summary:The ring $B(R)$ of all real-valued measurable functions, carrying the pointwise convergence, is a sequential ring completion of the subring $C(R)$ of all continuous functions and, similarly, the ring $\mathbb{B}$ of all Borel measurable subsets of $R$ is a sequential ring completion of the subring $\mathbb{B}_0$ of all finite unions of half-open intervals; the two completions are not categorical. We study $\mathcal L_0^*$-rings of maps and develop a completion theory covering the two examples. In particular, the $\sigma $-fields of sets form an epireflective subcategory of the category of fields of sets and, for each field of sets $\mathbb{A}$, the generated $\sigma $-field $\sigma (\mathbb{A})$ yields its epireflection. Via zero-rings the theory can be applied to completions of special commutative $\mathcal L_0^*$-groups

H. Herrlich

J. Novák

Ján Borsík

L. Paulík

M. Laczkovich

R. Frič

Roman Frič

summary:We define various ring sequential convergences on  $\mathbb{Z}$ and $\mathbb{Q}$. We describe their properties and properties of their convergence completions. In particular, we define a convergence $\mathbb{L}_1$ on  $\mathbb{Z}$ by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields  $\mathbb{Z}/(p)$. Further, we show that $(\mathbb{Z}, \mathbb{L}^\ast _1)$ is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D.  Dikranjan

Marcos, J. E.

Institute of Mathematics AS CR, v. v. i.

A completion of $\mathbb{Z}$ is a field

summary:It is known that the ring $B(\mathbb R)$ of all Baire functions carrying the pointwise convergence yields a sequential completion of the ring $C(\mathbb R)$ of all continuous functions. We investigate various sequential convergences related to the pointwise convergence and the process of completion of $C(\mathbb R)$. In particular, we prove that the pointwise convergence fails to be strict and prove the existence of the categorical ring completion of $C(\mathbb R)$ which differs from $B(\mathbb R)$

Borsík, Ján

Frič, Roman

Czech digital mathematics library

Pointwise convergence fails to be strict

Czechoslovak Mathematical JournalRoman FričRings of maps: sequential convergence and completionCzechoslovak Mathematical Journal, Vol. 49 (1999), No. 1, 111–118Persistent URL: http://dml.cz/dmlcz/127471Terms of use:© Institute of Mathematics AS CR, 1999Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documentsstrictly for personal use. Each copy of any part of this document must contain these Terms of use.This document has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://dml.czCzechoslovak Mathematical Journal, 49 (124) (1999), 111–118RINGS OF MAPS: SEQUENTIAL CONVERGENCEAND COMPLETIONRoman Frič, Košice(Received June 10, 1996)Abstract. The ring B(R) of all real-valued measurable functions, carrying the pointwiseconvergence, is a sequential ring completion of the subring C(R) of all continuous functionsand, similarly, the ring   of all Borel measurable subsets of R is a sequential ring completionof the subring  0 of all finite unions of half-open intervals; the two completions are notcategorical. We study L ∗0 -rings of maps and develop a completion theory covering the twoexamples. In particular, the σ-fields of sets form an epireflective subcategory of the categoryof fields of sets and, for each field of sets  , the generated σ-field σ( ) yields its epireflection.Via zero-rings the theory can be applied to completions of special commutative L ∗0 -groups.Keywords: Rings of sets, completion of sequential convergence rings, Z(2)-generation,Z(2)-completion, σ-rings of maps, epireflection, fields of events, foundation of probabilityMSC 2000 : Primary 54A20, 54B30; Secondary 54H13, 60A990. IntroductionWe continue our investigations of L ∗0 -rings of functions started in [BKF]. Recallthat an L ∗0 -ring is a ring (even though C(R) and   0 possess the unit element, ingeneral we do not assume its existence) carrying a sequential convergence havingunique limits and satisfying the Urysohn axiom and such that if sequences 〈xn〉and 〈yn〉 converge then their difference 〈xn − yn〉 and product 〈xnyn〉 converge tothe difference and product of their limits, respectively. Background informationon the completion of L ∗0 -rings can be found in [FKO] and [FKT]. Notice that thefield Q of rational numbers can carry an L ∗0 -field convergence having no completion([FZE]) and the usual convergence on Q admits exp expω nonhomeomorphic L ∗0 -field completions ([FCB]).Written during the author’s stay at Ehime University with the support from the JapanSociety for the Promotion of Science; partially supported by Grant GA SAV 1230/96111In [BKF] it was shown that the categoricalL ∗0 -ring completion of C(R) exists andis different from B(R). For the undefined categorical notions the reader can consult[HES].In the present paper we investigate completions of L ∗0 -rings of maps into a com-plete L ∗0 -ring. Section 1 is devoted to the L∗0 -ring   0 of half-open intervals, i.e.,the field of sets consisting of all finite unions of intervals of the form [a, b), (−∞, b),(−∞,+∞), [a,+∞), where a, b ∈ R, carrying the usual sequential convergence. InSection 2 we describe the categorical background of the completions of C(R) and   0 .In Section 3 we present two additional results: on rings of sets and the applicationof the completion of L ∗0 -rings to commutative L∗0 -groups via zero-rings.1. The L ∗0 -ring   0It is known that every ring of sets can be visualized as a ring of characteristic func-tions. Each set is identified with its characteristic function, the symmetric differenceas addition becomes the pointwise addition of characteristic functions modulo 2, theintersection as product becomes the usual pointwise product, and the usual conver-gence of sets (A = limAn means A = lim supAn = lim inf An) becomes the pointwiseconvergence of the corresponding characteristic functions; in this way   0 and   be-come L ∗0 -subrings of Z(2)R = {0, 1}R. Denote by   1 the first pointwise sequentialclosure of   0 in Z(2)R (the limits of sequences in   0 ) and, for α ∈ ω1, denote by α the α-th closure of   0 . Then each  α is a ring of sets and   =  ω1 , see [NOV],[LAC].In [BKF] it is proved that the pointwise sequential convergence on the L ∗0 -ringB1(R) of the first Baire class functions fails to be strict. We shall prove that thepointwise convergence on   1 fails to be strict, too. Recall the definition of strictness(cf. Definition 1.2 in [FSZ]).Let (X,) be an L ∗0 -ring ( denotes the sequential convergence as a subset ofX  × X) and let X0 be a subring of X . The first sequential closure X1 of X0 isa ring. Denote by 0 and 1 the restrictions of  to X0 and X1, respectively. Wesay that 1 is strict (more exactly it should be: strict with respect to X0) if thefollowing holds:(s) Let 〈xn〉 be a sequence ranging in X1 \ X0 which converges under 1 tox ∈ X1. Then there exist a subsequence 〈x′n〉 of 〈xn〉 and sequences 〈y(k)n 〉ranging in X0, k ∈ , such that the sequence 〈x(k)n 〉 converges under 1 tox′k and each diagonal sequence 〈y(n)d(n)〉, d :  → , converges under 1 to x.Proposition 1.1. The pointwise convergence in   1 fails to be strict.112 . Let p1, p2, p3, . . . denote the increasing sequence of all prime numbers.For each n ∈ , let An = {k/pn ; k = 1, 2, . . . , pn − 1}. Then An ∈   1 \   0 andthe sequence 〈An〉 pointwise converges to ∅. For each k ∈ , let 〈Bkn〉 be a sequencein   0 which pointwise converges to Ak. We show that there exists a mapping u of into  such that for each mapping v of  into , v(k) > u(k) for each k ∈ ,and for each strictly increasing mapping s of  into  the subsequence 〈B(s(n))v(s(n))〉 ofthe diagonal sequence 〈B(n)v(n)〉 does not converge to ∅. Clearly, then the pointwiseconvergence on   1 fails to be strict.So, since all sets Ak are finite, for each k ∈  choose u(k) ∈  such that Ak ⊂ B(k)nfor each n > u(k). Let v be a mapping of  into  such that v(k) > u(k) for eachk ∈  and let s be a strictly increasing mapping of  into . There exists aninterval I1 = [a1, b1), a1 < b1, such that ps(1) ∈ I1 ⊂ B(s(1))v(s(1)). Put t(1) = 1. Byinduction, define a strictly increasing mapping t of  into  and a sequence of half-open intervals In = [an, bn), an < bn, such that In ⊂ B(s(t(n)))v(s(t(n))) and In+1 ⊂ In.However, since the distance between two neighboring points of An tends to 0, andeach Bkn is a finite union of half-open intervals, the construction of 〈In〉 is trivial andwe omit its description. Because the intersection∞⋂n=1In is nonempty, the sequence〈B(s(n))v(s(n))〉 cannot pointwise converge to ∅. This completes the proof. Observe that we have proved a rather strong negation of (s) for   1 .It is known that a commutative L ∗0 -group can have many nonequivalent L∗0 -group completions and its Novák completion [NOV] yields its categorical L ∗0 -groupcompletion ([FKO]). On the one hand, the convergence in the Novák L ∗0 -groupcompletion is strict ([FSZ]), on the other hand, we show that the convergence in theNovák L ∗0 -group completion of   0 fails to be an L∗0 -ring convergence.Example 1.2. Let  denote the usual pointwise convergence on   0 . Then theNovák L ∗0 -group completion of   0 has   1 as its underlying group and is equippedwith an L ∗0 -group convergence ∗1 defined as follows: 〈Bn〉 converges to B under∗1 iff for each subsequence 〈B′n〉 of 〈Bn〉 there exists its subsequence 〈B′′n〉 such thatB′′nB = AnA, n ∈ , where 〈An〉 is a sequence in   pointwise converging toA ∈   1 . The sequence 〈[n, n + 1)〉 converges to ∅ and N ∈   1 , but their product〈N∩[n, n+1)〉 = 〈{n}〉 fails to converge under ∗1 . Indeed, otherwise there would exista sequence 〈Mn〉 in   0 converging pointwise to M ∈   1 such that {n} = MnMfor infinitely many n ∈ , which is impossible. Hence ∗1 fails to be an L ∗0 -ringconvergence.Interesting results concerning strictness can be found in [PAU].It is known that an L ∗0 -field need not have any L∗0 -ring completion ([FZE]).In [BKF] it was shown that if an L ∗0 -ring (X,) has a completion, then it has113the categorical one: the epireflection in the subcategory of all complete L ∗0 -rings;denote it by (X,) = (X̂, ̂). Further, by Theorem 3.3 in [BKF], (X,) is ω-densein (X̂, ̂). That means that at most ω iterations of the sequential closure of X withrespect to ̂ are sufficient to get X̂. But C(R) fails to be ω-dense in B(R) and hencethe latter L ∗0 -ring fails to be the categoricalL∗0 -ring completion of C(R) (Corollary3.4 in [BKF]). By the same argument we get the followingCorollary 1.3.   carrying the pointwise convergence fails to be the categoricalcompletion of   0 .Then the question arises whether there is any categorical construction of an L ∗0 -ring completion such that B(R) and   are the corresponding completions of C(R)and   0 , respectively. We show that the answer is affirmative.2. H-completionsLet (H, ) be a completeL ∗0 -ring. For eachL∗0 -ring (X,) denote by Hom(X, H)the set of all sequentially continuous ring homomorphisms of (X,) into (H, ).Recall the notion of an initial structure. Let X be a ring, {(Xa,a ) ; a ∈ A} aset of L ∗0 -rings and F = {fa : X → Xa ; a ∈ A} a set of ring homomorphisms. If is an L ∗0 -ring convergence and it is the coarsest of all L∗0 -ring convergences ′ onX (we do not assume that uniqueness of limits and the Urysohn axiom) such thatall fa : (X,′ ) → (Xa,a ) are sequentially continuous, then  is called the initialL ∗0 -ring convergence with respect to F . For example, the product convergence isthe initial structure with respect to the projections.Definition 2.1. We say that (X,) is H-generated if  is the initial L ∗0 -ringconvergence with respect to Hom(X, H).Note: Since we assume uniqueness of the limits, if (X,) is H-generated, thenevery pair of distinct points of X is separated by some h ∈ Hom(X, H); in fact, theinitial structure with respect to F exists whenever F separates points of X ; also,if 〈xn〉 does not converge to x under , then there exists h ∈ Hom(X, H) such that〈h(xn)〉 does not converge to h(x) under  ; hence if H is the real line (or sequentiallyregular), then each H-generated L ∗0 -ring is sequentially regular (see [FMR]).Example 2.2. For each r ∈ R and each f ∈ RR define evr(f) = f(r). Sincewe identify A ⊂ R with its characteristic function, we have evr(A) = 1 if r ∈ A andevr(A) = 0 otherwise. It is easy to see that evr is, in fact, a sequentially continuousring homomorphism of RR, equipped with pointwise convergence, into R and alsoa sequentially continuous ring homomorphism of Z(2)R into Z(2), where Z(2) is114the usual discrete ring {0, 1} with addition modulo 2. In the same way we defineevaluations evx(f) for each set M , x ∈ M and f ∈ RM , or f ∈ Z(2)M .The next assertion was proved in [FPA] via a straightforward calculation.Lemma 2.3. Let h be a sequentially continuous ring homomorphism of   0 intoZ(2). Then there exists r ∈ R such that h = evr.Lemma 2.4. Let h be a sequentially continuous ring homomorphism of C(R)into R. Then there exists r ∈ R such that h = evr. . Since h is a linear functional on C(R), it follows from Theorem 1 in[ITH] that h is a finite linear combination of evaluations at points of R. Using thefact that h is a ring homomorphism, we easily infer that h is an evaluation at somer ∈ R. Corollary 2.5. (i) C(R) is R-generated.(ii)   0 is Z(2)-generated.We omit the straightforward proofs of the next two propositions.Proposition 2.6. (i) Each L ∗0 -subring of an H-generated L∗0 -ring is H-generated. (ii) If each (Xa,a ), a ∈ A, is an H-generated L ∗0 -ring, then theirproduct is H-generated, too.Proposition 2.7. Let (X,) be an L ∗0 -ring. Then the following are equivalent:(i) (X,) is H-generated.(ii) The canonical map ϕ of (X,) into the power HHom(X,H), defined by ϕ(x) =(h(x);h ∈ Hom(X, H)), is a homeomorphic isomorphism of (X,) onto anL ∗0 -subring of HHom(X,H).(iii) There exists a homeomorphic isomorphism of (X,) onto an L ∗0 -subring ofa power HM of (H, ).Definition 2.8. Let (X,) be anL ∗0 -ring and let (Y,  Y ) be its L ∗0 -subring.If each sequentially continuous ring homomorphism of (Y,  Y ) into (H, ) can beextended to a sequentially continuous homomorphism of (X,) into (H, ), then(Y,  Y ) is said to be H-embedded in (X,).Definition 2.9. Let (X,) be an H-generated L ∗0 -ring. If X is sequentiallyclosed in each H-generated L ∗0 -ring (X′,′ ) in which it is H-embedded, then (X,)is said to be H-complete.115Definition 2.10. Let (X,) be an H-generated L ∗0 -ring. A sequence 〈xn〉 issaid to be H-fundamental if for each h ∈ Hom(X, H) the sequence 〈h(xn)〉 convergesin H under  .Proposition 2.11. Let (X,) be an H-generated L ∗0 -ring. Then the followingare equivalent:(i) (X,) is H-complete.(ii) Every H-fundamental sequence in (X,) converges.(iii) The image of (X,) under the canonical map ϕ is a sequentially closed subringof the L ∗0 -ring HHom(X,H).(iv) There exists a homeomorphic isomorphism of (X,) onto a sequentially closedL ∗0 -subring of some power HM of (H, ).Let A be the category of allL ∗0 -rings with sequentially continuous ring homomor-phisms as morphisms. Let H (H) and Hc(H) be the full and isomorphism closedsubcategories of all H-generated and H-complete L ∗0 -rings, respectively.Definition 2.12. Let (X,) be an H-generated L ∗0 -ring. We say that anH-generated L ∗0 -ring (X′,′ ) is an H-completion of (X,) provided that(i) (X,) is an H-embedded L ∗0 -subring of (X′,′ ) and X ′ is the smallest se-quentially closed subset of X ′ containing X ;(ii) (X ′,′ ) is H-complete.Proposition 2.13. Each H-generated L ∗0 -ring (X,) has an H-completion andis uniquely determined up to a homeomorphic isomorphism pointwise fixed on X . . According to Proposition 2.7, the canonical map ϕ of (X,) intoHHom(X,H) is a homeomorphic isomorphism of (X,) onto the corresponding L ∗0 -subring of HHom(X,H); for the sake of simplicity we identify (X,) with its ϕ-image. Denote by η(X,) = (XH ,H ) the smallest sequentially closed L ∗0 -subringof HHom(X,H) containing (X,). It follows from the categorical properties ofHHom(X,H) that (X,) is H-embedded in (XH ,H ). By Proposition 2.11, (XH ,H )is H-complete and hence an H-completion. By a standard categorical argument itfollows that (XH ,H ) is uniquely determined up to a homeomorphic isomorphismpointwise fixed on X .(Remember, we work with unique sequential limits and if two sequentially contin-uous maps agree on a topologically dense subset, then they are identical.) Corollary 2.14. Hc(H) is epireflective in H (H) and η yields the epireflector.116Corollary 2.15. (XH ,H ) is a maximalH-generatedL ∗0 -ring containing (X,)as a topologically dense H-embedded L ∗0 -ring.Corollary 2.16. (i) B(R) is the epireflection of C(R) into R-completeL ∗0 -rings.(ii)   is the epireflection of   0 into Z(2)-complete L ∗0 -rings.3. Additional resultsAssume that  is a ring of subsets of a set M , i.e.  is closed with respect to thesymmetric difference and the intersection. Unlike in the case of   0 , it can happenthat not every sequentially continuous ring homomorphism of  into Z(2) is anevaluation evx at some x ∈ M . Despite this fact, we show that the generated σ-ringσ( ) is the Z(2)-completion of  . The following construction has been suggested in[FPA].Proposition 3.1. An L ∗0 -ring (X,) is Z(2)-generated iff it is a ring of sets. . 1. Let (X,) be Z(2)-generated. Put M = Hom(X, Z(2)). Then thecanonical map ϕ sends x ∈ X into a subset A(x) = {h ∈ Hom(X, Z(2)) ; h(x) = 1}.Clearly, this yields a homeomorphic isomorphism of (X,) onto the correspondingring of subsets of M .2. If  is a ring of subsets of a set M , then An → A in  iff evx(An) → evx(A)for each x ∈ M . Obviously, the canonical map ϕ of  into Z(2)Hom( ,Z(2)) definedby ϕ(a) = {h(a) ; h ∈ Hom( , Z(2))} is a homeomorphic isomorphism. Proposition 3.2. A ring of sets is Z(2)-complete iff it is a σ-ring. . The assertion follows from Proposition 2.11. Corollary 3.3. Let  be a ring of sets. Then  ∈ H (Z(2)) and the generatedσ-ring σ( ) is its Z(2)-completion, i.e. the epireflection of  into Hc(Z(2)).Concerning rings of sets and the extension of sequentially continuous maps we callthe reader’s attention to the pioneering [NOE].We end with the following observation. Let (X,) be a commutative L ∗0 -group.Then X can be viewed as a zero-ring and (X,) as an L ∗0 -ring. The construction ofan H-completion therefore also yields a categorical completion theory of L ∗0 -groupsfor suitable commutative L ∗0 -groups (H, ). In particular, it might be interestingto investigate H-completions when H is the torus (cf. [FCA]).117References[BKF] Borsík, J. and Frič, R.: Pointwise convergence fails to be strict. Czechoslovak Math.J. 48(123) (1998), 313–320.[FCA] Frič, R.: On continuous characters of Borel sets. In Proceedings of the Conference onConvergence Spaces (Univ. Nevada, Reno, Nev., 1976). Dept. Math. Univ. Nevada,Reno, Nev., 1976, pp. 35–44.[FCB] Frič, R.: On completions of rationals. In Recent Developments of General Topol-ogy and its Applications, Math. Research No. 67. Akademie-Verlag, Berlin, 1992,pp. 124–129.[FKO] Frič, R. and Koutník, V.: Completions for subcategories of convergence rings. InCategorical Topology and its Relations to Modern Analysis, Algebra and Combina-torics. World Scientific Publishing Co., Singapore, 1989, pp. 195–207.[FKT] Frič, R. and Koutník, V.: Sequential convergence spaces: iteration, extension, com-pletion, enlargement. In Recent Progress in General Topology. North Holland, Am-sterdam, 1992, pp. 199–213.[FMR] Frič, R., McKennon, K. and Richardson, G. D.: Sequential convergence in C(X). InConvergence Structures and Application to Analysis (Frankfurt/Oder, 1978), Abh.Akad. Wiss. DDR, Abt. Math.-Naturwiss.-Technik, 1979, Nr. 4N. Akademie-Verlag,Berlin, 1980, pp. 57–65.[FPA] Frič, R. and Piatka, Ľ.: Continuous homomorphisms in set algebras. Práce Štud.Vys. Šk. Doprav. Žiline Sér. Mat.-fyz., 2 (1979), 13–20. (In Slovak.)[FZE] Frič, R. and Zanolin, F.: Coarse sequential convergence in groups, etc.. CzechoslovakMath. J. 40 (115) (1990), 459–467.[FZS] Frič, R. and Zanolin, F.: Strict completions of L ∗0 -groups. Czechoslovak Math. J.42 (117) (1992), 589–598.[HES] Herrlich, H. and Strecker, G. E.: Category Theory. 2nd edition, Heldermann Verlag,Berlin, 1976.[ITH] Isbell, J. R. and Thomas Jr., S.: Mazur’s theorem on sequentially continuous func-tionals. Proc. Amer. Math. Soc. 14 (1963), 644–647.[LAC] Laczkovich, M.: Baire 1 functions. Real Analysis Exchange 9 (1983/84), 15–28.[NOE] Novák, J.: Über die eindeutigen stetigen Erweiterungen stetiger Funktionen. Cze-choslovak Math. J. 8 (1958), 344–355.[NOV] Novák, J.: On completions of convergence commutative groups. In General Topol-ogy and its Relations to Modern Analysis and Algebra III (Proc. Third PragueTopological Sympos., 1971). Academia, Praha, 1972, pp. 335–340.[PAU] Paulík, L.: Strictness ofL0-ring completions. Tatra Mountains Math. Publ. 5 (1995),169–175.Author’s address: Matematický ústav SAV, Grešákova 6, 040 01 Košice, Slovakia, e-mail: fric@mail.saske.sk.118

English

Rings of maps: sequential convergence and completion

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Czechoslovak Mathematical JournalJán Borsík; Roman FričPointwise convergence fails to be strictCzechoslovak Mathematical Journal, Vol. 48 (1998), No. 2, 313–320Persistent URL: http://dml.cz/dmlcz/127418Terms of use:© Institute of Mathematics AS CR, 1998Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documentsstrictly for personal use. Each copy of any part of this document must contain these Terms of use.This document has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://dml.czCzechoslovak Mathematical Journal, 48 (123) (1998), 313–320POINTWISE CONVERGENCE FAILS TO BE STRICTJán Borsík1 and Roman Frič,2 Košice(Received September 14, 1995)Abstract. It is known that the ring B( ) of all Baire functions carrying the pointwiseconvergence yields a sequential completion of the ring C( ) of all continuous functions. Weinvestigate various sequential convergences related to the pointwise convergence and theprocess of completion of C( ). In particular, we prove that the pointwise convergence failsto be strict and prove the existence of the categorical ring completion of C( ) which differsfrom B( ).1.Consider the set    of all real functions carrying the pointwise (sequential) con-vergence. If we start with the ring C( ) of all continuous functions, then the ringB1( ) of all 1-st Baire class functions is the first sequential closure of C( ), the ringBα( ) of all α-th Baire class functions, α < ω1, is the α-th sequential closure ofC( ), and the ring B( ) of all Baire functions is the smallest subset of    contain-ing C( ) and closed with respect to the pointwise convergence, hence sequentiallycomplete, see [NOV], [LAC].Let  be a sequential convergence on B1( ) such that, for each sequence 〈fn〉of continuous functions, 〈fn〉 converges to f ∈ B1( ) under  iff it converges to fpointwise. Then  is said to be admissible. If  is compatible with the group or ringstructure of B1( ), then each Cauchy sequence of continuous functions convergesunder  and we get B1( ) as a group or ring (sequential) precompletion of C( ).Observe that , for example the pointwise convergence, need not be complete. Toget a completion, in such cases we have to iterate the precompletion process. Incase of the pointwise convergence the usual sequential completion of C( ) is the ring1Grant GA SAV 1228/952Grant GA SAV 1230/95313B( ). In the present paper we investigate admissible convergences and alternativeways of (pre)completing C( ).Strictness is a natural way how to control the convergence of sequences of idealpoints in an extension of a convergence space or a precompletion of a sequentialgroup or ring ([FZS], [FKE], [PAU]).An admissible convergence  on B1( ) is said to be strict if the following conditionis satisfied (see Definition 1.2 in [FZS]):(s) Let 〈fn〉 be a sequence ranging in B1( ) \ C( ) which converges under to f ∈ B1( ). Then there are a subsequence 〈f ′n〉 of 〈fn〉 and sequences〈g(k)n 〉, k ∈ , of continuous functions such that the sequence 〈g(k)n 〉 pointwiseconverges to f ′k and each diagonal sequence 〈g(n)d(n)〉, d :  → , pointwiseconverges to f .In [FZS] the authors asked whether the pointwise convergence is strict. We provethat the answer is “NO”.Theorem 1.1. The pointwise convergence on B1( ) fails to be strict. . Let p1, p2, p3, . . . denote the increasing sequence of all prime numbers.For each n ∈ , let An = {k/pn ; k = 1, 2, . . . , pn − 1} and let fn denote thecharacteristic function of An. Let f denote the constant zero function. Then fn ∈B1( )\C( ) for each n ∈  and the sequence 〈fn〉 pointwise converges to f . For eachk ∈ , let 〈g(k)n 〉 be a sequence of continuous functions which pointwise converges tofk. We show that there exists a mapping u of  into  such that for each mappingv of  into , v(k) > u(k) for each k ∈ , and for each strictly increasing mappings of  into  the subsequence 〈g(s(n))v(s(n))〉 of the diagonal sequence 〈g(n)v(n)〉 does notpointwise converge to f . Clearly, then the pointwise convergence on B1( ) fails tobe strict.So, since all sets Ak are finite, for each k ∈  choose u(k) ∈  such that g(k)n (x) >1/2 for each n > u(k) and each x ∈ Ak. Let v be a mapping of  into  such thatv(k) > u(k) for each k ∈  and let s be a strictly increasing mapping of  into .From g(s(1))v(s(1))(1/ps(1)) > 1/2 it follows that there exists a closed interval I1 ⊂ (0, 1)such that 1/ps(1) ∈ int I1 and g(s(1))v(s(1))(I1) > 1/2. Put t(1) = 1. By induction,define a strictly increasing mapping t of  into  and a sequence 〈In〉 of closedintervals such that int In ⊃ In+1 = ∅ and g(s(t(n)))v(s(t(n)))(In) > 1/2 for all n ∈ . Chooset(2) ∈  such that s(t(2)) ∈ {s(2), s(3), . . .} and As(t(2))∩ int I1 = ∅. Choose a closedinterval I2 such that int I2 = ∅, int I1 ⊃ I2 and g(s(t(2)))v(s(t(2)))(I2) > 1/2. Analogouslydefine t(3) and I3, . . . , t(n) and In, and so on. Now, choose x0 ∈∞⋂i=1Ii = ∅. Since314g(s(t(n)))v(s(t(n)))(x0) > 1/2 for all n ∈ , the sequence 〈g(s(n))v(s(n))〉 does not pointwise convergeto f . This completes the proof. 2.In this section we prove some simple facts about strict admissible convergences.Let f and fn, n ∈ , be functions in   . We say (cf. [FKE]) that the sequence〈fn〉 and the function f are linked if there are sequences 〈g(k)n 〉, k ∈ , in    suchthat for each k ∈  the sequence 〈g(k)n 〉 pointwise converges to fk and each diagonalsequence 〈g(n)d(n)〉, d :  → , pointwise converges to f . Note: condition (s) canbe reformulated as “if 〈fn〉 converges to f under  and fn ∈ B1( ) \ C( ) for alln ∈ , then there exists a subsequence 〈f ′n〉 of 〈fn〉 which is linked to f via a doublesequence, i.e. sequence of sequences, of continuous functions”.Lemma 2.1. Let 〈fn〉 be a sequence of functions linked to a function f . Then〈fn〉 converges pointwise to f . . Assume that, on the contrary, for some x ∈   the sequence 〈fn(x)〉 doesnot converge to f(x). Then there exists a positive number ε such that |fn(x)−f(x)| >ε for infinitely many n ∈ . Clearly, this is a contradiction with the assumption that〈fn〉 and f are linked. Corollary 2.2. Every strict convergence on B1( ) is finer than the pointwiseconvergence.Next, we describe the coarsest and the finest strict convergences on B1( ) com-patible with the ring structures of B1( ).Construction 2.3. Denote by s the set of all pairs (〈fn〉, f) such that 〈fn〉is a sequence of functions of B1( ), f ∈ B1( ) and for each subsequence 〈f ′n〉 of〈fn〉 there exists its subsequence 〈f ′′n 〉 which is linked to f via a double sequenceof continuous functions. As a rule, (〈fn〉, f) ∈ s means that the sequence 〈fn〉converges to f under s .Claim 2.3.1. s is a strict L∗0-ring convergence. . It follows easily from Lemma 2.1 that each sequence s -converges toat most one limit. The remaining axioms of convergence follow directly from thedefinition of s . Indeed, each constant sequence converges, each subsequence ofa convergent sequence converges to the same limit, and s satisfies the Urysohnaxiom: if 〈fn〉 and f are such that for each subsequence 〈f ′n〉 of 〈fn〉 there exists its315subsequence 〈f ′′n 〉 such that (〈f ′′n 〉, f) ∈ s , then (〈fn〉, f) ∈ s . Further, sums andproducts of convergent sequences converge to the corresponding sums and productsof their limits, hence s is compatible with the ring structure of B1( ). It followsfrom Lemma 2.1 that s is admissible and since s is clearly strict, the proof iscomplete. Claim 2.3.2. Let  be a strict L∗0-ring convergence on B1( ). Then s iscoarser than . . If 〈fn〉 converges to f under , then some subsequence 〈f ′n〉 of 〈fn〉 islinked to f via a double sequence of continuous functions and hence 〈f ′n〉 convergesto f under s . Since s satisfies the Urysohn axiom, it follows that  ⊂ s . Construction 2.4. Denote by N the set of all sequences in B1( ) of theform〈 k∑i=1(fin − fi)gi〉, where k ∈ , 〈fin〉 is a sequence of continuous functionspointwise converging to fi ∈ B1( ), i = 1, . . . , k. Trivially, N is closed with respectto subsequences and finite sums. Since〈(f1n − f1)g1〉〈(f2n − fn)g2〉 = 〈(f1nf2n − f1f2)g1g2〉− 〈(f1n − f1)f2g1g2〉 − 〈(f2n − f2)f1g1g2〉,it follows that N is closed with respect to finite products, too. By Lemma 2 in [FZE],there exists a unique L-ring convergence under which a sequence 〈fn〉 converges tothe constant zero function  iff 〈fn〉 ∈ N . Denote by r its Urysohn modification.Recall that 〈fn〉 converges to  under r iff for each subsequence 〈f ′n〉 of 〈fn〉 thereexists its subsequence 〈f ′′n 〉 belonging to N .Claim 2.4.1. r is a strict L∗0-ring convergence. . Obviously, r is finer than the pointwise convergence on B1( ) andhence the limits of r -convergent sequences are uniquely determined. Thus r isan L∗0-ring convergence. If 〈fn〉 is a sequence of continuous functions pointwiseconverging to f ∈ B1( ), then (〈fn〉, f) ∈ r . Consequently, r is admissible.The proof of strictness of r is straightforward. Hint: if (〈hn〉, h) ∈ r and hn ∈B1( ) \ C( ) for all n ∈ , then there exists a subsequence 〈h′n〉 of 〈hn〉 such that〈h′n − h〉 ∈ N ; hence 〈h′n〉 is of the form〈 k∑i=1(fin − fi)gi + h〉, where fi, gi, h ∈B1( ), and 〈fin〉 is a sequence of continuous functions pointwise converging to fi,i = 1, . . . , k; the rest is trivial. Claim 2.4.2. Let  be a strict L∗0-ring convergence on B1( ). Then r ⊂ .316 . Since  is admissible and compatible with the ring structure of B1( ),〈(fn − f)g〉 converges under  to  whenever 〈fn〉 is a sequence of continuous func-tions pointwise converging to f ∈ B1( ) and g ∈ B1( ). Hence r ⊂ .It is known that each commutative L∗0-group can have many nonequivalent L∗0-group completions and its Novák completion ([NOV]) yields its categorical L∗0-groupcompletion ([FKO]). We show that the Novák L∗0-group completion of C( ) fails tobe an L∗0-ring completion. Example 2.5. Let  denote the pointwise convergence on C( ). Then theNovák L∗0-group completion of C( ) has B1( ) as its underlying group and isequipped with an L∗0-group convergence ∗1 defined as follows: 〈fn〉 converges tof under ∗1 iff for each subsequence 〈f ′n〉 of 〈fn〉 there exists its subsequence 〈f ′′n 〉such that f ′′n − f = gn − g, n ∈ , where 〈gn〉 is a sequence of continuous func-tions pointwise converging to g ∈ B1( ). Let h be the characteristic function ofthe singleton {0} and let fn be the constant function with value 1/n, n ∈ . Thenh ∈ B1( ) and the sequence of continuous functions 〈fn〉 pointwise converges to  ,but their product 〈hfn〉 fails to converge under ∗1 . Hence ∗1 fails to be an L∗0-ringconvergence and clearly ∗1  r .Note: it is known that an L∗0-ring need not have an L∗0-ring completion ([FZE]) andthere are known sufficient conditions guaranteeing the existence of the categoricalL∗0-ring completion ([FKO]); C( ) fails to be a field and hence does not satisfy theconditions.We finish this section by mentioning some problems. First, we do not knowwhether s or r is complete. Second, if not, then we can ask whether B1( )equipped with s or r has an L∗0-ring completion, at all.3.Our final goal is to construct an L∗0-ring completion of C( ) having a universalextension property or, in categorical terms, an epireflection of C( ) into completeL∗0-rings. Since C( ) is not a field, we cannot use the construction due to J. Novák.The interested reader is referred to [FKO] for the background information about L∗0-ring completions and to [HES] about categorical notions. To make the paper moreself-contained, we briefly recall some related notions.Let  be an L∗0-convergence on a set Y = ∅. For A ⊂ Y , denote by clA the setof all  -limits of sequences ranging in A. Define 0-clA = A and, by induction, foreach ordinal number α  ω1 define α-clA =⋃β<αcl(β-clA). Then 1-clA = clA and317each α-cl yields a closure operator on Y . Further, ω1-cl is idempotent and hence atopology; it is the finest of all topologies on Y coarser than cl.Note: if ω1-clA = Y and f, g are continuous maps of (Y, ) into an L∗0-space(Y ′, ′ ) such that f(x) = g(x) for each x ∈ A, then f = g; consequently a morphismwith a topologically dense range is an epimorphism in L∗0-spaces.Let Y be a ring (commutative, not necessarily possessing a unit element) andlet  be an L∗0-ring convergence on Y . A sequence 〈xn〉 is said to be Cauchy if〈x′n − x′′n〉 converges under  to zero whenever 〈x′n〉 and 〈x′′n〉 are subsequences of〈xn〉. If each Cauchy sequence converges, then we speak of a complete L∗0-ring. LetX be a subring of Y . Then, for each ordinal number α  ω1, the set α-clX is asubring of Y and if Y is complete, then the subring ω1-clX is complete, too. If Yis complete and Y = ω1-clX , then (Y, ) is said to be an L∗0-ring completion of Xcarrying the restriction of  to X . Finally, for L∗0-convergences the coordinatewiseconvergence on products is the categorical one and the product of complete L∗0-ringsis a complete L∗0-ring.Let A,B, and C denote the categories of all L∗0-rings, all L∗0-rings having a com-pletion, and all complete L∗0-rings, respectively. A straightforward application of theusual categorical tricks yields the followingTheorem 3.1. C is an epireflective subcategory of B. . Let X be a ring carrying an L∗0-ring convergence  and let (X,) beits L∗0-ring completion. We show that there exists its L∗0-ring completion (X,) =(X̂, ̂) having the following universal extension property: for each continuous homo-morphism f of (X,) into a complete L∗0-ring (Y, ) there exists a unique continu-ous homomorphism f̂ of (X̂, ̂) into (Y, ) such that f(x) = f̂(x) for each x ∈ X .Since X̂ is the smallest sequentially closed subset containing X , the embedding id:(X,) → (X̂, ̂) is an epimorphism and  yields an epireflector of B into C. Theconstruction of (X̂, ̂) is divided into two parts. The first has an auxiliary character.Part 1. There exists a nonempty set S = {fa : (X,) → (Xa,a ) ; a ∈ A} ofcontinuous homomorphisms such that each (Xa,a ) is a complete L∗0-ring and iff is a continuous homomorphism of (X,) into a complete L∗0-ring (Y, ), thenthere exists a ∈ A and a homeomorphic isomorphism g of (Xa,a ) onto a subring(Yf ,  Yf ) of (Y, ) such that f is a composition of g ◦ fa and the embeddingof (Yf ,  Yf ) into (Y, ). Indeed, each f determines a complete L∗0-subring of(Y, ) the underlying set of which is Yf = ω1-cl f(X). Since card(1-cl f(X)) cannotexceed the cardinality of the set of all countable infinite subsets of f(X) and ω1-cl f(X) =⋃β<ω1cl(β-cl f(X)), it follows that card(Yf )  exp card(X). Hence there isa set {(Xb,b ) ; b ∈ B} of complete L∗0-rings such that card(Xb)  exp card(X) and318each (Yf ,  Yf ) is homeomorphic and isomorphic to some (Xb,b ), b ∈ B. Note:S yields a so-called solution set for (X,) with respect to the inclusion functor of Cinto B.Part 2. The product∏a∈A(Xa,a ) is a complete L∗0-ring and, via the canonicalembedding sending x ∈ X into ϕ(x) = (fa(x), a ∈ A), (X,) can be view as thecorresponding L∗0-subring of∏a∈A(Xa,a ) (remember, (X,) is an L∗0-subring of itscompletion (X,)). Denote by (X̂, ̂) the smallest sequentially closed L∗0-subring of∏a∈A(Xa,a ) containing (X,). It is easy to see that (X̂, ̂) has the desired properties.This completes the proof. Lemma 3.2. Let (Y, ) be a complete L∗0-ring and let X be a subring of Y . PutX = ω-clX and define  ⊂ X × X as follows: (〈xn〉, x) ∈  whenever (〈xn〉, x) ∈ and there exists a natural number k such that xn ∈ (k-clX) for each n ∈ .Then (X,) is a complete L∗0-ring and the identity mapping on X is a continuousisomorphism of (X,) onto (X,  X). . It is easy to verify that  is an L∗0-ring convergence on X finer than  X . If 〈xn〉 is a Cauchy sequence in (X,), then there exists k ∈  such thatx ∈ (k-clX) for each n ∈ . Thus (X,) is complete. Theorem 3.3. Let (X,) be an L∗0-ring in B and let (X̂, ̂) be its categoricalL∗0-ring completion. Then ω-clX = X̂. . The assertion follows from Lemma 3.2. Putting X̂ = Y and ̂ =  , weeasily infer that X = X̂ and  = ̂ . Corollary 3.4. B( ) carrying the pointwise convergence fails to be the categor-ical completion of C( ). . Indeed, ω-clC( ) = Bω( )  B( ), while C( ) is ω-dense in itscategorical L∗0-ring completion. Problem. Describe the categorical L∗0-ring completion of C( ).References[FKE] Frič, R. and Kent, D. C.: Regularity and extension of maps. Math. Slovaca 42(1992), 349–357.[FKO] Frič, R. and Koutník, V.: Completions for subcategories of convergence rings. In:Categorical Topology and its Relations to Modern Analysis, Algebra and Combina-torics. World Scientific Publishing Co., Singapore, 1989, pp. 195–207.[FZE] Frič, R. and Zanolin, F.: Coarse sequential convergence in groups, etc. CzechoslovakMath. J. 40 (115) (1990), 459–467.319[FZS] Frič, R. and Zanolin, F.: Strict completions of L∗0-groups. Czechoslovak Math. J.42 (117) (1992), 589–598.[HES] Herrlich, H. and Strecker, G. E.: Category Theory. 2nd edition, Heldermann Verlag,Berlin, 1976.[LAC] Laczkovich, M.: Baire 1 functions. Real Analysis Exch. 9 (1983/84), 15–28.[NOV] Novák, J.: On completions of convergence commutative groups. In: General Topol-ogy and its Relations to Modern Analysis and Algebra III (Proc. Third PragueTopological Sympos., 1971). Academia, Praha, 1972, pp. 335–340.[PAU] Paulík, L.: Strictness of L0-ring completions. Tatra Mountains Math. Publ. 5 (1995),169–175.Authors’ address: Matematický ústav SAV, Grešákova 6, 040 01 Košice, Slovakia, e-mail: borsik@linux1.saske.sk, fric@linux1.saske.sk.320

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Rings of maps: sequential convergence and completion

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