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On a universal property of the final topology
Chaos, Solitons and Fractals 32 (2007) 212–214 www.elsevier.com/locate/chaos
On a universal property of the final topology Diname´rico P. Pombo Jr. Instituto de Matema´tica, Universidade Federal Fluminense, Rua Ma´rio Santos Braga, s/n, 24020-140 Nitero´i, RJ, Brazil Accepted 26 October 2005
Abstract A criterion for the equicontinuity of sets of mappings on a set with a final topology is established. 2005 Elsevier Ltd. All rights reserved.
In this note we obtain a criterion for the equicontinuity of sets of mappings defined on a set endowed with an arbitrary final topology with values in an arbitrary uniform space, as well as a few consequences of it. Before stating our main result, let us recall that a set X of mappings from a topological space X into a uniform space Y is equicontinuous at x0 2 X if for each vicinity V of Y there exists a neighborhood U of x0 in X such that the relations x 2 U, f 2 X imply (f(x0), f(x)) 2 V; X is equicontinuous if it is equicontinuous at each element of X. Theorem 1. Let (Xa)a2I be a non-empty family of topological spaces, X a set and, for each a 2 I, let ga be a mapping from Xa into X. Consider X endowed with the final topology for the family (Xa, ga)a2I [1, Chap. 1, Section 2, no. 4]. Then, for each uniform space Y and for each set X of mappings from X into Y, the following conditions are equivalent: (a) X is equicontinuous; (b) X ga ¼ ff ga ; f 2 Xg is equicontinuous for all a 2 I. X
Y X ! ga % X ga Xa Proof. Obviously, (a) implies (b), because ga is continuous for all a 2 I. In order to prove that (b) implies (a), let Y be a uniform space and let X be a set of mappings from X into Y such that X ga is equicontinuous for all a 2 I. Consider the set FðX; Y Þ of all mappings from X into Y endowed with the uniform structure of uniform convergence [2, Section 1, no. 1], and define g : X ! FðX; Y Þ by g(x)(f) = f(x) for x 2 X and f 2 X. It is easily seen that X is equicontinuous if and only if g is continuous.
E-mail address: [email protected]ff.br. 0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.10.101
D.P. Pombo Jr. / Chaos, Solitons and Fractals 32 (2007) 212–214
213
We claim that g ga is continuous for all a 2 I. Indeed, let a 2 I g
X ! ga -
FðX; Y Þ % g ga
Xa and xa 2 Xa be given, and let V be a vicinity of Y. By the equicontinuity of X ga at xa, there exists a neighborhood Ua of xa in Xa such that the relations ya 2 Ua, f 2 X imply ((f ga)(xa), (f ga)(ya)) 2 V, that is, the relation ya 2 Ua implies ((g ga)(xa)(f), (g ga)(ya)(f)) 2 V for all f 2 X. Thus g ga is continuous at xa. Therefore g ga is continuous for all a 2 I, and hence g is continuous by Proposition 6, p. 31 of [1]. Consequently, X is equicontinuous, proving (a). This completes the proof. h Corollary 2. Let (Xa)a2I, X and (ga)a2I be as in Theorem 1. Let (Yk)k2J be a non-empty family of uniform spaces, Y a set and, for each k 2 J, let hk be a mapping from Y into Yk. Consider Y endowed with the initial uniform structure for the family (Yk, hk)k2J [1, Chap. 2, Section 2, no. 3]. Then, for each set X of mappings from X into Y, the following conditions are equivalent: (a) X is equicontinuous; (b) hk X ga ¼ fhk f ga ; f 2 Xg is equicontinuous for all (a, k) 2 I · J.
ga
X " Xa
X
! !
hk Xga
Y # Yk
hk
Proof. Obviously, (a) implies (b), because ga is continuous for all a 2 I and hk is uniformly continuous for all k 2 J. In order to prove that (b) implies (a), let a 2 I be given. Since, by hypothesis, hk ðX ga Þ is equicontinuous for all k 2 J, Proposition 3, p. 27 of [2] ensures that X ga is equicontinuous. Therefore X is equicontinuous by Theorem 1, proving (a). h Corollary 3. Let (Xa)a2I, X and (ga)a2I be as in Theorem 1. Let T be a set, M a set of subsets of T, Z a uniform space, FðT ; ZÞ the set of all mappings from T into Z and H FðT ; ZÞ. Consider H endowed with the uniform structure of Mconvergence [2, Section 1, no. 2]. Then, for each set X of mappings from X into H, the following conditions are equivalent: (a) X is equicontinuous; (b) fðf ga ÞjA; f 2 Xg is equicontinuous for all ða; AÞ 2 I M. Proof. For each A 2 M let hA be the mapping u 2 H 7!ujA 2 FðA; ZÞ; and consider FðA; ZÞ endowed with the uniform structure of uniform convergence. Since the uniform structure of Mconvergence on H is the initial uniform structure for the family ðFðA; ZÞ; hA ÞA2M , the result follows immediately from Corollary 2. h Corollary 4. Let (Xa)a2I, X and (ga)a2I be as in Theorem 1. Let (Gk)k2J be a non-empty family of topological groups, G a group and, for each k 2 J, let hk: G ! Gk be a group homomorphism. Consider G endowed with the initial topology for the family (Gk, hk)k2J ([1, Chap. 1, Section 2, no. 3]; G is a topological group by Theorem 1.9 of [11]). Then, for each set X of mappings from X into G, the following conditions are equivalent: (a) X is equicontinuous; (b) hk X ga is equicontinuous for all (a, k) 2 I · J. Proof. Since the uniform structure derived from the topology of G is the initial uniform structure for the family (Gk, hk)k2J (each Gk being endowed with the uniform structure derived from the topology of Gk), the result follows immediately from Corollary 2. h
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D.P. Pombo Jr. / Chaos, Solitons and Fractals 32 (2007) 212–214
Corollary 5. Let (Xa)a2I, X and (ga)a2I be as in Theorem 1. Let F be either a locally convex space over R or C or a locally convex space over a non-Archimedean non-trivially valued complete field, and let C be a set of seminorms defining the topology of F [6,7]. For each q 2 C let Fq be the vector space F seminormed by q and 1q: F ! Fq the identity mapping. Then, for each set X of mappings from X into F, the following conditions are equivalent: (a) X is equicontinuous; (b) 1q X ga is equicontinuous for all (a, q) 2 I · C. Proof. Since the topology of F is the initial topology for the family (Fq, 1q)q2C, the result follows immediately from Corollary 4. h Remark 6. (a) Corollary 5 is applicable, in particular, to the locally convex spaces F over C which occur in the theory of distributions [9], some of which play an important role in the study of the physical sciences [8], and also to the locally convex spaces F over C which occur in p-adic mathematical physics and p-adic quantum mechanics [5,10]. (b) Corollary 5 is applicable, in particular, to the locally convex space F ¼ Qp over Qp (Qp being the field of p-adic numbers), whose relevance in E-infinity theory [3] has been emphasized in [4].
References [1] Bourbaki N. Topologie ge´ne´rale. Actualite´s Scientifiques et Industrielles, Quatrie`me e´dition, vol. 1142. Hermann; 1965 [Chapitres 1 et 2]. [2] Bourbaki N. Topologie ge´ne´rale. Actualite´s Scientifiques et Industrielles, Deuxie`me e´dition, vol. 1084. Hermann; 1967 [Chapitre 10]. [3] El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy physics. Chaos, Solitons & Fractals 2004;19:209–36. [4] El Naschie MS. A few hints and some theorems about Witten’s M theory and T-duality. Chaos, Solitons & Fractals 2005;25: 545–8. [5] First international conference on p-adic mathematical physics, Proceedings of the Steklov Institute of Mathematics, vol. 245, 2004. [6] Robertson AP, Robertson W. Topological vector spaces. 2nd ed. Cambridge University Press; 1973. [7] Schikhof WH. Locally convex spaces over nonspherically complete valued fields. Bull Soc Math Belg Se´r B 1986;38:187–224. [8] Schwartz L. Me´thodes mathe´matiques pour les sciences physiques. Enseignement des sciences, Deuxie`me e´dition, vol. II. Hermann; 1965. [9] Schwartz L. The´orie des distributions. Publications de l’Institut de Mathe´matique de l’Universite´ de Strasbourg, vol. IX–X. Hermann; 1966. [10] Vladimirov VS, Volovich IV, Zelenov EI. P-adic analysis and mathematical physics. Series on Soviet and East European mathematics, vol. 1. World Scientific; 1994. [11] Warner S. Topological fields. Notas de Matema´tica, vol. 126. North-Holland; 1989.
On a universal property of the final topology Diname´rico P. Pombo Jr. Instituto de Matema´tica, Universidade Federal Fluminense, Rua Ma´rio Santos Braga, s/n, 24020-140 Nitero´i, RJ, Brazil Accepted 26 October 2005
Abstract A criterion for the equicontinuity of sets of mappings on a set with a final topology is established. 2005 Elsevier Ltd. All rights reserved.
In this note we obtain a criterion for the equicontinuity of sets of mappings defined on a set endowed with an arbitrary final topology with values in an arbitrary uniform space, as well as a few consequences of it. Before stating our main result, let us recall that a set X of mappings from a topological space X into a uniform space Y is equicontinuous at x0 2 X if for each vicinity V of Y there exists a neighborhood U of x0 in X such that the relations x 2 U, f 2 X imply (f(x0), f(x)) 2 V; X is equicontinuous if it is equicontinuous at each element of X. Theorem 1. Let (Xa)a2I be a non-empty family of topological spaces, X a set and, for each a 2 I, let ga be a mapping from Xa into X. Consider X endowed with the final topology for the family (Xa, ga)a2I [1, Chap. 1, Section 2, no. 4]. Then, for each uniform space Y and for each set X of mappings from X into Y, the following conditions are equivalent: (a) X is equicontinuous; (b) X ga ¼ ff ga ; f 2 Xg is equicontinuous for all a 2 I. X
Y X ! ga % X ga Xa Proof. Obviously, (a) implies (b), because ga is continuous for all a 2 I. In order to prove that (b) implies (a), let Y be a uniform space and let X be a set of mappings from X into Y such that X ga is equicontinuous for all a 2 I. Consider the set FðX; Y Þ of all mappings from X into Y endowed with the uniform structure of uniform convergence [2, Section 1, no. 1], and define g : X ! FðX; Y Þ by g(x)(f) = f(x) for x 2 X and f 2 X. It is easily seen that X is equicontinuous if and only if g is continuous.
E-mail address: [email protected]ff.br. 0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.10.101
D.P. Pombo Jr. / Chaos, Solitons and Fractals 32 (2007) 212–214
213
We claim that g ga is continuous for all a 2 I. Indeed, let a 2 I g
X ! ga -
FðX; Y Þ % g ga
Xa and xa 2 Xa be given, and let V be a vicinity of Y. By the equicontinuity of X ga at xa, there exists a neighborhood Ua of xa in Xa such that the relations ya 2 Ua, f 2 X imply ((f ga)(xa), (f ga)(ya)) 2 V, that is, the relation ya 2 Ua implies ((g ga)(xa)(f), (g ga)(ya)(f)) 2 V for all f 2 X. Thus g ga is continuous at xa. Therefore g ga is continuous for all a 2 I, and hence g is continuous by Proposition 6, p. 31 of [1]. Consequently, X is equicontinuous, proving (a). This completes the proof. h Corollary 2. Let (Xa)a2I, X and (ga)a2I be as in Theorem 1. Let (Yk)k2J be a non-empty family of uniform spaces, Y a set and, for each k 2 J, let hk be a mapping from Y into Yk. Consider Y endowed with the initial uniform structure for the family (Yk, hk)k2J [1, Chap. 2, Section 2, no. 3]. Then, for each set X of mappings from X into Y, the following conditions are equivalent: (a) X is equicontinuous; (b) hk X ga ¼ fhk f ga ; f 2 Xg is equicontinuous for all (a, k) 2 I · J.
ga
X " Xa
X
! !
hk Xga
Y # Yk
hk
Proof. Obviously, (a) implies (b), because ga is continuous for all a 2 I and hk is uniformly continuous for all k 2 J. In order to prove that (b) implies (a), let a 2 I be given. Since, by hypothesis, hk ðX ga Þ is equicontinuous for all k 2 J, Proposition 3, p. 27 of [2] ensures that X ga is equicontinuous. Therefore X is equicontinuous by Theorem 1, proving (a). h Corollary 3. Let (Xa)a2I, X and (ga)a2I be as in Theorem 1. Let T be a set, M a set of subsets of T, Z a uniform space, FðT ; ZÞ the set of all mappings from T into Z and H FðT ; ZÞ. Consider H endowed with the uniform structure of Mconvergence [2, Section 1, no. 2]. Then, for each set X of mappings from X into H, the following conditions are equivalent: (a) X is equicontinuous; (b) fðf ga ÞjA; f 2 Xg is equicontinuous for all ða; AÞ 2 I M. Proof. For each A 2 M let hA be the mapping u 2 H 7!ujA 2 FðA; ZÞ; and consider FðA; ZÞ endowed with the uniform structure of uniform convergence. Since the uniform structure of Mconvergence on H is the initial uniform structure for the family ðFðA; ZÞ; hA ÞA2M , the result follows immediately from Corollary 2. h Corollary 4. Let (Xa)a2I, X and (ga)a2I be as in Theorem 1. Let (Gk)k2J be a non-empty family of topological groups, G a group and, for each k 2 J, let hk: G ! Gk be a group homomorphism. Consider G endowed with the initial topology for the family (Gk, hk)k2J ([1, Chap. 1, Section 2, no. 3]; G is a topological group by Theorem 1.9 of [11]). Then, for each set X of mappings from X into G, the following conditions are equivalent: (a) X is equicontinuous; (b) hk X ga is equicontinuous for all (a, k) 2 I · J. Proof. Since the uniform structure derived from the topology of G is the initial uniform structure for the family (Gk, hk)k2J (each Gk being endowed with the uniform structure derived from the topology of Gk), the result follows immediately from Corollary 2. h
214
D.P. Pombo Jr. / Chaos, Solitons and Fractals 32 (2007) 212–214
Corollary 5. Let (Xa)a2I, X and (ga)a2I be as in Theorem 1. Let F be either a locally convex space over R or C or a locally convex space over a non-Archimedean non-trivially valued complete field, and let C be a set of seminorms defining the topology of F [6,7]. For each q 2 C let Fq be the vector space F seminormed by q and 1q: F ! Fq the identity mapping. Then, for each set X of mappings from X into F, the following conditions are equivalent: (a) X is equicontinuous; (b) 1q X ga is equicontinuous for all (a, q) 2 I · C. Proof. Since the topology of F is the initial topology for the family (Fq, 1q)q2C, the result follows immediately from Corollary 4. h Remark 6. (a) Corollary 5 is applicable, in particular, to the locally convex spaces F over C which occur in the theory of distributions [9], some of which play an important role in the study of the physical sciences [8], and also to the locally convex spaces F over C which occur in p-adic mathematical physics and p-adic quantum mechanics [5,10]. (b) Corollary 5 is applicable, in particular, to the locally convex space F ¼ Qp over Qp (Qp being the field of p-adic numbers), whose relevance in E-infinity theory [3] has been emphasized in [4].
References [1] Bourbaki N. Topologie ge´ne´rale. Actualite´s Scientifiques et Industrielles, Quatrie`me e´dition, vol. 1142. Hermann; 1965 [Chapitres 1 et 2]. [2] Bourbaki N. Topologie ge´ne´rale. Actualite´s Scientifiques et Industrielles, Deuxie`me e´dition, vol. 1084. Hermann; 1967 [Chapitre 10]. [3] El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy physics. Chaos, Solitons & Fractals 2004;19:209–36. [4] El Naschie MS. A few hints and some theorems about Witten’s M theory and T-duality. Chaos, Solitons & Fractals 2005;25: 545–8. [5] First international conference on p-adic mathematical physics, Proceedings of the Steklov Institute of Mathematics, vol. 245, 2004. [6] Robertson AP, Robertson W. Topological vector spaces. 2nd ed. Cambridge University Press; 1973. [7] Schikhof WH. Locally convex spaces over nonspherically complete valued fields. Bull Soc Math Belg Se´r B 1986;38:187–224. [8] Schwartz L. Me´thodes mathe´matiques pour les sciences physiques. Enseignement des sciences, Deuxie`me e´dition, vol. II. Hermann; 1965. [9] Schwartz L. The´orie des distributions. Publications de l’Institut de Mathe´matique de l’Universite´ de Strasbourg, vol. IX–X. Hermann; 1966. [10] Vladimirov VS, Volovich IV, Zelenov EI. P-adic analysis and mathematical physics. Series on Soviet and East European mathematics, vol. 1. World Scientific; 1994. [11] Warner S. Topological fields. Notas de Matema´tica, vol. 126. North-Holland; 1989.