X, Y, Z be me ic spaces . Mo eo e , suppose ha X is com- pac . Then, 1) <D : Cúp (X x, Y, Z) --> Cup (X, Cúp (Y, Z)) is up-isomo phism . 2) The se o pa h-componen s o Cú p (Y, Z) is in bijec i e co espondence wi h [Y, Z] up . 3) 1 induces a bijec i e co espondence 1 : [X x Y, Z]up -i [X, Cú p (Y, Z)] . Rema ks andExamples . I we conside only p ope maps, he na u al mapD : C p (X Z Y, Z) ---) C p (X, C p (Y, Z)) whe e C p (Y, Z) is endowed wi h he compac -open opology, is easily checked o be well de ined and injec i e . How- e e , se e al oubles appea : (1) The compac ness o X is necessa y in o de o p o e ha 1 is on o, e en i C p (Y, Z) is endowed wi h he uni o m con e gence opology . The ollowing example shows his ac : Le g : R -> C p (I, R) = C(I, R) be he map gi en by g(x) ( ) = x -(1- )x3 . I is easy o check he con inui y o g . In o de o p o e ha g is p ope , we ake a compac K C C(I,R) and a sequence {x  } C g -1 (K) . Then he e exis s a subsequence {z } o {x  }, such ha {g(zn)} con e ges o B E K . In pa icula , limg(z  )(1) = limz  = B(1) and we conclude ha g E Cp(R,C(I,R» . Bu he con inuous map (x, ) = g(x) ( ) is no p ope because «1' ) 21 , ) E -1 (0) o each E [0,1) . (2) Al hough X is compac , we canno ensu e ha 1D is on o i we conside he compac -open opology on C p (Y, Z) : Le g : I -)C p (R, R) gi en by g(0) (x) = go (x) = x and g( ) (x) = g (x) _ { ? x + 11  i 0 1/2 < x 1 / 2 _  (0 < < 1) . I is clea ha g E C p (R, R) o each E I . The con inui y o í ---> g ollows om he ac ha lim  = o in I implies ha {g } con e ges uni o mly on he compac subse s o Z o g o . Bu ( , x) -- g (x) is no a p ope map because ( ,1l ) E -1 (0) o each E (0,1] . (3) The p oo o he Theo em assu es ha 1 is on o i we conside he uni o m con e gence opology, u, on C p (Y, Z) and we assume he compac ness o X . Bu in such si ua ion, 1 :C p (X xY,Z) --+C(X,CP (Y, z» 12 6  R . AYALA, E . DOMINGUEZ, A . QUINTERO is no well de ined as shows he nex example : Le : I x R -> R 3 be he map ( , x) = ( , x, x) . This map is a p-map, and i 1( ) was con inuous and ,, -i o, gi en E > 0 he e would exis n o E N such ha i n > no x I I n - 0  1 :51 ( n , x, n x) - ( o' x, o x)  I < E o each x E R . Taking  x 1 la ge enough would yield he con adic ion E < x I I n  - o  I < E . The nex p oposi ion shows ha he e is no any possible duali y up- iso- mo phismwhen X is no compac . P oposi ion . Le Z be he open in e al (-1,1) . The e exis s no up-homo- opy equi alen e be ween C úp (R 2 , Z) and CúP (R, CúP (R, Z)) . We will need he ollowing lemma : Lemma .  The majo .1 : [Rn, Z]uP -> [Rn, Z] P gi en by A([ ]++p) = [ ] p is bijec i e . 1 P oo . I , g : Rn -> Z a e p-homo opic up-maps, he homo opy H : Rn x I -> Z gi en by H(x, ) = (x) + (1- )g(x) is a u-map . Now,we a e going o show ha H is p ope : Le K C Z be a compac subse and { n = (xn, ' ,» a seiluence in H -1 (K) . We may assume ha { n } con e ges o o . I we suppose ha { n }has no' clus e poin s we ha e lim x n = oo . Because and g a e p homo opic we ge lim (x n ) = limg(x n ) E {-1,1} . I his common limi s is 1 and U is an euclidean neighbou hood o 1 missing K, he e exis s n o such ha (xn),g(xn) E U o each n >_ n o . In pa icula , H(xn, n) E U  (n >_ no) con adic s he assump ion {(xn, n)} C H -1 (K) . So, A is an injec i e map . In o de o p o e ha A is on o we ecall ha [Rn, Z]p = [Sn -1 , S ° ], and i s elemen s a e he p-classes o he maps g_ 1 , g 1 : Rn -a Z gi en by g ; ( ) = ( j/(1+ 1 1)  (j = -1,1) i n > 2, o h( ), 1 h( ) 1, -h( ), - 1 h( ) i n = 1, whe e h( ) = (2/7 ) a c an( ) . This ollows, o ins an e, om he embedding heo ems o Edwa ds-Has ings, see [2, 6 .2 .7] . Now, a is on o because all he ep esen a i es a e up-maps . P oo o P oposi ion 4 : I su ices o p o e ha hose spaces ha e no he same numbe o pa h-componen e . As a consequence o Co olla y 2 .2) he pa h- componen s o Cú P (R 2 , Z) a e in bijec i e co espondence wi h [R 2 , Z]up . Bu [R 2 , Z]up - [R2, Z]p, by lemma 5, and he la e se has wo elemen s . Now,we a e going o show ha Cúp (R, Cúp (R, Z)) has a leas ou pa h- componen s . As aboye, Cúp (R, Z) has ou pa h-componen e, and hey a e he componen e o go (x) = (2/7 ) a c an(x),  91 = - 90, 92 =I g o I and g 3 = =- 1 go 1 .  Since Cúp (R, Z) is me izable and g o is a u-map,  o : R -~ Cú P (R, Z) de ined by o ( ) (x) = g o( + x) is a u-map .  Also, by using he UNIFORMILY CONTINUOUS PROPER MAPS  127 heo em o Ascoli i is easy o check ha o is a p-map . So, we ha e go a up-map o such ha o (R) lies in he pa h-componen o g o . In a simila way, we ge up-maps i wi h i (R) lying in he pa h-componen o gi (i = 1, 2, 3) . In pa icula , i and i a e no up-homo opic (0 <_ i =, 4 j < 3) . We conclude, applying co olla y 2 .2) again, ha he up-maps { ;}o< ;<3 de ine ou dis inc pa h-componen e . Rema k . I he me ic on Z is no bounded, lemma 5 is alse . Indeed, o each pai o eal numbe s a l , a 2 > 0, he up-maps l , 2 : R  ) R gi en by i ( ) = a  (i = 1, 2), a e p-homo opic, bu no up-homo opic : I H : R xI ---~ R is a up-homo opy be ween i and 2 , by [3, 111 .10] he e would exis e > 0 such ha H (x, ) - H (y, ') 1< max{e 1 (x, ) - (y, ') 1, e} o each couple (x, ), (y, ') E R x I . The e o e, la, -a2 1 x=1 i (x) - 2(x) j=j H(x, 0) - H(x,1) 1 :5E and aking x la ge enough he abo e inequali y would yield he con adic ion e<x1 al-a2 j< c . In ac , we ha e p o ed ha ca d [R, R]up > ca d R . Re e ences 1 .  N . BOURBAKI, in "Gene al Topology," He mann,1966 . 2 .  D . A . EDWARDS, H . M . HASTINGS, Cech and S een od Homo opy The- o y wi h applica ions o Geome ic Topology, Lec . No es 542 Sp inge (1976) . 3 .  J . R . ISBELL, Uni o m spaces, Ma h . Su eys, 12 AMS (1964) . R . Ayala : Dp o . d e Geome ía y Topología Facul ad de Ma emá icas 41012-Se illa, SPAIN . E . Dominguez : Dp o . d e Ma emá icas Facul ad de Ciencias Ciudad Uni e si a ia 50009-Za agoza, SPAIN . A . Quin e o : Dp o . d e Geome ía y Topología Facul ad de Ma emá icas 41012-Se illa, SPAIN . Rebu el 30 de Jung de 1987