A QUASITOPOS CONTAINING CONV AND MET AS FULL SUBCATEGORIES

. We show that convergence spaces with continuous maps and metric spaces with contractions, can be viewed as entities of the same kind. Both can be characterized by a "limit function" % which with each filter associates a map % from the underlying set to {ne extended positive real line. Continuous maps and contractions can both be (htacterized as limit function preserving maps. lhe properties common to both the convergence and metric case serve as a basis for tne definition of the category, CAP. We show that CAP is a quasitopos and that, apart from the categories CONV, of convergence spaces, and MET, of metric spaces, it also contains the category AP of approach spaces as nicely embedded subcategorles.

I. INTRODUCTION.In [17] the categories TOP of topological spaces and continuous maps and pq-MET of extended pseudo-quasi-metric spaces and non-expansive maps were embedded in a common supercategory.The idea behind this embedding being that topological spaces and metric spaces can be viewed as objects of the same type, in the sense that they both can be described by a "distance between points and sets".Starting with a pq-MET space (X,d) this distance is the usual one given by 6(x,A) := inf d(x,a).Starting with a topolo- aeA gical space (X,) a distance can be defined by 6(x,A) := 0 if x A and 6(x,A) := if x A. A notion of distance has been axiomatized in [17] in such a way as to general- ize both the metric and topological cases and resulted in the definition of the category AP of approach spaces and contractions.
There are several advantages to this.
In the first place that of unification, e.g. the notions of compactness (in TOP)   and of total boundedness (in pq-MET) which turn out to be special cases of a measure of compactness in AP [18] and which in turn makes a concept introduced by C. KURATOWSKI in [16] a canonical categorical notion.A similar situation presents itself for the notions of connectedness (in TOP) and Cantor's "kettenzussamenhang" (in MET) [2], [19].
In the second place there are several classes of important topological spaces, e.g.spaces of measures with the weak topology and spaces of random variables with the topo- logy of convergence in measure which can more naturally be equipped with AP-structures such that the topological structures are their TOP-coreflections [17].
In order to study these concepts and spaces it however soon became clear that we would need a theory of convergence in AP.We develop such a concept of convergence by means of assigning "limit functions" to filters, and moreover we show that AP can be completely characterized by four axioms about limit functions; two fundamental axioms -one on limit functions of principal ultrafilters and another on limit functions of comparable filtersa third axiom of a pretopological nature on limit functions of in- tersections of filters and a fourth one on limit functions of (Kowalsky-) diagonal fil- ters [15].Using this convergence-description of AP we obtain a very elegant characterization ef initial structures in AP.AP is a topological construct in the sense of [i], [I0], [ii].However from a categorical point of view some desirable properties are missing.
The existence of nice function space objects is indeed an important advantage in homo- topy, topological algebra, and infinite dimensional differential calculus.The topo- logical construct becomes extremely nice to work in when apart from being cartesian closed it also is hereditary, i.e. a quasitopos [5], [22], [23].AP is neither carte- sian closed nor hereditary.The situation is similar to the classical ones.Neither TOP nor pq-MET" is a quasitopos.By weakening the axioms "bigger" categories with nicer properties result.For example CONV is a quasltopos containing TOP, and pqs-MET is a quasitopos containing pq-MET By dropping the diagonal axiom and weakening the pre- topological axiom on limit functions we introduce the supercategory CAP of convergence approach spaces.CAP is a quasitopos and moreover it contains both quasitopoi CONM and pqs-MET as nicely embedded subcategorles.From this embedding it then follows that convergence spaces and extended pseudo-quasl-semi metric spaces can be viewed as enti- ties of the same type, both being characterized by means of limit functions of filters.Moreover also AP is nicely embedded in this supercategory.

PRELIMINARIES.
If X is a set then the set of all filters on X shall be denoted F(X).
If e F(X), then the set of all ultrafilters finer than shall be denoted U().
If {X} then we write shortly U(X).
For any collection of subsets of X we denote stack X := {B c XI A e A B}.If consists of a single element A we put shortly stackxA and if moreover A consists of a single point a then we put stackxa.If no confusion can occur, we drop the subscript and simply write stack a.s.o..If (Sj)jmj is a family of sets, then elements of their product S. shall jeJ J sometimes be denoted in a functional notation, e.g.s where for all j e J s(j) S.. JR+ stands for [0,] and all supreme and infima are taken in]R+. .eU() U PROOF.Suppose not, in that case the family u {x\=(%)l /-u()} has the finite intersection property and thus is contained in some Q0 U().Then however X\o( 0) 0 which is a contradiction.
Given two -pq-metric spaces (X,d) and (X',d') a function f X X' is called non-expansive if d' o (ff) <. d.
Given two approach spaces (X,6) and (X' ,6') a function f X X' is called a contrac- tion if for all x X and A 2 X 6'(f(x),f(A)) .<6(x,A) or equivalently, if for all A c 2 x 6f(A) o f <. 6.
3. CHARACTERIZATION OF AP VIA A CONVERGENCE THEORY.
In this section we shall give alternative characterizations of both approach spaces and contractions.
Let X e ISETI.We recall the Kowalsky diagonal operator (D [15] defined as fol- lows.For any index set J, any collection of filters (j)jeJ on X, and filter 3 on J (D (( j)jj,) := V n F jF In the case the collection of filters is a selection on X in the sense that we have a map 5 ,x--> x--> (x) then we put shortly(D (,) for (D((S(y))yeX, )" In the sequel we require the following results. reader.
Easy proofs are left to the PROPOSITION 3. I.
Fe jF 2 If (l)le is a family of filters on J and 0 i then(D((;)jjj leL N d)((j)jej, i )" leL 3 d(J)J eJ') (q/j).n U(j) JJ' JeJ jeJ 4 If each j, j e J is ultra and is ultra then 0((j)jj,) is ultra.
For any family (j)jej of filters on X ( n 3sup (2j).For any e F(X) and any selection of filters ((Y))yeX A(0"J(,)) <_ A() + sup ((y))(y).yeX Moreover, for any x e X and A c X onX PROOF.qL EU stack A) (CALl) follows from (DI) whereas (CAL2) follows from the fact that implies U() c U().

6(x,A)
To prove (PRAL), let (j)jeJ c F(X).One inequality follows from (CAL2), to show the other one observe that for all I U( j) and U q there exists j J and U(j) such that U .Consequently we have sup jJ J sup sup sup (:j).
sup 6 u To prove (AL) let us first suppose that U(X) and that for all y X (y) U(X) too.Now put e := s.. X((y))(y).Then for any D (D(,), by Pro- position 3.1.1there exists F Y" such that for all y F D (y), and conse- quently Thus D () Second, let us now suppose and all (y), y X are arbitrary filters on X, let again e := yeSU ((y))(y) and for each ( yX U($(y)) let 6 := yexSUp X(6(y))(y).
Then by straightforward verification we have sup .
To prove the final claim of the theorem, first from the fact that for any  /)) LeU(stack A) q/eU 8 which proves the remaining inequality.
Moreover, for any e F(X) ()(x) sup sup 5U" eu(3 ue'u. PROOF.(DI) follows from (CALl), (D2) follows from the fact that the infimum over an empty set is infinite, and (D3) follows from the fact that for any A,B c X U(stack ADB) U(stack A) D U(stack B).Before tackling (D4), we prove the final claim of the theorem.Let the map l' be defined as -X ' F(X) -->R+ > sup sup 6 U.

LeU() Ue
Now. let % e U(X).Then first we have '() Now for any 8 e N U(stack U) and any U % we have U e 8(U) and thus we have U-qL n 8(U) c %[ and it follows from (CAL2) that (t) ( n e(u)).
By the arbitrariness off St ollows in combination with (3.5) that Together with (3.4) this shows that and coincide on ultrafilters.
By definition of ' and the fact that fulfils (PL) it then follows that ' .In order to prove now (D4) let A c X, + and choose any e U(A(e)).Now suppose that for some y e A () and for all e U(stack A) < (%t)(y) x,(/)(y) sup (y.U).

ue %
This implies that for all e U(stack A) there exists Ut e such that < 6(y,U%).
n By Proposition 2.1 we can then find /l''"'n e U(stack A) such that A c U UL i and then it follows from (D3) that i=l n < inf 6(y,U i=l n (y, u u% i) .

< (y,A)
which is in contradiction to the choice of y, (y) e U(stack A) such that Thus for all y A () we can find (S(y))(y) s .
For y A (e) put (y) :ffi stack y and then put , :m sup A((y))(y).From the arbitrariness of e U(stack A(e)) and the definition of 6 it then finally follows that 6(x,A) 6(x,A(e)) + e.
The combined results of Theorems 3.1 and 3,2 give yet another way to describe the ob- jects of the category AP.
In what follows objects of AP shall then often also be denoted (X,A) where then is a map on F(X) fulfilling (CALl), (CAL2), (PRAL) and (AL).We shall characterize the morphisms of AP using this new description of objects.THEOREM 3.3.If (X,l), (X',%') e IApI and f X X' is a function then the following are equivalent
In the sequel, a convergence-approach limit and a convergence-approach space will be denoted shortly a CAP-limit and a CAP-space respectively.
We recall that a category of structured sets which is fibre-small and has the pro- perty that all constant maps between objects are morphisms is called a construct [i], [i13.
If we denote CA__P the category with objects all CAP-spaces and morphisms all con- tractions, then we obtain the following result, the verification of which is quite tri- vial.
A construct is called topological [Ii] if it is finally (or equivalently initially) complete.
PROOF.In order to show that CAP is initially complete consider the source f. (x > (xj,.j))jeJ where all items have their obvious meaning.

Let
be defined by -X sup . .jeJ (stack fj([)) o f..j To show that is a CAP-limit on X is quite simple.
(CAL3) follows from the observation that for any j e J and any , e F(X), we have stack f.j(0) stack f j() 0 f j().
To show that is initial, let (X',A') e ICAPI and let g X' X be a function such that for all j e J f. o g is a contraction.Then for any e F(X') we have (CALl) and (CAL2) are trivial and stack g()) o g sup A(stack f (stack g())) o f J J jeJ sup j(stack fjog()) o (fjog) jeJ .<,( ).
o g Consequently g too is a contraction and we are done.
Before proceeding we now need some further notational conventions and definitions.
If X,Y e ICAPI then HOMcAp(X,Y) stands for the set of all morphisms i.e. contrac- tions from X to Y.If no confusion can occur concerning the category under study we often omit the subscript and simply write HOM(X,Y).Clearly, stack () e F(Y).
Next for any f e HOM(X,Y) if AX and Ay are the CAP-llmlts on X and Y respectively, we defin,.
is a CAP-limit on HOM(X,Y).
PROOF.We leave the details to the reader.(CALl) follows from the fact that for any f e HOM(X,Y) and any F(X) stack((stack f)) stack f().( CAL2) and (CAL3) follow from the facts that for any f HOM(X,Y) and any ,# F(HOM(X,Y)) respectively, if @ then L(,f) c L(@,f) and if and # are arbitrary then L(0,f) L(,f) 0 L(,f).
If is a topological construct, then is called cartesian closed if for all objects A,B G I, the set HOM G (A,B) can be endowed with a G -structure such that the evaluation map ev A HOM (A,B) B defined by ev(a,f) := f(a) is co-universal with respect to the endofunctor A For more information on cartesian closedness, we refer to [7], [9], [20], [21].
PROOF.The assertion we have to prove breaks up in two parts (I) For any two objects (X,Ix) and (Y,ly) in CAP and as defined in Proposition 4.2, the evaluation ev (X,% x) (HOM(X,Y),%) --> (X,ly) is a contraction.
In order to verify (2) notice that for any F(Z), F(X), z Z and x X, since f HOM(XZ,Y), we have f* Ry(stack ()( 9) The arbitrariness of and z shows that f is indeed aain a contraction.This ends the proof of the theorem.
A topological construct is called hereditary provided final epi-sinks are heredit- ary, or equivalently as was shown in [ii], if partial morphisms are representable.ff is a construct and A,B [[ then a partial morphism from A to B is a morphism f HOM (C,B) where C is a subobject of A.
If G has subobjects then partial morphisms are representable if every object B can be embedded via the addition of a single point =B into an object B @ I such that for every partial morphism f C We shall use this characterization to prove our next result.
PROOF.Let (X,Xx),(Y,X) ICAPI and let Z X.The subobject determined by Z we shall denote (Z,z) where then for any F(Z) kZ () x(stackx ).It is rather dreary but straightforward to verify that (Y#,l#) ICAPI and that (Y,) is embedded in (Y,) by inclusion, so we omit this.Now we define f# (x, x) --> by f#(x) f(x) if x Z and f#(x) y if x X\Z.To show that f# is a contraction, let e F(X) and x X.If has a trace on Z then it is clear that stack ,# f#() has a trace on Y equal to stacky f(IZ).If then x Z it follows that f#(x)Y= f(x) Y and by definition of # and the fact that f HOM(Z,Y) we then obtain l#(stacky If x X\Z the same inequality results at once from the definition of and from f#(x) y.If does not have a trace on Z then again the same inequality holds for any x X by definition of and the fact that stack.#f#() stack y@ (R)y.By Theo- rem [II] this proves the theorem.
Since by definition a quasitopos is a hereditary cartesian closed topological con- struct [i0] our main result now is an immediate consequence of the foregoing theorems.THEOREM 4.4.CAP is a quasitopos, m 5. THE HEREDITARY TOPOLOGICAL CONSTRUCT PRAP DEFINITION 5.1.Given X ISET a map -X F(X) -->/ is called a pre-approach limit (or PRAP-limit for short), if it fulfils (CALl), (CAL2)  and (PRAL).The pair (X,l) is then called a pre-approah space (or PRAP-spa.cefor short).
Clearly each pre-approach space is a convergence-approach space.The full subcate- gory of CAP with objects all pre-approach spaces shall be denoted PRAP.From Proposi- tion 4.1 we at once obtain the next result.PROPOSITION 5.1.PRAP is a construct.In Theorems 3.1 and 3.2, we proved that giving a distance on a set X is equivalent to giving an approach limit on X.A simple inspection of the proofs of these two theo- rems reveals that (DI), (D2) and (D3) are equivalent to (CALl), (CAL2) and (PRAL).Con- sequently, if we call a map 6 X 2 x-->+ fulfilling (DI), (D2) and (D3) a pre-distance, then without further proof we can state the following two results.THEOREM 5.1.If X e ISETI and 6 is a pre-distance on X, then the map -X F(X) -->R+ --> sup sup U is a pre-approach limit on X.Moreover, for any x X and A c X Moreover, for any F(X) x(U)( As was the case for approach spaces the structure on a pre-approach space shall be de- termined either by a pre-approach limit or by a pre-distance, whichever is more conve- nient.
PROOF.Since PRAP contains all indiscrete CAP-objects, it will suffice to show that PRAP is initially closed in CAP.Let (Xj,j)jej be a collection of PRAP-spaces and con- sider the source fo (X ] > (Xj '" 3 jeJ" Let be the initial CAP-limit on X given by Theorem 4.1.To prove that fulfils (PRAL), let (k)keK be a collection of filters on X then REMARK.It is easily verified that the PRAP-reflection of a CAP-space (X,%) is given by id X (X,E) > (X,lp) where the pre-distance associated with kp is given by 6(x,A) := inf k(q)(x).eU(stack A) COROLLARY 5.1.PRAP is a topological construct.
We shall later give a simple reason why PRAP is not cartesian closed, it is how- ever hereditary as we shall now prove.THEOREM 5.4.PRAP is an hereditary topological construct.
PROOF.The proof goes exactly the same as that of Theorem 4.3., the only difference being that now one starts with (X,x),(y,) IPRAPI and one has to show that (Y,) IPRAPI We leave this to the reader.

EMBEDDING CONV IN CAP
A convergence space [6], [15] is a pair (X,q) where X ISETI and q F(X) X fulfils (CI) for all x e X (stack x, (2) For all F(X) and x X (,x) e q, => (,x) q.
Given convergence spaces (X,q), (X',q') a function f X X' is called continuous if for all (,x) q we have (stack f(),f(x)) e q'.
The class with objects all convergence spaces and morphisms all continuous maps, is a quasitopos [I0], denoted CON.
The proof of the following result is quite straightforward and so we omit it.THEOREM 6.1.CON-V is embedded as a full subcategory in CAP by the functor CON---> CAP (X,q) -> (X k q where for all F(X), and x X [ 0 if (i,x) q q()(x) := otherwise.
We shall now show that this embedding actually is extremely nice, but first we mention the following useful characterization of CONV in CAP, similar to that of TOP in AP [17].
As the formulation of this proposition suggests we shall not differentiate between the notion of a convergence space and of a CAP-space fulfilling the condition of Propo- sition 6.1.This is after all entirely justified by Theorem 6.1.THEOREM 6.2.CONV is a bireflective subcategory of CAP.

EMBEDDING PRETOP IN PRAP
A pre-topoloKical space [4], [6] is a convergence space (X,q) where instead of (C3) q fulfils the stronger condition (PR) For any collection (j,x)jeJ c q we have n , The full subcategory of CONV with objects all pre-topological spaces is denoted PRETOP.It is quite easy to see that precisely the same results hold for PRETOP w.r.t.PRAP, as those proven in Section 6 for CONV w.r.t.CAP.We therefore list them without further explanation.THEOREM 7.1.PRETOP is embedded as a full subcategory in PRAP by the functor PRETOP --> PRAP (X,q) --> (x,x) q where for all e F(X) and x e X f q 0 if (,x) e q otherwise.PROPOSITION 7.1.A space (X,i) e IPRAPI is a pre-topological space, if and only if for all e (X) %([)(X) c {0,=}, or equivalently, if 6 is the pre-distance associated with , if and only if 6(Xx2X) c {0,}.THEOREM 7.2.PRETOP is a bireflective subcategory of PRAP, the bireflection of any PRAP-space beinB the same as its CONV-bireflection.
THEOREM 7.3.PRETOP is a bicoreflective subcategory of PRAP, the bicoreflection of any PRAP-space being the same as its CONV-bicoreflection.
Again, we shall not differentiate between pre-topological spaces and PRAP-spaces fulfilling the condition of Proposition 7.1.

EMBEDDING AP IN PRAP
From Section 3 it is quite clear that AP is embedded as a full subcategory of PRAP.
PROOF.Since AP contains all indiscrete CAP-objects, it will suffice to show that AP is initially closed in CAP.Let (Xj,j)jeJ be a family of AP-spaces and consider the source f.

(X -]
> (Xj,%_j))jeJ" Let i be the initial CAP-limit on X.From Theorem 5.3 we already know that I fulfils (PRAL).To show that it also fulfils (AL), let F(X), let ((Y))yeX be a selec- tion of filters on X an put e := sup %((y))(y).Now, for all J define the fol- yeX lowing selection of filters on X.
We leave to the reader the straightforward verification that for all j J (D(O'j,stack f.()) stack f.((D(S 2)).

EMBEDDING pqs-MET IN PRAP
The most general kind of map measuring a distance between points of a set X is an extended pseudo quasi-semimetric (shortly -pqs-metric).An -pqs-metric d XX]R+ need only fulfil d(x,x) 0 for all x e X.The pair (X,d) then is called an -pqsmetric space.Given-pqs-metric spaces (X,d) and (X',d') a function f X X' is Let pqs-MET stand for the category with objects all -pqs-metric spaces and morphisms all non-expansive maps.
Fe yeF PROOF.That Ad fulfils (CALl) and (CAL2) is clear.That it also fulfils (PRAL) is seen as follows.Let (j)jej c F(X) then for any x e X we have ld 0 )( Ipqs-METl and f (X,d) (X',d') is non-expansive it is easily verified that f (X,d) (X',d,) is a contraction.The converse is equally simple upon noticing that from the definition of d' for any x,y e X d(x,y)= d(stack y)(x).REMARK.By Theorems 5.1 and 5.2, the pre-approach space (X,d) is identical to (X,6d) where 6 d is the pre-distance derived from d' i.e. for all x X and A X 6d(X,A) inf inf sup d(x,y).
(9.1) QeU(stack A) U t yeU This rather complicated expression for 6 d can however be much simplified using the fol- lowing lemma.
LEMMA 9.1.Given (X,d) e Ipqs-METl, q e U(X) and x X we have sup inf d(x,y) inf sup d(x,y).
It is clear that (X,d 6) Ipqs-METl.The remainder of the proof now is exactly the same as in Theorem 6.7 [17], where it was shown that pq-MET is a bicoreflective subca- tegory of AP, and so we omit this.m Analogous to the characterization of pq-MET in AP [17], we have the next result, the verification of which we leave to the reader.PROPOSITION 9.1.A space (X,6) IPRAP is an (R)-pqs-metric space, if and only if for any x X and A c X 6(x,A) inf 6(x,{a}).aeA THEOREM 9.4.pqs-MET is a hereditary topological construct.
PROOF.This is an immediate consequence of Corollary 5.1, Theorem 9.
y-X Now consider the diagonal filter(&,W) then A e (y) and thus too yea () A e(D(,/).From Proposition 3.1.4it then also follows that d)(,/) e U(stackA) and from the definition of and (AL) it then follows further that for any x e X (x,A) (e(S,#))(x) s ()(x) + ' ()(x) + .
inequality & follows from the fact that for any UI,U 2 q7_ inf d(x,y) & sup d(x,y).To show the other one, suppose YeU YeU 2 inf sup d(x,y) > = 0.
3 and of Theorem 6 [11].REMARK.Initial structures in pqs-MET are obtained as follows.Let fo (X ] > (Xj,dj))jej be a source, then the initial -pqs-metric on X is given by