Exponential objects and Cartesian closedness in the ... - Springer Link

Applied Categorical Structures 1: 345-360, 1993. (g) 1993 Kluwer Academic Publishers. Printedin the Netherlands.

345

Exponential Objects and Cartesian Closedness in the Construct Prtop E. L O W E N - C O L E B U N D E R S and G. SONCK*

Departement Wiskunde, Vrije UniversiteitBrussel, Pleinlaan 2, 1050 Brussel, Belgium; * Aspirant NFWO (Received: 14 May 1993; accepted: 3 July 1993) Abstract. We give an internal characterization of the exponential objects in the construct Prtop and investigate Cartesian closedness for coreflective or topological full subconstructs of Prtop. If $ is the set {0} U {1; n > 1} endowed with the topology induced by the real line, we show that there is no full coreflective subconstruct of Prtop containing $ and which is Cartesian closed. With regard to topological full subconstructs of Prtop we give an example of a Cartesian closed one that is large enough to contain all topological Frtchet spaces and all TI pretopological Frtchet spaces. Mathematics Subject Classifications (1991). 54B30, 18D15, 54A05. Key words: Pretopological space, finitely generated, Frtchet space, sequential space, exponential object, Cartesian closedness.

1. I n t r o d u c t i o n

The well-known fact that the well-fibred topological construct Prtop is not Cartesian closed, has led to the investigation of better behaved subconstructs or superconstructs. Whereas the situation for superconstructs is well understood, only partial results are available with respect to subconstructs. The problem of f n d i n g Cartesian closed epireflective subconstructs of Prtop was settled by Schwarz in [15]: such a category is trivial, i.e. consists only of indiscrete spaces. Here we treat subconstructs which are coreflective in Prtop. In the process of looking for coreflective Cartesian closed subconstructs we first give a characterization of the power in Prtop with basis Y and exponent X , which is actually the Prtop-bireflection of the pseudotopological space ( C ( X , Y ) , c) consisting of the set of all continuous functions from X to Y , endowed with continuous convergence. Using this description we are able to characterize the exponential objects in Prtop. They are exactly the finitely generated pretopological spaces. This result disproves the conjecture formulated in [16] that the exponential objects in Prtop can be characterized by some filter-theoretic description of core-compactness. The full subconstruct of Prtop whose objects are the finitely generated pretopological spaces is coreflective in Prtop; it is in fact isomorphic to the construct Rere [1]. If we look for a larger and more useful subconstruct, it is reasonable to include the topological space $ consisting of the set {0} U {~; n _> 1} endowed with the

346

E. LOWEN-COLEBUNDERS AND G. SONCK

topology induced by the real line. We will prove that there is no coreflective full subconstruct of Prtop containing $ which is Cartesian closed. This result implies for instance that the full subconstruct FrPrtop of Prtop whose objects are the Fr6chet spaces is not Cartesian closed and that the subconstructs consisting of compact Hausdorff pretopological spaces or locally compact pretopological spaces are not exponential in Prtop. Finally, in case the full subconstruct of Prtop is only required to be topological, we present a positive result by constructing a bireflective subconstruct of FrPrtop containing all T1 Fr6chet pretopological spaces and all topological Fr6chet spaces that is Cartesian closed. The following notational conventions will be adopted. T'(X) will denote the power set of a set X. When z is an element of a set X, we shall denote ~ the ultrafilter on X generated by the subset {z}. The same notation will be used for the constant sequence zW ~ X : n --+ x. If f : X ~ Y is a map between sets, and .T is a flter on X, f (.T) is the filter on Y generated by the sets f ( F ) with F E .T. If ~ is a sequence in a set X, we put ~,~ = ~(n) for all n E ~r, and ~ will also be denoted by ((n) ; .T(~) is the Fr6chet-filter of ~ on X, i.e. the filter generated by the sets {~n; n _> k} with k E ZTV. Finally, we agree to indicate a structured set often by its underlying set only, or by its structure only. Categorical terminology follows Ad~imek, Herrlich, Strecker [1]. Reference for the results on Pstop, the construct of all pseudotopological spaces and continuous maps, can for instance be found in the survey paper Herrlich, Lowen-Colebunders, Schwarz [9]. In that paper further reference to the original sources is given. For results on L*, the construct of sequential spaces satisfying the Urysohn-axiom, good references are the survey papers Fri~, Koutnik [5] and [6] where further reference to the original sources can be found. Original sources on Cartesian closedness are [7] and [10]. We recall some of the notions that will frequently be needed in the sequel. A pseudotopology on a nonempty set X is a function q assigning to each element x of X a set of proper filters on X such that the following properties are Satisfied:

(P1) Vx E X : & E q(x) (P2) If J7z and ~ are filters on X with :7: C ~ and ~ E q(x), then ~ E q(x) (P3) jz E q(x) wheneverLt E q(x) for all ultrafiltersLt on Z with F C Lt X

(X, q) is called a pseudotopological space. We also write .T" ~ x, 9r ---+x or.T" ~ x instead of .%" E q(x), and we say that .T" converges to x (in q). A continuous map between pseudotopological spaces X and Y is a function from X to Y that preserves convergence: if .T x x then f(iT') Y f(x). The category of pseudotopological spaces and continuous maps is denoted by Pstop. It is a well-fibred topological construct and it is Cartesian closed. Cartesian closedness means that the functor

EXPONENTIAL OBJECTS AND CARTESIAN CLOSEDNESS

X ×-

347

: Pstop --~ Pstop has a right adjoint. In the context of Pstop this is

characterized by the existence of canonical function spaces: for each pair X, Y of pseudotopological spaces the set C(X, Y) of all continuous functions from X to Y can be endowed with a pseudotopological structure c(X, Y) (called continuous convergence and also simply denoted c), such that a. the evaluation map evx,y : X × (C( X, Y), c) --~ Y is continuous (where × denotes the product in Pstop) b. for each pseudotopological space Z and each continuous h : X × Z --~ Y the map h* : Z -~ (C(X, Y), c) defined by h*(z)(x) = h(x, z) is continuous. An explicit description of c is given by: a filter ~ on C(X, Y) converges to f E C(X, Y) in c if and only if for all x E X and all filters .T on X converging to x in X we have evx,y(.T × ~) --~ f ( x ) in Y, where .T × ~ is the usual product of filters. With regard to the natural order on the set of all pseudotopologies o n a set X our notation deviates from the one used in [1] and [16]; ifp and q are pseudotopologies on X we put p

_

1

}

aL(pz)®CY_AL(Pz)

C,,(0,0)

-

Ly

xA 0 .

Now the first case can't occur since (1/k(n)) does not converge to 0 in L:y and the second case can't occur either since for all n we have 1/k(n) f[ aL(pz) ~o • c. (I/m} converges to 0 in (Z, pz) and i/m converges to 0 in (Y, py) for all m. Then, (Y, py) being the Prtop-quotient of the Frrchet space (X, Px), it is itself a Fr6chet space. So we can conclude that (i/m) also converges to 0 in (Y, py). But then ( ( l / m , 1/m))~>_l converges to (0,0) in (Z × Y, pz[:3py), and then also in (Z x Y, pz[]cpy). [] 4.9. COROLLARY. FrPrtop is not Cartesian closed.

[]

4.10. COROLLARY. There is no exponential full subconstruct of Prtop that con-

tains $. Proof This follows immediately from Theorem 5 in [14].

[]

In contrast with the situation in Top the subconstructs of compact Hausdorff pretopologies and locally compact pretopologies, although finitely productive in Prtop, are not exponential in Prtop. 5. Cartesian Ciosedness for Topological Full Subconstructs of Prtop In order to give an example of a Cartesian closed topological full subconstruct of Prtop which is larger than the one consisting of all finitely generated spaces, we

358

I~. LOWEN-COLEBUNDERS AND G. SONCK

use a strengthening of the axiom (F). This stronger axiom was introduced in [2]. 5.1. DEFINITION. A sequential convergence space (X,/Z) i s said to be strongly Fr~chet if it satisfies the condition

for all sequences ~, r/in X and all x E X. Remark that every T1 sequential space (5: -+ y ~ x = y) and every space (X, L(p)) (with (X, p) a topological space), satisfies ( S F ) and that every strongly Fr6chet sequential space is a Fr6chet sequential space. We denote by SFL* the full subconstruct of L* whose objects are precisely the strongly Fr6chet sequential spaces. 5.2. PROPOSITION. SFL* is finally dense and bireflective in L*. Proof This follows immediately from the well-known fact that the space $ (considered as a sequential space) is finally dense in L*, and since $ is T1, it is a strongly Fr6chet sequential space. It is moreover easily seen that L*-initial [] structures preserve the property (SF). 5.3. PROPOSITION. If X and Y are L*-spaces and Y is a strongly Frgchet space, then ( C (X, Y ) , P) (where r is the canonicalfunction space-structure on C (X, Y ) ) also is a strongly Frdchet space. Proof Let (fn) and (g~) be sequences in C(X, Y), and suppose that P --..+

g

and V n E J~V : fn "r' +

gn.

Then suppose (¢n} ~ x. Since ~n ~ Cn, we have f~('~n) ~ g~(¢~) for every n C ZW. Moreover, (gn(~,~)) --~ g(x). Since Y is a strongly Fr6chet sequential space, this implies that (f~(~,~)) ~ g(x). [] From 5.2 and 5.3 we have the following result: 5.4. THEOREM. SFL* is a Cartesian closed topological subconstruct of FL*. [] Applying the functor P to SFL* we obtain the next result. 5.5.THEOREM. The full subconstruct of Prtop whose objects are the Frdchet pretopologies in which the convergence of sequences satisfies ( SF) is Cartesian closed and topological It is a bireflective subconstruct of FrPrtop containing all

EXPONENTIAL OBJECTS AND CARTESIAN CLOSEDNESS

359

T1 Frdchet pretopologies and all topological Frdchet spaces (and in particular $). [] If however we impose on a topological subconstruct of Prtop to be finitely productive in Prtop and to contain $, then it cannot be Cartesian closed. This follows from an argument which is quite similar to the one used by Herrlich in [8]. 5.6. THEOREM. No fMl topological subconstruct of Prtop which contains $ and

is closed under the formation of squares in Prtop is Cartesian closed. Proof Suppose C is a full topological subconstruct of Prtop that contains $ and is closed under the formation of squares in Prtop. We can repeat the argument used in (5) of Proposition 6 in [8] to conclude that the assumptions made on C imply that C contains all zerodimensional topological Fr6chet spaces. In particular, Qis a C-object. We identify all natural numbers in Qby setting Q = ~ f v - u {oo} (where oo ~ Q) and f : Q ~ Qthe function defined by f(x) = x for x E Q\EV and f(x) = oo for x E zW. We endow Qwith the pretopological quotient-structure for f . Remark that this structure coincides with thetopological quotient-structure on Qand that Qis zerodimensional and Fr6chet, so Qis a C-object too and f • Q --+ is a quotient in C. Since C is closed under the formation of squares in Prtop, also Q x Qand Q x belong to C. In order to prove that f x f • Q x Q--+ Q x Qis not a quotient in C, our argument slightly differs from the one used in Herrlich [8]. Let (rn) be a sequence of irrationals in ]0, 1[ strictly decreasing and convergent (for the usual topological structure on ]0, 1[) to 0. For n _> 1 let (sn,,~)m_>2 be a sequence of rationals in ]rn, r,~_ 1 [ strictly decreasing and convergent to rn. Every point (Sn,m,n -Jr-l/m) in Q x Qwith n _> 1 and m _> 2 will be surrounded by a small rectangle in the following way: for n _> 1 and m _> 3 let OZn,m,1 O~2n,m,OZn,2,2 1 2 2 /3~, m , fi/z,,,~, flz,2 be irrational numbers such that

{

8n,m ~ Oln,rn ~ Otn,m ~ 8n,m--1 1

forn>

2 _l,m>2

and B = f x f(A). The set A is open andclosed in Q x Qand it is saturated for f × f. The set B however is not closed in Q x Q since

(oo, ec) E (clg~xg~B)\B.

360

E. LOWEN-COLEBUNDERS AND G. SONCK

N e x t consider g • Q x Q --+ {0, 1 } where {0, 1} is discrete and hence belongs to C and where g is 0 on B and 1 outside B. Then g o ( f x f ) is a m o r p h i s m in C but 9 is not a morphism. Finally we can conclude that f x f is not a quotient in C. This implies the result that C is not Cartesian closed. []

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

J. Adfimek, H. Herrlich, and G.E. Strecker: Abstract and Concrete Categories, John Wiley (1990). P. Antosik: On a topology of convergence, Colloq. Math. 21 (1970), 205-209. G. Choquet: Convergences, Ann. Univ. Grenoble Sect. Sei. Math. Phys. 23 (1948), 57-112. B.J. Day and G.M. Kelly: On topological quotient maps preserved by pullbacks or products, Proc. Camb. Phil. Soc. 67 (1970), 553-558. R. Fri~ and V. Koutn~: Sequential convergence: iteration, extension, completion, enlargement, in Recent Progress in General Topology (1992), 201-213. R. Fri~ and V. Koutn~: Sequential structures, Abh. Akad. Wiss. DDR 4 (1980), 37-56. H. Herrlich: Cartesian closed topological categories, Math. Colloq. Univ. Cape Town 9 (1974), 1-16. H. Herrlich: Are there convenient subcategories of TOP? Topology Appl. 15 (1983), 263-271. H. Herrlich, E. Lowen-Colebunders, and F. Schwarz: Improving TOP: PRTOP and PSTOP, pp. 21-34, Category Theory at Work, Heldermann Verlag (1991). H. Herrlich and L.D. Nel: Cartesian closed topological hulls, Proc. Amer. Math. Soc. 62 (1977), 215-222. K.H. Hofmann and J.D. Lawson: The spectral theory of distributive lattices, Trans. Amer. Math. Soc. 246 (1978), 285-310. D.C. Kent: Decisive convergence spaces, Fr6chet spaces and sequential spaces, The Rocky Mountain J. of Math. 1 (1971), 367-374. V. Koutnfk: Many-valued convergence groups, Polska Adad. Nauk (1980), 71-75. L.D. Nel: Cartesian closed coreflective hulls, Quaestiones Math. 2 (1977), 269-383. E Schwarz: Cartesian closedness, exponentiality and final hulls in pseudotopological spaces, Quaestiones Math. 5 (1982), 289-304. E Schwarz: Powers and exponential objects in initially structured categories and applications to categories of limit spaces, Quaestiones Math. 6 (1983), 227-254. O. Wyler: Function Spaces in Topological Categories, volume 719 of Lecture Notes in Math. (1979), 411-420.