Let C be a ring of (not necessarily bounded) real-valued functions with a common domain X such that C includes all the constant functions and if f   and lt; C then | f | ε C

Abian, Alexander

Salbany, Sergio

Spanish

Universidad Nacional De Colombia - Repositorio Institucional UN

Rev~~ta Cotomb~ana de Matemat~ea~Voto XIII (1979) pag~o 39 - 48RINGS OF REAL-VALUED FUNCTIONS AND THEFINITE SUBCOVERING PROPERTY*byAlexander ABIAN and Sergio SALBANYAbstract. Let C be a ring of (not necessari-ly bounded) real-valued functions with a commondomain X such that C includes all the constantfunctions and if f (;.C then I fie C. Without re-sorting to any topological notions and using onlyalgebraic techniques, we prove that X can beextended to a set X' and every f £: C can be extended to a function f' on X', such that the resuIting set C' of the extended functions is aring isomorphic to C, and such that if E' isany bounding s~bset of C' with the property thatfor every zeX' there exists an f'eE' with...n The first author acknowledges an F.L.I. grant fromIowa State University. The second author acknowledg-es a grant from C.S.I.R., South Africa.39f I (z) ¢ 0 , t h'en -i:h 'ere exis t s af init e sub set 0 fE' with the same property (E' is bounding if forevery fie G' there exist a constant functionc 'c G' and e' E: E' such that c ' ~ If'lte')"When X is a completely regular Hausdorff spaceand C its ring of continuoos real-valued functions(or - bounded continuous real-valued functions )then X' is the Hewitt realcompactification ( or-toe Stone-eech compactification~ since in thiscase the requirement that E' must be bounding canbe dropped) of Xo In effect~ our result shows thatthe Hewitt real-compactification (or -the Stone-~Cech compactification) theorem can be formulatedpurely algebraically requiring neither the conti-nuity of functions nor a topology on XoIn what follows all the functions are real~va-lued. Also~ all the ring-theoretical statementswhich are made in connection with a set F offunctions (with a common domain) refer to the poi~twise addition and multiplication of the elementsof r; similarly~ statements pertaining to orderamong the elements of F refer to the pointwisecomparisono Moreover~ if f is a function thenwe let If I Istand for the function whose valuesare the absolute values of fo Let c be a func-tion whose domain is Xo Then~ as expected, c isca~d a eon~tant function on X if and only ifcIx ) := r for every x E: X, In particular, if r :::1then c is called the unit 6unetion on Xo40Let E be a set of functions with a common domain X and let u be the unit function on X.We say that u i¢ cove~ed by the element¢ 06 Eif and only if for every z E:: X we have:(1) fez} t- 0 for some f c E.·lemma 1.Let F be a ¢ub¢et 06 a !Ling C On nunc-tion¢ with a common domain X and let u E: C wheJLe ui¢ the unit nunction on X. In u i~ not covelLed bythe element¢ 06 F then the ideal J On C geneJLatedby F i¢ pJLOpelL.Proof. Since u is not covered by the elementsof F, from (1) it follows that for some z E:. X itis the case that fez) - 0 for every f E: F. Butthen if g E: J we see that g(z) = o . Hence~ u ¢: Jand J is a proper ideal. IIFrom the Lemma we have immediately:Corollary. Let c be a lLing a¢ mentioned in Le-mma 1 and E be a ~ub¢et On c • In u i¢ covelLed byno 6inite numbett..On element¢ On E then the idealT On c genett..atedby E i~ plLopelL.Let C be a set of functions with a common do-main X. A set E of functions on X is calledbounding set (or a bounding subset in case E<; C)of C if and only if for every f€ C there exists41a constant function c con X such that(2 ) c ~ If 1 + e for some e E: E 0Lemma 2, Le~ e be a ~~ng 06 6une~1on~ w1~h aeommon domain X ~ueh ~ha~ eve~y eon~~an~ nune~~onon X i~ an elemen~ 06 e and 16 f€e ~hen Ifl~ CoLe~ M be bo~h a maximal ideal and bounding 4ub4e~06 e 0 Then M i4 a ~eat ideal 06 e (i.e" e!M 1~i~omo~ph~e to ~he ~eal~).P roo f 0 Let us observe that e is a latticewhere fv g and ft. g are equal, respectively~to ~(f+g + If-gl ) and 1.2(f+g - If-gl ). Thus~by [1J , po66, the maximal ideal M is an absolu-\.tely convex ideal of e and e!M is a totally or-dered field, However, M is bounding subset ofe and therefore for every f E:. e, there existsa constant c such that, in view of (2) andusing an obvious notation, we have c+M ~ Ifl+M.Hence, e!M is also Archimedeano Moreover, sinceevery ~onstant function is an element of e wesee that e!M has a subfield isomorphic to thereals. But then, as such, e!M itself is iso-morphic to the reals and M is a real ideal of e,as desired .•Based on the above, we have:Theorem. Le~ e be a ~ing 06 (~eal-valued) 6une\42tion~ with a common domain X ~ueh that eve~y eon~tant 6unction on X i6 an element 06 C and i6 f~ Cthen Ifl£ C and whe~e u i~ the ounction on Xo Then.X can be extended to a ~et x'and eve~y element f00 C can be extended to a 6unetion f'with x'a~ it~domain ~uch that:(i) .the ~e~ulting ~et eroo the extended 6unction~i~ a ~ingo(ii) the co~~e~pondenee f ~ f'i~ a ~ing i~omo~phi~m6~om C 0nto c' c(iii) i6 uti~ eove~ed by the element~ 06 a bounding~ub~et Er06 c'then u' i~ al~eady eove~ed by 6inite-ly many element~ 06 E' 0~roofo Since C contains all the constant functions on X, then the set M given byxM ={flfE:Cxand f(x) =0 }is a real ideal of C for every x E: X. Let{ MYY E: Y }be the set of all the real ideals not of the form(3), and consider( 5 ) f'X = XUy43Clearly, x' is an extension of 'X, and(6 ) { M zis the set of all the real ideals of C.To every fcC let us make correspond a func-tion f" on X defined as:( 7 ) f'::: f on and f'( y) := f(Mod M ) ony Y.Obviously, r' is an extension of f , Moreover,My = {f If€:: C and fl( y) :: O} for every y E: Y,which by (3) implies(8 ) fCC and f~z) = O} for everyAgain, from definition (7) it readily follows that:f' + s' = (f+g)i and f' s' = (fg)land sin c e Cis a r in g , we see t hat C' g i ven by ~( 10) fCC}is also a ringo Hence (i) is established.Clearly, if f'::; gt then f:;; g which by (9)implies that the correspondence f~~· is a ring44isomorphism from C onto C~ Hence (ii) is alsoestablishedoNext, let the unit function ul be covered bythe elements of a bounding set E' of CI, i. e.,as in (1), for every xE:.X we have:for some f'e E'.To prove (iii) we must show that ul is already covered by some finitely many elements of E~Let us assume to the contrary thatby no finite number of elements ofthe Corollary, the ideal II of C'E' is p r op er-, Also, since Ef is aIiset of C,u' is coveredE'" Thus, bygenerated bybounding sub-(12 ) Et,; Ii' and I~ is a bounding subset of C~Since ut' is the extension of u and f t-+ f'is a ring isomorphism, in view of (12), we see thatthe subset I of C defined by:(13 )IfiE:.II}is both 'a proper ideal and a bounding subset of C .As a proper ideal, I is contained in a maximalideal M of C . But then, since I is a boundingsubset of C we see that M is both a maximalideal a d a bounding subset of C. Consequently,45from Lemma 2 it follows that M is a real idealof C and~ in view of (6)~ we have:(14) I£ M = Mz for some Z E: Xi.Now , from (12), (13), (14), (8) it follows thatf'( z ) :: 0 for everywhich contradicts (11)0 Hence our assumption isfalse and (iii) is establishedo -Remark. We observe that X i(as well as X )can be topologized with subbasic open sets of theform {x f Ix ) ¢ oj for some fE: C (as well asfor some f'E:C' )~which in fact form a base. Itcan be readily verified tha with respect to thisop ogy all the elements of C (as ell as of C~become continuous functions (where reals are topo-logized as usual)o If the elements of C separatepoints and closed subset of X then X becomescompl~tely regular Hausdorff and X' becomes real-compact (since every real ideal in C' is fixed).Moreover, if C is the ring of all continuousfunctions on X then X is C-embedded in X'andIhence Xi is the Hewitt realcompactification of XoIf in the above, C was the ring of all bound.'.ed continuos functions on X then X is C"-em-bedded in x' and hence X' o v1S the Stone-Cech com-46pac~fication of X {since in this case the requi-rement that E' must be a bounding subset of Cican be dropped)oOn the other hand, if X had a completely re-gular topology to sta~with~ this topology coin-cides with the one defined above for any ring Cof continuos functions separating points and clo~sed subsets. Therefore~ the above Remark showsthat the topological structures involved in theHewitt realcompactification or the Stone-eec, compactification can be fully recovered from theunderlying algebraic structureso***BIBLIOGRAPHY•[1] Gillman, La and Jerison, M., "Rings of Conti-nuous Functions~ Princeton, Van Nostrand,(1960) 0Vepa~tment 06 Mathemat~c~Iowa State Un~ve~~~tyAm e s , Iowa 5 0 0 1 1U. S. A.Mathemat~caf In~t~tuteUn~ve~~~ty 00 Ox6o~d24-29 St. G~fe~Oxoo~d, ENGLANV47The authors thank the referee for some impro-vements in the texto(Recibido en septiembre de 1978)48

Rings of real-valued functions and the finite subcovering property

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Rings of real-valued functions and the finite subcovering property

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