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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 <_ q if and only if 1x : (X, q) --~ (X, p) is continuous; q is said to
be finer (or larger) than p and p coarser (or smaller) than q. This notation conforms
the original paper by Choquet [3] and later literature on convergence theory.
Prtop is the full subconstruct of Pstop whose objects are the pseudotopological
spaces (X, q) satisfying the strengthening (P3 ') of axiom (P3) above:
(P3') Vx E X : ['lq(x) E q(x),
so-called pretopological spaces. We put V(X,q)(X) = N q(x) and call );(X,q)(X)
(also denoted F x (x), Vq (x) or simply V(x) the neighborhoodfilter in x in the space
(X, q). Axioms (P2) and (P3') imply that a pretopological space is completely
determined when the neighborhoodfilter in each point is given. If (X, q) is a
pretopological space and A is a subset of X, we say that V C X is a neighborhood
of A if V E ~;(x) for all x E A.
In any pretopological space X a Cech-closure operator cl is defined by
x E cI A .c~ '¢V E V(x) : V N A ~ O
and the other way around, a Cech-closure operator cl on X determines a unique
pretopology on X by
V(x) = {V C X :
x ¢ c/(X\V)}.
Sometimes we will be concerned with convergent sequences in a pretopological
space X: if ~ is a sequence in X, we say that ~ converges to x E X if V(x) C 3v(~)
X
and denote this ~ ~ x.
Prtop is finally dense and bireflective in Pstop. In fact any pseudotopological space (X, q) is the infimum in Pstop of the set of all pretopological spaces
(X, p) with p finer than q. If (X, q) is a pseudotopological space, the bireflection
l x : (X, q) --~ (X, F(q)) of (X, q) in Prtop is defined by
.T ~ x in F(q) ~ .T D ['1 q(x).
348
E. LOWEN-COLEBUNDERSAND G. SONCK
Prtop is a well-fibred topological construct; it inherits the products of Pstop. Prtop
has an initially dense object, namely.the.space 3 with unde.rlying set {0,1,2} and
neighborhoodfilters V(0) = )2(2) = 0 M 1 fq 2 and )2(1) = 1 N 2.
Another construct, although not directly related to Prtop, will be used extensively in several proofs of results about Prtop. It is the construct L* of sequential
convergence spaces satisfying the Urysohn-axiom. A sequential convergence space
(X, £) consists of a nonempty set X and a relation £ C X nv × X satisfying the
following conditions:
(L1) Vz e X • (3c,z) E £
(L2 ) If ( ~ , z) E C and ~ o k is a subsequence of ~, then ( ~ o k, :c) E ~.
(L3) If ~ is a sequence in X and z E X such that every subsequence ~ o k of ~ has
a subsequence ~ o k o I with (~ o k o l, :c) E £ then (~, z) E C
£ is then called a sequential convergence on X. The third axiom is the Urysohn£
axiom. As usual, (~, z) E Z~is written ~ ~ z and we say that ~ converges to z (in
Z~). A function f • (X, Z~) ---+ ( X I, £~) between sequential convergence spaces is
£1
sequentially continuous if it preserves convergence: if ~ • z then f o ~ ~ f ( z ) .
The construct of all sequential convergence spaces and sequentially continuous
maps is denoted by L*. It is a well-fibred topological construct. If (fi : X
(Xi, £i))~eI is a source in L*, then the initial structure £ on X is defined as
follows: a sequence ~ in X converges in £ to z if and only if for all i ¢ I we have
fi o ( -~ A(x).
When (X, £) and (X', fJ) are sequential convergence spaces, their product in
L* is denoted by (X × X ~,/~ ® Z:').
If (X, £) is a sequential convergence space and f • X --~ X ~is surjective, then
f • (X, £) --~ (X ~, £') is a quotient in L* if and only if the following condition
holds for all sequences r/in X ~ and all elements x ~ of XI: ~ converges to x ~ in fJ if
and only if for every subsequence 71o k ofT1 there exist x E f -1 ( x') and a sequence
in X such that ~ converges to x in f-, and such that f o ~ is a subsequence o f t o k.
L* is Cartesian closed. For each pair (X, £), (X ~, £:') of sequential convergence
spaces the set C ( X , Y ) of all sequentially continuous functions from .(X, £) to
(X', £') can be endowed with a sequential convergence P(X, Y) (or simply F)
defined by: a sequence (f~) in C ( X , Y ) converges to f E C ( X , Y ) in P(X, Y)
if and only if (fn(~n)) converges to f ( x ) in £' whenever ~ converges to x in
£. The sequential convergence F(X, Y) is called continuous convergence and
(C(X, Y), F(X, Y)) is the canonical function space of (X, £) and (X',/::') in L*.
2. Internal Characterization of the Power in Prtop
In the general context of topological constructs, Wyler introduced the notions of
admissible and proper structures [17]. We recall their definitions when applied to
the particular case of Prtop.
EXPONENTIAL OBJECTS AND CARTESIAN CLOSEDNESS
349
2.1. DEFINITION. If X and Y are pretopological spaces, C(X, Y) will denote
the set of all continuous functions from X to Y, evx,y " X x C(X, Y) ~ Y
is the evaluation function and for a continuous map h • X x Z ~ Y (with Z
a pretopological space), the function h* • Z ~ C(X, Y) is defined as in the
introduction. Now we call a pretopological structure v on C(X, Y)
admissible if ex,g " X x (C(X, Y), v) ~ Y is continuous,
proper if for all pretopological spaces Z and all continuous maps
h • X x Z ~ Y the map h* • Z ~ (C(X, Y), v) also is continuous.
Remark that using the order on the set of all pseudotopological structures on
C(X, Y) the previous notions can be expressed as follows:
v is admissible if and only if c < v,
v is proper if and only if v <__c.
If we denote by ( C ( X , Y ) , E ( X , Y ) ) the pretopological bireflection of
(C(X, Y), c(X, Y)), then E ( X , Y) obviously is proper. So it is clearly the largest
proper pretopological structure on C(X, Y). Following Schwarz [16] we call
(C(X, Y), E ( X , Y)) the power in Prtop with basis Y and exponent X. Sometimes we write E instead of E ( X , Y). Moreover since the structure c of continuous
convergence is the infimum in Pstop of the set of all admissible pretopologies on
C(X, Y), the structure E being the bireflection, is the infimum in Prtop of the same
set. So we can conclude that E coincides with the infimum in Prtop of the set of
all admissible pretopologies on C(X, Y).
In order to give an internal description of the powerobject we relate the continuous functions from a pretopological space X to 3 to couples in 7'(X).
2.2. NOTATIONS. Let X be a pretopological space. We put
O ( X ) = { (A, V) E (7'(X))2; V is a neighborhood of A }.
Further let r/s : C(X, 3) --~ O(X) be the function defined by
~lx(f) = ( f - l ( 1 ) , f - l ( { 1 , 2 } ) ) ,
then clearly r/x is a bijection. We put ~x = ~x -1. Let or(X) be the (unique)
pretopology on O(X) such that
: (o(x),
--,
(c(x, 3),
3))
is an isomorphism (in Prtop). In the sequel the pretopology or(X) on O(X) will play
the same role as the Scott-topology on the lattice of all open sets of a topological
space (see [4] and [11]). The idea is to make use of O(Y) in order to find first a
description of continuous convergence on C(X, Y) and then of the powerobject
(C(X, Y), Z(X, Y)).
If X and Y are pretopological spaces and f E C(X, Y) we put
7")y = { ( x , ( B , W ) ) e X x O(Y);x e f - l ( B ) } .
350
E, LOWEN-COLEBUNDERS AND G. SONCK
2.3. PROPOSITION. Let X and Y be pretopological spaces, • a filter on C ( X, Y )
and f E C ( X , Y). The following properties are equivalent:
1. k~ -+ f in c ( X , Y ) ,
2. V ( x , ( B , W ) ) E Pf, 3Jl E if2 such that f-i g - l ( W ) E Vx(x).
ge.4
Proof Since Pstop is Cartesian closed and 3 is an initially dense object in Prtop
it follows from Herrlich, Nel [10] that the source
(~Ph : (C(X, Y), c(X, Y)) -+ (C(X, 3), c(X, 3)))heC(Y,3 )
where ~h(f) = h o f , is an initial source in Pstop. It follows that
~ f in c ( X , Y )
Vh E C(Y, 3) • qOh(~) --+ qoh(f) in c(X, 3)
V(B,W) E O(Y),Vx E f - ' ( B ) " evx,Y(Vx(x) x %w(B,W)(~)) -+ 1 in 3
¢
~
**
i~
v(z, (B, W)) ~ PS, 3A E ~, ?V E Vx(x) • evx,y(V x ~y(8,w)(A))
C {1,2}
Now finally observe that e v x , y ( V x ~y(B,W)(¢4)) C {1,2} can be equivalently
expressed by the inclusion V C Agc.4 9-1(W) •
[]
2.4. THEOREM. Let X and Y be pretopological spaces, f E C(X, Y) and
7-g C C(X, Y). Then 7-[ is a E(X, Y)-neighborhood o f f if and only ifT-I satisfies
the following condition: "if for all (x, ( B, W) ) E 72S we choose a neighborhood
V(x, (13, W)) of x inside f - l ( W ) , then there exists a finite subset Q of~PI such
that all 9 E C ( X , Y) that satisfy
V(x,(B,W))
E e
•
V(.,(B,W)) c 9-1(w)
belong to 7-l".
Proof Let 7-I be a E(X, Y)-neighborhood of f, and choose for all (z, (B, W)) E
P I a neighborhood V(m, (B, W)) ofm in X inside f - l ( W ) , and define
B ( m , ( B , W ) ) = {9 E C(X,Y); V ( x , ( B , W ) ) C 9 - ' ( W ) } .
EXPONENTIALOBJECTS AND CARTESIANCLOSEDNESS
35 l
Then we have f E B(x, (B, W)) for all (x, (B, W)) E 7~S and so the family
{13(x, (B, W)); (x, (B, W)) E 79f} of subsets of C(X, Y) generates a filter ¢~ on
C(X, Y). From 2.3 it is easily seen that q? converges continuously to f. So the
pretopology v¢,y on C(X, Y) defined by V(f) = • N f and 12(9) = ) for all
g E C(X, Y ) \ { f } is admissible. Moreover we have
By definition of q~ there exists a finite subset Q of 79y with
N
a x , ( B , w ) ) c ~.
(x,(S,W))EQ
This proves the first implication.
Conversely, we have to prove that if 7-/ satisfies the quoted condition, it is a
E(X, Y)-neighborhood of f, or, equivalently that it is a u-neighborhood of f for
all admissible pretopologies v on C(X, Y). For such a v, we have
v~(f)-~ f in 4 x , Y)
and so
V(x, (B, w)) e vf,~A(x,(B,W)) e V~(f) :
(']
g-~(W) e 12x(x).
geA(x,(B,W))
Thus we can use the supposition with
V(x, (B, W)) -~ N { g - l ( w ) ; 9 e .At(x, (B, W))}
for all (x, (B, W)) E 79f. So we obtain a finite subset Q of 7)I such that all
9 E C(X, Y) that satisfy
V(X,(]~,W)) ~ Q : V ( x , ( B , W ) ) C g - l ( w )
belong to 7-/. This clearly implies
N{A(x, ( B , W ) ) ; ( x , ( B , W ) ) e Q} c
and so "H E 12v(f).
[]
Via the isomorphism ~x : (O(X), ~r(X)) ~ (C(X, 3), 2 ( X , 3)) we can now
use the previous theorem to characterize the pretopology ~r(X) on O(X).
2.5. COROLLARY. Let (A, V) E O ( X ) and 7-( C (9(X). Then ~ is a o'(X)neighborhood of (A, V) if and only if ~ satisfies the following condition: "If for
all x ~ A we choose a neighborhood V~ of x inside V, then there exists a finite
subset A' of A such that all (C, Z) E O(X) that satisfy
352
E. LOWEN-COLEBUNDERS AND G. SONCK
U
xEA t
belong to 7-[".
[]
2.6. COROLLARY. For (A, V), (B, W) E O ( X ) with A C B and W C V we
have
V~(x)(A, V) c V~(x)(B, W).
In particular, ifT-[ is a a( X)-neighborhood of(A, V) E O( X ) then 7-[ contains all
(B, W ) E O ( X ) with A C B and W C V.
[]
3. Exponential Objects in Prtop
We first recall the definition of an exponential object in the construct Prtop: a
pretopological space X is exponential in Prtop if the functor X × - : Prtop --+
Prtop has a right adjoint. Application of Theorems 3.1, 3.2 and 3.3 in Schwarz
[16] gives the following useful characterization of the exponential objects in Prtop.
3.1. PROPOSITION. For a pretopological space X the following are equivalent:
1. X is exponential in Prtop
2. E(X, 3) is admissible on C ( X , 3)
3. ~(X, Y ) is admissible on C ( X , Y ) f o r all pretopological spaces Y
4. c(X, 3) is apretopologyon C(X, 3)
5. c( X, Y ) is a pretopology on C( X, Y ) for all pretopological spaces Y.
[]
In order to give an internal characterization of exponential objects in Prtop we
need the following observations.
3.2. PROPOSITION. For a pretopological space X the following are equivalent:
1. every point in X has a smallest neighborhood,
2. for every point x in X we have ['1V(x) E V(x),
3. ifx E c I A ( w i t h x E X , A C X)thenthereexistsa E Asuchthatx E cl{a}.
[]
3.3. DEFINITION. A pretopological space X is said to befinitely generated if it
satisfies one (and then all) of the properties in 3.2.
3.4. THEOREM. For a pretopological space X the following are equivalent:
1. X is exponential in Prtop
2. X is finitely generated
Proof Suppose X is exponential in Prtop. Take x in X and put
=
({x }, x ) ) .
353
EXPONENTIAL OBJECTS AND CARTESIAN CLOSEDNESS
Then • ---+ ax({x}, X) in c(X,3) because by 3.1, E ( X , 3 ) is admissible. From
2.3 we now know
~A E •
: ['1 {9-1({1,2});9 E .A} E ])(x).
Because ~x(.A) E ])~(x)({x}, X), it follows from 2.6 that ({x}, V) E ~Tx(.A) for
all neighborhoods V of x. This implies ['1V(x) E Y(x) because
NV(x) = N
{(~x({x},V)I-I({1,2});V
E V(x)}
D ["1 {9-1({1,2}); 9 E .A}.
Conversely suppose that X is a finitely generated pretopological space. Then
take an arbitrary pretopological space Y, f E C(X,Y) and { ~ ; / E I} a
family of filters on C(X, Y) all converging continuously to f. In view of 3.1
it suffices to prove that • = AiEz ~I'i converges continuously to f. Therefore we take (x, (B, W)) E 7~y. For all / E I there exists .Ai E ~i such that
Vi = ('1 {9 -1 (W); 9 E .Ai} is a neighborhood of x. Then .A = Ui~I .Ai belongs to
and
N {9-1(w);9 e A} = A
iEI
This is a neighborhood of x, because all 17i are and X is finitely generated.
[]
Let Fing be the full subconstruct of Prtop whose objects are the finitely generated
pretopological spaces. It is easily seen that Fing is bicoreflective in Prtop; the
bicoreflection in Fing o f a pretopological space (X,p) is I x : (X,p ~) --+ (X,p)
where p~ is the pretopology on X in which each element x of X has ~ Vp(x) as
its smallest neighborhood.
The Sierpinski topological space 2, and in fact any finite pretopological space,
is finitely generated. Moreover we have: Fing is the bicoreflective hull of the class
of all finite spaces in Prtop; it is also the bicoreflective hull of the class {2} in
Prtop.
Fing is finitely productive and closed for subspaces in Prtop.
Since Fing is an exponential subconstruct of Prtop, the results of Theorem 5 in
Nel [14] can be applied. This implies:
3.5. PROPOSITION. Fing is Cartesian closed.
[]
When X and Y are finitely generated pretopological spaces, the Fing-funcfion
space is defined as follows: for f E C(X, Y) the smallest neighborhood of f is
given by
The result stated in 3.5 also follows from the observation that Fing is isomorphic
to the topological universe Rere of reflexive relations.
354
E. LOWEN-COLEBUNDERS AND G. SONCK
4. Cartesian Closedness for Coreflective Full Subconstructs of Prtop
In the process of searching for Cartesian closed coreflective full subconstructs of
Prtop that are larger and more useful than Fing, it is reasonable to assume that
they contain as an object the space $, with underlying set {0} tO { n1 ; n > 1 } and
(pre)topological structure induced by the usual topology of the real line.
An important role will be played by the subconstruct of Prtop with as objects
the Fr6chet pretopological spaces. We recall the definition given in Kent [12].
4.1. DEFINITION. A pretopological space X is called a Frgchet pretopological
space if its closure operator is completely determined by the convergent sequences;
more precisely, this means that for each A C X and z C X , x C c I x A if and only
if there is a sequence ranging in A and converging in X to x.
Let FrPrtop be the full subconstruct of Prtop whose objects are the Fr6chet
pretopological spaces. It is easily seen that coproducts in Prtop preserve Fr6chet
spaces; moreover theorem 3 in Kent [12] implies that quotients in Prtop also
preserve Fr6chet spaces. Hence FrPrtop is coreflective in Prtop. An argument
similar to the one used to prove that the bicoreflective hull in Top of {$} is the
subconstruct of Top of all sequential spaces leads to the following result.
4.2. PROPOSITION. FrPrtop is the bicoreflective hull of{S} in Prtop.
Proof Let X be a Fr6chet pretopological space. Let I be the set of all couples
((, x) with x C X and ( a sequence in X converging to x in X. For all i E X, let
Xi be the space $, and let fi : Xi --* X be the function defined by
i f / = (~,x). Then (fi : Xi --~ X)iEI isafinalsinkinPrtop.
[]
4.3. NOTATIONS. When (X, p) and (II, q) are Fr6chet spaces, we use the notation
( X x Y, pt2q) for their product in FrPrtop.
4.4. DEFINITIONS. If E is a sequential convergence on a set X, we define a
pretopological structure P ( E ) on X by taking the closure operator cl defined by
x E cI A ~ 3~ c A zv : ~---* x.
For each sequential space (X, E), P ( E ) is a Frdchet-pmtopology on X.
I f p is a pretopology on a set X, we define a sequential structure L(p) on X in
the usual way by
A sequential convergence space (X, E) is said to be Frdchet if it satisfies the
condition
355
EXPONENTIAL OBJECTS AND CARTESIAN CLOSEDNESS
--+X=~--+X
for all sequences ~ in X and all x E X . F L * is the full subconstruct of L* with as
objects all Fr6chet sequential convergence spaces.
If (X, p) is a pretopological space, ( X , L(p)) is always Fr6chet.
For an L*-space (X,/2), we have £ _> L ( P ( £ ) ) .
Now we define functions t9o : L* ~ F r P r t o p , P • F L * -+ F r P r t o p and
L " F r P r t o p -~ F L * by
Po((X,/2)) = P ( ( X , / 2 ) ) = (X, P(/2))
L((X,p)) = (X,L(p))
and by letting morphisms unchanged as set-functions.
In Koutnfk, [13], the Fr6chet condition (F) is called "axiom A". The following
proposition merely restates Koutnl'k's results in categorical terms.
4.5. PROPOSITION.
a. Po, P and L are fimctors,
b. P is an isomorphism with L as its inverse,
c. L o Po : L* --~ F L * is a bireflector.
[]
For later use we also need a direct isomorphic description of the bireflection of an
L*-space in F L * .
4.6. DEFINITION. If (X,/2) is an L*-space and x E X, we put
A~ =
v ~ X ; i;--+ x
and let Z~be the sequential convergence on X defined by: ~ ~ x if and only if every
£
subsequence of ~ has a subsequence ~ o k satisfying ~ o k ~ x or ~ o k(fV) C A~.
4.7. PROPOSITION. For an L*-space ( X , £), l x
: ( X , £) --* ( X , £ ) is the
bireflection o f ( X , /2) in FL*.
Proof Clearly every sequence that £-converges to a point also £-converges
to that point and a constant sequence £-converges to a point if and only if it £converges to that point. Moreover, (X, £) clearly is an L*-space and satisfies the
axiom (F), so it is a Fr6chet sequential space.
Now let f : (X,/2) --+ (X ~, £~) be a sequentially continuous function to a
Fr6chet-space ( X ~, U). We show that f : (X,/2) ~ (X ~, U ) is again sequentially
continuous. Therefore take ~ c_~ x. An arbitrary subsequence of f o ~ is of the form
f o ~ o k. Either we can find a subsequence ~ o k o l which £-converges to x and
£1
then f o ~ o k o l --+ f ( x ) , or we can find a subsequence ~ o k o l in A~ and
356
E. LOWEN-COLEBUNDERS AND G. SONCK
then f(~(k('/(n)))) ---* f ( x ) for all n e JTV. In both cases we can conclude that
ZY
f o ~ o k o 1 ~ f ( x ) and since (X ~,/Y) satisfies the Urysohn-axiom this finally
£1
implies that f o ~ --, f(x).
1:3
4.8. THEOREM. There is no Cartesian closed coreflective full subconstruct of
Prtop containing $.
Proof Let C be a coreflective full subconstruct of Prtop containing $. From
4.2 it follows that FrPrtop C C. When (X,p) and (Z, q) are C-objects, we
use the notation (X x Z, pDcq) for their product in C. Since for Fr6chet spaces
( X x Z, pC3q) and ( X x Z, pmcq ) are the coreflections of (X x Z, p x q) in FrPrtop
and in C respectively, we clearly have the inequalities
p x q <_pDcq <_pDq.
Next consider the following sets:
U
m>l
u {m+ n
and
1
-;n>l
n
}.
X and Z are endowed with the metrizable topological structures Px and Pz
respectively, induced by the usual topology of the real line.
Let f : X --~ Y be the surjection defined as follows:
Vm_> 1, V n _ > 2 ; f
m+
--~.
Let Y be endowed with the pretopological structure pg for which
f : (X, p x ) ~ ( Y ,
py)
is a quotient in Prtop. On the other hand l e t / : y be the L*-structure on Y such that
f • ( X , L ( p x ) ) -* (Y,£.y)
is a quotient in L*. We will prove:
a. l z x f : (Z x X, pzDcPx) --+ (Z x Y, P(L(pz) ® £ y ) ) is a quotient in C
1
b. The sequence ( ( ~ , ~))m_>l
does not converge to (0,0)in (Z x Y, P ( L ( p z ) ®
cy))
1
c. The sequence ( ( 1 , ~)),~___,
converges to ( 0 , 0 )in
( Z x r, pzmcpy)
Then we can conclude, from b. and c., that the structures pzrncp Y and P(L(pz) 63
ZSy) on Z x Y are different. From a. it then follows that
357
EXPONENTIAL OBJECTS AND CARTESIAN CLOSEDNESS
l z x f • (Z x X, pzDcpx) ~ (Z x Y, pz[3cpy)
is not a quotient in C. Since f • (X, Px) ~ (]1, PY) is a quotient in C, the construct
C is not Cartesian closed. So we only have to prove the assertions a., b. and c.
a. Note that on Z x X the structures Pz x Px, Pz • P x and Pz [:]cPx all coincide.
Since L* is Cartesian closed, we know that
1z x f • (Z × X, L(pz) ® L(px)) --+ (Z x Y, L(pz) ® Ey)
is a quotient in L*. Then FL* being bireflective in L*, we know that L(pz) 63L(px)
is a FL*-structure on Z x X and that the FL*-quotient is obtained by applying
the bireflector L o P0 to the codomain. Using the fact that FL* is isomorphic to
FrPrtop, we can apply the functor P on the domain and codomain and use the
equality P(L(pz) 63 L(px)) : pz[3px.
b. Suppose that ((1/m, 1/m)) mE 1would converge to (0,0) in (Z x ]f, P (L (Pz ) 63
£ y ) ) , then it would also converge to (0,0) in L(P(L(pz) 63/:y)), which coincides
with the structure L(pz) 63 £ y described in 4.6. So there would exist a strictly increasing map k • /N\{0} --+ £V\{0} such that either ((Ilk(n), 1/k(n)))
converges to (0,0) in L(pz) G3£ y or
{(
1
1 )
k(n)' k ~ )
;n>
_
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
<&m
<Zi,m <n+
m--1
1, m_> 3 a n d
Sn,2 < 0z2,2 "( rn-1
<Z ,2<n+l
for n _> 1. Further put
A ----
U
1
1
2
(]Otn,mq_l,
o~2,m[f"lQ) x (]/~n,m+l,/J~,m[nQ)
n>_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.
[]
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