Conditions of angelic type in function spaces

Pub . Ma . UAB Vol . 30 Ns 1 Maig 1986 CONDITIONS OF ANGELIC TYPE IN FUNCTION SPACES by Miguel Ba celó  and Miguel A . Canela ABSTRACT : This pape deals wi h a class o opological spaces in which a-compac ness and compac ness coincide and he igh ness c a compac subse is less o equal han a, a being an in ini e ca dinal numbe . This class is a na u al ex ension o he class o s ic ly angelic spa- ces, in oduced by W . Go ae s . Su icien condi ions a e gi en o a space o con inuous unc ions o belong o his class, and some esul s on locally con exspaces a e cb ained as an applica ion . 1 . In oduc ion - The aim o his pape is o gi e a desc ip ion u a class o opolo- gical spaces in which a-compac ness and compac ness coincide, a being an in ini e ca dinal numbe , which will emain ixed h oughou his pape . This class is a-p oduc i e (closed by p oduc s o amilies o ca d .i iali- y <a) . The class desc ibed will be deno ed by Aa , whe e he "A" s ands o , angelic . Indeed, o a = a 0 , ou class coincides wi h he class o s ic ly angelic spaces in oduced by W . Go ae s [2] . These spaces a e a subclass o he angelic spaces s udied by P yce in [5] . The p ope ies o he class o angelic spaces can be ound in In.Sec ion 2 we ecall some opological no ions which will be used h oughou his wo k . In Sec ion 3 we in oduce he class A n , p o ing some 'p ope ies . The main esul is he s abili y o p oduc s o ca di nali y < a, and he key o his esul is a heo em c V .I . Malyhin [41 . In Sec ion 4 we gi e su icien condi ions o a space o con i- nuous unc ions o belong o A , , endowed wi h he opology c poin wise con e gence . In Sec ion 5 we es ic ou sel es o locally con ex spa- ces, ob aining wi h he ools gi en in he p e iousSec ions some e- sul s appea ed in a o me pape by M . Valdi ia [6] . 2 . Some opological - no ions .- a will deno e a ixed in ini eca dinal numbe . A subse S o a opological space X (all he spaces in ol ed a e Hausdo ) is a- compac e e y ne (xcon ained in S, wi h  III < a has a clus e , poin x E S .  S  is  ela i ely  a- compac  i e e y  ne  (x i ) iEI'  wi h  III <a  has a clus e  poin  xEX .  E e y  ( ela i ely)  compac  subse  is  ( ela i ely) a-compac , bu no con e sely . A coun e example can be ob ained modi- ying he usual example o a sequen ially compac space which is no compac [1, 1 .2(7)] . i I X is a opological space, he igh ness o X, (X),is he mini- mal ca dinal numbe wi h he ollowing p ope y : i S is a subse o X and xE cl(S)  is a closu e poin o S,  he e is a subse  MC S  wi h 1M1 < and  x E cl (M) . . The  densi y  cha ac e  o  X,  d (X), is  he  minimal ca dinali y o a . dense subse o X . The weigh o X, w(X), is he mini- mal ca dinali y o a basis o open subse s o he opology o X . The weigh  o  X  a  he  poin  x E X,  w(X),  is  he minimal  ca dinali y o a x  > basis  o neighbou hoods o x . Clea ly wx (X) < w(X),  bu no con e sely . - Fo he e and c he ca dinal unc ions o he Gene al Topology, [3) can be used as a s anda d e e ence . 3 . The class A a .- The class A a will be he class o all opological spaces X sa is- ying he ollowing condi ions : (i) E e y ela i ely a-compac o Xis ela i ely compac . (ii) E e y compac subse c X has igh ness < a . (iii) I a subse SCX is ccmpac and d(S)< a, hen w x (S) < a o e e y xES . We gi e nex he p ope ies o his class . 3 .1 . P oposi ion : The condi ion (ii) o he p eceding de ini ion can be eplaced by : (ii)' I SCX is ela i ely compac and xEcl(S), he e isane (xi)iEI con ained in S, wi h  III < a and lim x i = x, ob aining an equi alen de ini ion . P oo :  (ii) '  implies  (ii) .  Con e sely,  i (S) < a  o a compac subse SC X,  we  ake  MC S  wi h  x E cl (M)  and  1 M1 < a .  Then cl (M)  is  compac  and has densi y cha ac e < a, and, by (iii) he weigh a x is < a . The e o- I X is a opological space, he igh ness o X, (X),is he mini- mal ca dinal numbe wi h he ollowing p ope y : i S is a subse o X and  xE cl (S)  is  a  closu e  poin  o  S,  he e  is  a  subse  MC S  wi h ¡MI < and  x E cl(M) .  The  densi y  cha ac e  o  X,  d(X), is  he  minimal ca dinali y o a dense subse o X . The weigh o X, w(X), is he mini- mal ca dinali y o a basis o open subse s o he opology o X . The weigh o X 2. he poin xEX, wx(X), is he minimal ca dinali y o a basis  o neighbou hoods o x .  Clea ly wx(X) < w(X),  bu no con e sely . - Fo he e end c he ca dinal unc ions o he Gene al Topology, [3] can be used as a s anda d e e ence . 3 . The class A a .- The class A a will be he class o all opological spaces X sa is- ying he ollowing condi ions : (i) E e y ela i ely a-compac o X is ela i ely compac . (ii) E e y compac subse c X has igh ness < a . (iii)  I  a  subse S C X  is  compac and d(S)< a,  hen w x (S)< a o  e e y XES . We gi e nex he p ope ies o his class . 3 .1 . P oposi ion : The condi ion (ii) o he p eceding de ini ion can be eplaced by : (ii)' I SCX is ela i ely compac and xEcl(S), he e isane (xi)ic-I con ained in S, wi h  III <a and . lim x i = x, ob aining an equi alen de ini ion . P oo :  (ii) ' implies  (ii) .  Con e sely,  i (S) < a  o a compac subse SC X,  we  ake  MC S  wi h  x E cl (M)  and  1 M1 < a .  Then  cl (M)  is  compac  and has densi y cha ac e < a, and, by (iii) he weigh a xis <a . The e o- 52 We ha e (i) . Suppose  ha  S C U X j is  compac ,  and  we  a e going o show ha ¡EI (S)< a . We can suppose, wi hou loss o gene ali y, S =  ni(S) . ¡El Acco ding o a esul o V .I . Malyhin [4, Theo em 4), he ini e p oduc o compac spaces wi h igh ness <a has igh ness < a . I x is a closu e  poin  o a subse D C S,  we can  ind,  o each  J C 1  ini e,  a subse MJC J (D) wi h 1MJ1 < a and iT J (x)E cl(MJ) . We can choose now M J CD wi h nj(MJ ) = MJ and x E cl(M) . We ha e (ii) . Finally,  i  S C U X 1 .  is  co pac  and d (S ) < a ,  hen  d (i i (S) ) <a,  and ¡El - hus n (S) has weigh < a in e e y poin . Keeping in mind he cons uc- i  - ion o a basis o neighbou hoods o he p oduc opology, and e- calling III< a, i is easy o see ha 11 n 1 .(S) has weigh < a a e e- iE I y poin . Q .E.D . and 1 M J j<a . Ac ually, 4 . Spaces o con inuous unc ions .- M- U M J J has ca dinali y <a We a e going o see ha ce ain a .ssump ions on a opological space X imply ha he space C(X) = C(X,1R) o con inuous eal unc ions be- longs o he class A a , endowed wi h he opology o poin wisecon e gen ce W X . Indeed, we ob ain esul s analogous o hose ob ained by W . Go- ae s [c]  o he case a =¡{ 0 . 4 .1 . P oposi ion :. Le X be a compac space . Then (C(X),w x ) belongs o Aa . I is well-known ha e e y subse S C C(X), wX ela i ely coun a- compac , is wX - ela i ely compac and coun able igh néss (e .g .[1]) . we can, eplacing i necessa y X by a sui able quo ien , P oo : bly To check (iii), suppose ha S sepa a es he poin s o X . I S i wX -compac and DCS is dense,  wi h  ID 1 < a,  X  admi s a basis o uni o mi yo ca dinali y < a , and hence d(X) < a . Re e sing he a gumen , we ha e a se o con inuous eal unc ions, o ca dinali y <a , on S, which sepa a es he po .in s o S, and he e o e w(S) <el . Q .E .U . 4 .2 .  P oposi ion :  Le X be a opological space wi h a dense subse D C X which is ela i ely a-compac . Then (C(X),w x ) belongs o Aa . P oo : Condi ions (i) and (ii) can be ob ained as in 4 .1 om known e- sul s (see [1]) . To check (iii), we ema k ha , i S C C(X) is w X -co n- pac , we can conside . he mapping 01 : X - C(S) de ined by 0 1 (x)( ) _ = (x),  which is con inuous wi h espec o he opology w S .  Then 0 1 (D) is ela i ely a-compac in (C(S),w S ), and i s closu e FI is a compac subse o (C(S),w S ) . Now m2 : S 1 C(H), de ined in an analogous way, is injec i e and con inuous, and he e o e a homeomo phism, and, using 4 .1, we a e done . Q .E .D . 4 .3 . P oposi ion : Le X,Y be opological spaces . a)  I X  admi s  a  dense  subse D C X such ha  (C(D),w D )  is  in A,,,  hen (C(X),W X ) is in A a . b)  I X  U  X i ,  whe e  (C(Xi),wx .)  is  in  Aa and  111< a,  hen  (C (X),wX) iEI  1 is in Aa c) I 4) : Y - " X is con inuous and su jec i e, and (C(Y),w y ) is in A a , hen (C(X),w x ) is in Al . P oo : Fo a) Take he es ic ion map C(X) - C(D) and epply 3 .3 . Fo b), cons uc an injec ion C(X) - HC(X .) in a na u al way and apply iEI 1 3 .4, ollowed by 3 .3 . Fo c), ake he mapping 0* : C(X) - C(Y) de ined by ~*( ) = o~ and use 3 .3 . Q .E .D . 4 .4 . Co olla y : Le X be a opological space wi h a amily (X i ) iEI o ela i ely a -compac subse s  such  ha  i s  union  is  dense  and  ~11< a . Then (C(X),W X ) is in Aa . Finally, we ha e he ollowing esul , which is a na u al ex ension o a heo em o D .H . F emlin ([1,3 .5], [2, P oposi ion 9]) : 4 .5 . P oposi ion : Le X be a opological space, and Z a me ic space, and suppose ha (C(X),w x ) is in á . Then (C(X,Z),wx) is in A a . P oo : The a gumen gi en in [2) o he coun able case can be used . Q .E .U . 5 . Applica ions o locally con ex spaces .- The esul s which ha e been s a ed he e ha e a pu ely opological na u e . Ne e heless, some pa icula cases ha e been p o ed by o he me hods, o ins an e o he case o a weak opology on a locally con- ex case . I X,Y a e eal locally con ex spaces, he space L(X,Y) o con i- nuous linea ope a o s is a subspace o he space C(X,Y) o con inuous unc ions,  closed wi h espec o wX .  The  opology induced by  w X  on L(X,Y) is usually called simple opology . F om 4 .5 and ° .2, we ob ain di ec ly : 5 .1 . P oposi ion : Le X,Y be locally con ex spaces, X being he union o a amily (Di)iEI o ela i ely a-compac subse s, wi h 111< a, and Y me izáble .  Then L(X,Y), endowed . wi h he simple opology, is in Al . We can conside , in pa icula , he case in which Y =IR and X is he dual E', endowed wi h he weak opology a(E',E) . We ob ain hus he ollowing esul : 5 .2 . Co olla y (M . Valdi ia [6]) : Le E be a locally con ex space,  E' being he union o a amily (D i ) ¡ EI o a(E',E)- ela i ely a-compac subse s, wi h I II <a . Then e e y weakly ( ela i ely) a-compac subse o E is weak- ly ( ela i ely) compac . NOTE : In [6], his esul is s a ed unde an ex a assump ion, he con- exi y o he D1 .'s, bu his assump ion is supe luous . Using 5 .1 and 3 .3, we ha e : 5 .3 . Co olla y (M . Valdi ia (6Í) : Le E be a ec o space and T and T' wo locally con ex opologies on E, T ine han T' . Suppose ha T' admi s a ze o-neighbou hood basis o ca dinali y < a , and deno e by E' _ _ (E,T)'.I A C Eis a(E,E')-( ela i ely)a-compac , A is a(E,E')-( ela i- ely) compac . FINAL NOTE : We ha e desc ibed a class o spaces in which compac ness and a-compac ness a e he same, wi h good s abili y p ope ies, which allows us o ob ain he esul s o his Sec ion . I can be ema ked ha almos all is he same i we eplace condi ion (iii) o he de ini ion by he ollowing  s onge condi ion :  I S CX  is compac and  d(S)<a,  hen W(S) ~ a .  Ne e heless,  he  class  ob ained  in  his  way  will .b e mo e es ic ed, because he e a e sepa able, i s coun able compac spaces which a e no me izable (e .g . he Helly compac ) . REFERENCES 1 . K . Flo e : Weakly compac se s . Sp inge , Be lin-Heidelbe g-New Yo k, 1980 . 2 .  W . Go ae s : A p oduc i e class o angelic spaces . J . London Ma h . Soc . 22 (1980), 355-364 . 3 .  I . Juhasz : Ca dinal unc ions in Topology . Ma h . Cen e T ac s 34, Ams e dam,1971 . 4 . V .I . Malyhin : The igh ness and Suslin numbe in exp X and in a p oduc o spaces . So ie Ma h . Dokl . 13 (1972), 496=499 . 5 . J .D . P yce : A de ice o R .J . Whi ley's applied o poin wise com- pac ness in spaces o con inous unc ions . P oc . London Ma h . Soc . (3) 23 (1971), 532-546 . 6 . M . Valdi ia : Some c i e ia o weak compac ness . J . eine angew . Ma h . 225 (1972), 165-169 . Rebu el 15 d'ac ubne del 1985 MiguelBa celó EscolaTécnica Supe io d'Enginye s Indus ialsde Ba celona . A inguda Diagonal Ba celona SPAIN . MiguelA . Canela Depa amen de Teo ia de .Funcions, Uni e si a d Uni e si a de Ba celona G an Via, 585 08007 Ba celona SPAIN .