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C AHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES M ARIA C RISTINA P EDICCHIO FABIO ROSSI Monoidal closed structures for topological spaces : counter-example to a question of Booth and Tillotson Cahiers de topologie et géométrie différentielle catégoriques, tome 24, no 4 (1983), p. 371-376 <http://www.numdam.org/item?id=CTGDC_1983__24_4_371_0> © Andrée C. Ehresmann et les auteurs, 1983, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Vol. XXI V - 4 (1983 ) CAHIERS DE TOPOLOGIE GÉOMÉTRIE DIFFÉRENTIELLE ET MONOIDAL CLOSED STRUCTURES FOR TOPOLOGICAL SPACES: COUNTER-EXAMPLE TO A QUESTION OF BOOTH AND TILLOTSON by Maria Cristina PEDICCHIO and Fabio ROSSI * INTRODUCTION. problem of studying monoidal clo sed structures [2] on the category Top of topological spaces and continuous maps has interested many authors (see, for instance, Wyler [6], Wilker [5] , Greve [3], and Booth & Tillotson [1]). In particular, the method of Booth & Tillotson is to topologize, by an arbitrary class d of spaces, both the set of continuous maps and the set-product of topological spaces (A-open topology and d.product, respectively). If A satisfies suitable conditions, then the d-open topology and the 8-product are related by an exponential law and determine a monoidal closed structure on Top ([I], Theorem In recent years, the , 2.6, page 42). In the list of their for sequential spaces, examples, the authors study the case, important of A = {N,oo}, where N 00 is the one-point compact- ification of the discrete space N of natural numbers. This class does satisfy the hypothesis of Theorem 2.6 of [I.]. not question they raise is to determine whether this 8-structure is monoidal closed on Top ([1], Section 6 (i), page 46). The aim of our note is to answer this question negatively, proving, by an example, that the considered structure is not monoidal closed. To obtain this result we will use some properties, due to Isbell [4], that we recall in Section 1. One 1. NOTATION. X, Y, will Z hom ( X , Y ) will indicate the X i. e. * Y. 1 will stand for to a Work space consisting a of set always topological of all continuous functions singleton two denote space, points partially supp orted by the Italian 0,1 C N R . 371 spaces and (maps) from and 2 for a Sierpinski space, with 0, {1}, {0, 7 ! as its open sets. Let first summarize us notions and results stated in some [1] and open set [4] . Lest 8 DEFINITION in Z, then be an I. 1 [1]. W(f, U) arbitrary If A E class of A, f : will denote the topological A - Y is a following spaces. map and U is subset of an horn ( Y, Z ) : By requiring the family of all these sets to be an open subbase, a topology on hom ( Y, Z , called the Q-open topology. we intro- duce The M A( - , - ) function space will be denoted by bifunctor from To pop X Top to Top . corresponding is a DEFINITION 1.2 the set [1]. We define the X X Y with the final (f-product X X A Y topology with respect to M A ( Y, Z ) ; of X and all Y incoming as maps of the forms : It is easy to see DEFINITION 1.3 that - [1]. class if for each AE neighborhoods in (i . which there i s (t - is a bifunctor from The class 3 A every are point of A has closed in A and 1.4 Furthermore the exponential is a continuous bijection. Top X Top of spaces will be said are [1]. I f A is a regular adjunction PROPOSITION an x law 372 a to to Top. be regular fundamental system of epimorphic images class a of spaces of compact spaces, then [41. A topology on the set Op Y of open sets of a topological space Y is a topological topology iff it makes finite intersection and arbitrary union continuous operations. DEFINITION 1.5 PROPOSITION F - 1.6 (4]. If then G (2)= Y* is G, F and G adjoint endofunctors of Top topological topology on the set Op Y, where a are Y = F(1). Conversely, if Y is an arbitrary space and Y* is a topological lapology on Op Y, then there is (up to isomorphism) precisely one pair of adjoint endofunctors F-l Y and G(2) = Y*. Z) is hom ( Y’, Z ) with the initial topology with respect Moreover G ( all the functions of the form where BU(f) Now, F(1) = Top - Top such that G: = f- 1 (U) we are able to for each f E hom ( Y, Z) . to prove the following proposition, of compact spaces, then on Op Y. a) For any Y c Top , M(i( Y, 2) is a topological topology Y* with b) Mél( Y, Z) is the set hom ( Y, Z) with the initial topology respect to all the functions PROPOSITION 1. 7. where BU(f) = c) Let Wo If A is regular class f -1(U), f hom(Y, Z). {Ua} be family o f open E = a on and let Y, Z > Hom ( Y, Z) with respect to all the functions is the set o f all finite intersec tions of Ua topology (see b). 1 f W set o f all arbitrary denote the initial Xua’ Ua E W 0 and W2 is the a unions of sets of Z W0 Va, then the initial topologies coincide. d) We topologize the all the sets set Op ( X X (f Y), the subbasic open of the form 373 sets being We denote this 1f the topology by class (i determines For any X , Y E a monoidal closed structure Top , on we have Top, then, ne- cessarily PROOF. a and b follow from 1.4 and 1.6. c) It is obvious that Conversely, is let U E W2 , then . It follows that commutative, where identity function of hom (Y,Z). Since Y* is a the union operation U is continuous, then B U. « 1» is also continuous for each U E W2 . It follows, from th e universal property of initial topology, that «1>> is continuous. A similar proof applies to the family W1 . d) From 1.3 the exponential law is a continuous bijection A is the canonical diagonal map and topological topology, From b and c, M (t( X, Y*) «1» the has the initial topology with respect to all the maps where V is a subbasic open set of Y*. It is easy 374 to see that M (1( X, Y*) and X X AY> commutative 2. Let d homeomorphic. diagram : are be the class whose only We obtain the thesis from the following element is compact- N 00’ the one-point ification of the discrete space N of natural numbers. In this section we shall prove that the d-open topology and the 8- determine a monoidal closed structure on Top ; this gives product a negative answer to the question posed by Booth and Tillotson in [1 , Section 6, Example ( i ) page 46. Clearly, for 1.7 d it suffices to find two spaces X and Y such that do not I Noo, it is easy to see that N, x (i N. = N 00 x N 00 ’ where x is the topological product. From the definition of the A-open topology, the open sets of (Noo x Noo) * are generated by the sets of the form Let X = Y = where s is any continuous map Noo- Noo. For 1.7 d the subbasic Noo - Noo x Noo’ with comp onents open sets of Noo x Noo> are all the r, t: sets of the form where r , t E hom ( Noo, Noo) . ever, it is not open in (Noo x Noo)* ider a is clearly for every non-empty open set in one-point Th e ( Noo x Noo )*, contains elements other than basic open set N DO x N 00. To see this, cons- m is 00. set Choose m E N such that each ri (oo) for 1 375 i b is either or Th en each has only some n a finite number of in points common E N such that ( m , n ) lies in contains N.)Q x Noo / I (m, n )} . This no {m} J x N 00 . with Thus the open si ( Noo) . completes the So there is proof. set I 0 li-structure, for A = {Noo}, is not monoidal closed it fold-product is not associative and 0-1 is not continuous; fur- Since the lows that the ther, M A ( Y, Z) ( cs-open topology is generally different from topology) 8, Example (VII), convergent sequence-open = ([1], MCHC ( Y, Z ) Section page 50 and Section 9 ( i ) page 52). The authors would like suggested to thank Max to Kelly for the improvements the first version of the paper. REFERENCES. J. TILLOTSON, Monoidal closed, cartesian closed and convencategories of topological spaces, Pacific J. Math. 88 ( 1980), 35 - 5 3. 1. P. BOOTH & ient KELLY, Closed categories, Proc. Conf. Algebra (La Jolla, 1965 ), Springer 1966, 421- 562. 2. S. EILENBERG & G. M. 3. G. GREVE, How many monoidal closed Math. 34, Basel ( 1980 ), 538-539. 4. J. R. ISBELL, Function spaces and structures categories there in Categorical Top : Arch. adjoints, Math. Scand. 36 (1975 ), 317-339. 5. P. WILKER, Adjoint product and hom functors in Math. 34 (1970), 269- 283. 6. O. WYLER, Convenient 225-242. are on for general topology, Pacific J. topology, Gen. Top. and Appl. 3 (1973), Istituto di Mate mati ca Universita di Trieste Piazzale Europa 1 34100 TRIESTE. IT AL IE 376