© Hindawi Publishing Corp.
COMPLETION OF A CAUCHY SPACE WITHOUT THE
T
2-RESTRICTION ON THE SPACENANDITA RATH
(Received5 April 1999 andin revisedform 27 April 1999)
Abstract.A completion of a Cauchy space is obtainedwithout theT2restriction on the space. This completion enjoys the universal property as well. The class of all Cauchy spaces with a special class of morphisms calleds-maps form a subcategory CHYof CHY. A com-pletion functor is defined for this subcategory. The comcom-pletion subcategory of CHYturns out to be a bireflective subcategory of CHY. This theory is appliedto obtain a characteri-zation of Cauchy spaces which allow regular completion.
Keywords and phrases. Cauchy space, Cauchy map,s-map, stable completion, completion in standard form, regular completion.
2000 Mathematics Subject Classification. Primary 54D35, 54A20, 54B30.
1. Introduction. The completion of Cauchy spaces is already well known and famil-iar to most of us. In fact, since Keller [5] introduced the axiomatic definition of Cauchy spaces a very rich andextensive completion theory has been developedfor Cauchy spaces during the last three decades [2, 3, 7, 10, 12]. It seems that Cauchy space rather than uniform convergence space is a natural generalization of completion of uniform space. But the completion theory developed so far is not so general in nature as the Cauchy spaces considered areT2Cauchy spaces [3, 6, 7, 8]. So the natural question arises, “whether a satisfactory completion theory can be developed for a Cauchy space without theT2-restriction on the space.” This question has been partially answered in this paper. A completion functor has been constructedfor a subcategory of CHY. This new subcategory is constructedby taking all the Cauchy spaces as objects and morphisms as certain special type of Cauchy maps which we calls-maps.
2. Preliminaries. The following are some basic definitions and notations which we will use throughout the paper. Afilteron a setXis a nonempty collection of nonempty subsets ofXwhich is closedunder finite intersection andformation of supersets. Let F(X)be the set of filters on X. If F,G∈ F(X), thenG ≥F if andonly if for each
F ∈F, ∃G∈G such thatG⊆F. This defines a partial order relation on F(X). If B
is abase[11] of the filterF, then we writeF=[B]andFis saidto begenerated by
B. A filter F∈F(X)is saidto have atrace on A⊆X, ifF∩A=φ, the empty set, ∀F ∈F. In this case,FA=[{F∩A|F ∈F}]is calledthetrace ofFonA. ˙x=[{x}]
is the filter generatedby the singleton set{x}and F∩G=[{F∪G|F ∈F, G∈G}]. IfF∩G=φ, ∀F ∈Fand∀G∈G, thenF∨Gis the filter[{F∩G|F∈F, G∈G}]. If ∃F∈FandG∈Gsuch thatF∩G=φ, then we say thatF∨Gfails to exist.
Definition2.1. ACauchy structureonXis a subsetC⊆F(X)satisfying the fol-lowing conditions:
(c1)x˙∈C,∀x∈X.
(c2)F∈C,G≥Fimply thatG∈C.
(c3)F,G∈CandF∨Gexists imply thatF∩G∈C.
The pair(X,C)is calledaCauchy space. IfCandDare two Cauchy structures on
XandC⊆D, thenCisfiner thanD, writtenC≥D. For a Cauchy structureConX
we define an equivalence relation “∼” byF∼Gif andonly ifF∩G∈C. Let[F]denote the equivalence class determined byF. Also, there is a convergence structureqc[7] associatedwithCin a natural way:
F q→c x if andonly ifF∼x.˙ (2.1)
A Cauchy space(X,C)is saidto be
•T2orHausdorff if andonly ifx=y, whenever ˙x∼y˙.
•Regularif andonly if clqcF∈C, wheneverF∈C, where “clqc” is theclosure operator forqc.
•Completeif andonly if eachF∈C qcconverges.
Note that most of the literature on Cauchy spaces including the completion theory [3, 6, 7, 8, 10] deals exclusively withT2 Cauchy spaces. The underlying reason for this is the existence of the unique limits in T2 spaces, which guarantees pleasant consequences. But theT2condition is quite restrictive, so our object is to construct a completion theory for Cauchy spaces in general.
A mapf :(X,C)→(Y ,K)is calledaCauchy map, if f (F)∈K whenever F∈C. The mapf is ahomeomorphism, iff is bijective andf,f−1are both Cauchy maps.
IfA⊆XthenCA= {G∈F(A)| ∃F∈Csuch thatG≥FA}is a Cauchy structure onA,
calledasubspace structureonA. The pair(A,CA)is asubspaceof(X,C). The mapping
f:(X,C)→(Y ,K)is anembeddingof(X,C)into(Y ,K), iff:(X,C)→(f (X),Kf (X))
is a homeomorphism, whereKf (X)is the subspace structure onf (X).
3. Cauchy space completion. Note that throughout this section(X,C)denotes a Cauchy space (not necessarilyT2), unless otherwise stated. Acompletionof a Cauchy space(X,C)is a pair((Y ,K),ϕ)consisting of a complete Cauchy space(Y ,K)andan embeddingϕ:(X,C)→(Y ,K)satisfying the condition clqkϕ(X)=Y.
We construct a completion space((X,˜C),j)˜ of the Cauchy space(X,C)in the fol-lowing way:
˜
X=x˙|x∈X∪[F]|F∈Cisqcnon-convergent, j:X →X˜is defined byj(x)=x,˙
˜
C=A∈F(X)˜ | ∃F∈C qcconvergent such thatA≥j(F)or∃F∈C qcnon-convergent such thatA≥j(F)∩[˙F].
(3.1)
Definition3.1. A completion((Y ,K),ϕ)of a Cauchy space(X,C)is saidto be instandard form, ifY=X˜,ϕ=j, andj(F) q→k [F]for eachqc non-convergent filterF
This notion of completion was given by Reed[10] forT2Cauchy spaces. Also, there is an equivalence relation defined between T2 completions. For a T2 Cauchy space
(Z,D), aT2completion((Y1,K1),ϕ1)≥((Y2,K2),ϕ2), if there exists a Cauchy map
h:(Y1,K1)→(Y2,K2)such thath◦ϕ1=ϕ2, andthe two completions are equivalent, if each of these two completions is greater than or equal to the other. Note that in case of equivalence,his a unique Cauchy homeomorphism.
The same definition leads to an equivalence relation between the non-T2completions of a non-T2Cauchy space(X,C), but it is not a categorical equivalence in the sense of Preuss [9, Proposition 0.2.4], because, as shown in the following examples,his not necessarily a unique Cauchy homeomorphism.
Example3.2. Let(R,C)be the set of real numbers with the usual Cauchy structure [1], letY=R∪{a}, wherea∈R, andZ=Y∪{b}, whereb∈Y. LetHbe an ultrafilter onRfiner than the filter generatedby the sequence of natural numbers. IfF∈F(R), let
FandFdenote the filters generated byFonY andZ, respectively. LetC=C∪{H}
be a Cauchy structure onR. Note thatqcis the usual topology onRandHis notqc convergent.
LetDbe the Cauchy structure onY generatedby{F|F∈C}∪{H∩a˙}andKbe the
Cauchy structure onZ generatedby{F|F∈C} ∪ {H∩a˙∩b˙}. Observe that(Y ,D)
and(Z,K) are equivalent completions of(R,C), if Reed’s definition of equivalence
describedabove is generatedto non-T2completions. But they are not Cauchy homeo-morphic, since(Y ,D)isT2while(Z,K)is not. Furthermore, the functions establishing this equivalence are not unique, sinceh1,h2:(Y ,D)→(Z,K)will both work, where
h1(y)=
y, ya, y∈∈R,R,
h2(y)=
y, yb, y∈∈R,R.
(3.2)
Example3.3. Let(R,C)andYbe as in Example 3.2. LetDbe the Cauchy structure
onY generatedby{F:F∈C}∪{˙0∩a˙}. Since(R,C)is complete, it is trivially a comple-tion of itself. Furthermore,(Y ,D)is another completion of(R,C)equivalent to(R,C)
in the sense described in the preceding example, but not Cauchy homeomorphic to
(R,C).
Both Examples 3.2 and 3.3 provide the motivation for introducing the following notion ofstablecompletion to ensure appropriate categorical behavior for our com-pletion for the whole class of Cauchy spaces.
Definition3.4. A completion((Y ,K),ϕ)of(X,C)is saidto bestableif whenever
z∈Y\ϕ(x)and ϕ(F) q→k zfor someF∈C, it follows thatzis the unique limit of ϕ(F)inY.
Definition3.5. A stable completionκ1=((Y1,K1),ϕ1)of a Cauchy space(X,C)
Cauchy maph:(Y1,K1)→(Y2,K2)such that the following diagram commutes:
(X,C) ϕ2
/
/
ϕ1
Y2,K2
Y1,K1.
h
9
9
s
s
s
s
s
s
s
s
s
s
(3.3)Ifκ1is finer thanκ2, we writeκ1≥κ2andκ2≥κ1. Ifκ1≥κ2, then we say that the two completionsκ1andκ2areequivalent.
Observe that by taking stable completions in defining the equivalence relation, we get the unique limit of filters which converge to points inY1\ϕ1(X)andY2\ϕ2(X). This ensures thathis a unique Cauchy homeomorphism.
Proposition3.6. ((X,˜ C),j)˜ is the finest stable completion of (X,C)in standard form.
Proof. It is easy to see thatj is injective and ˜Csatisfies(c1)and(c2)of Defini-tion 2.1. To prove(c3)letA,B∈C˜andA∨Bexist. We consider the following three cases to show thatA∩B∈C˜
(1) A≥j(F)andB≥j(g), whereF,g∈Care bothqcconvergent,
(2) A≥j(F)∩[˙F], whereFisqc non-convergent, andB≥j(g), wheregisqc con-vergent,
(3) A≥j(F)∩[˙F]andB≥j(g)∩[˙g], where bothFandgareqcnon-convergent. Proof of (1) is easy and(3) can be provedthe same way as (1) and(2), so we prove only (2).
In case (2),A∨Bexists impliesj(F)∨j(g)exists orj(g)∨[˙F]exists. Since the latter is an impossibility,j(F)∨j(g)exists. This implies that theqc non-convergent filter
F∩g∈CandsinceA∩B≥j(F∩g)∩[F∩˙g],A∩B∈C˜. We conclude that ˜Cis a Cauchy structure on ˜X.
It is routine to show that((X,˜C),j)˜ is a completion of(X,C)in standard form. To prove that this is also a stable completion, letF∈Cbeqc non-convergent. If there exists[g]=[F]∈X˜\j(X)such thatj(F) q˜→c [g], then[˙F]∩[˙g]∈C˜. This implies that there exists aqcnon-convergentH∈Csuch that[˙F]∩[˙g]≥j(H)∩[H˙]. So[F]=[H]=
[g], which leadto a contradiction. Next, letF∈C beqc convergent. Ifj(F) q˜→c [g], where[g]∈X˜\j(X), thenj(F)∩[˙g]∈C˜. Ifj(F)∩[˙g]≥j(T), for some convergent filterT∈C, then[g]=x˙, for somex∈X which leads to a contradiction. If, on the other hand,j(F)∩[˙g]≥j(L)∩[˙L], whereLisqcnon-convergent, thenF≥L. But since this implies thatLqcconverges, we have a contradiction. This proves that(X,˜C)˜ is a stable completion. Also it can be easily shown that it is the finest stable completion in standard form.
This completes the proof of Proposition 3.6. We call((X,˜ C),j)˜ theWyler completionof(X,C)
In fact, in this case if we identify ˙x with its equivalence class[x]˙ , then the Wyler completion coincides with((X∗,C∗),j)in [10]. Henceforth, we will refer to the
com-pletion((X∗,C∗),j)as theT2-Wyler completion of aT2Cauchy space(X,C).
Proposition3.8. Any stable completion ((Y ,K),ϕ)of a Cauchy space (X,C)is equivalent to one in standard form.
Proof. Defineh:(Y ,K)→(X,˜ C)˜ as follows
h(y)=
[F], ify∈Y\ϕ(X), ϕ(F) qk
→y,
˙
x, ify=ϕ(x). (3.4)
Note that such a non-convergent filterF∈Cexists, since((Y ,K),ϕ)is a completion of(X,C). Since this is also a stable completion, it follows thathis well defined and bijective.
Let ˜Ckbe the Cauchy structure on ˜Xgeneratedby{h(A)|A∈K}. Clearly, the dia-gram
(X,C) j
/
/
ϕ
˜ X,Ck˜
(Y ,K) h
:
:
u
u
u
u
u
u
u
u
u
(3.5)commutes and((X,˜Ck),j)˜ is a completion of(X,C)in standard form.
Also, it can be shown by [9, Propositions 1.2.2.5, 1.2.2.4, and0.2.7] thathis a Cauchy homeomorphism andtherefore, the two completion((Y ,K),ϕ)and ((X,˜Ck),j)˜ are equivalent. This proves Proposition 3.8.
In view of Proposition 3.8 all stable completions are equivalent to completions in standard form.
Definition3.9. A Cauchy mapf :(X,C)→(Y ,D) between two Cauchy spaces
(X,C) and (Y ,D) is saidto be ans-map if andonly if the following condition is satisfied:F∈Cconverges to at most one point inXimplies thatf (F)∈Dconverges to at most one point inY.
It is easy to see that the embedding map in any stable completion is an s-map, in particular, the mapj in the Wyler completion((X,˜C),j)˜ is ans-map. In fact, any Cauchy map with aT2codomain is ans-map. Also, the identity map on any Cauchy space is ans-map andcomposition of two s-maps is ans-map. So the class of all Cauchy spaces together with thes-maps as morphisms forms a category, which we call CHY. Henceforth, the termCauchy categorywill be usedto denote a categoryC
in which the object are Cauchy spaces andthe morphisms ares-maps. In this sense, CHYandT2CHY are Cauchy categories.
(R,C)(whereRand Care as in Example 3.2) onto a Cauchy space with only two el-ements, it follows that CHYis not closedunder the formation of final structure [4]
andtherefore, it is not a topological category.
But the category CHY has other nice properties a few of which we will discuss subsequently. Fric andKent [3] have shown that any Cauchy map on aT2Cauchy space can be uniquely extended to itsT2Wyler completion. The next proposition shows that the Wyler completion((X,˜C),j)˜ also enjoys this extension property with respect to thes-maps.
Proposition3.10. Letf:(X,C)→(Y ,K)be ans-map between two Cauchy spaces (X,C)and(Y ,K). Thenf has a unique extensionf˜:(X,˜ C)˜ →(Y ,˜ K)˜ which is also an s-map and the following diagram commutes:
(X,C) f
/
/
jX
(X,K)
jY
˜ X,C˜ ˜
f
/
/
Y ,˜K˜.(3.6)
Proof. f˜:(X,˜C)˜ →(Y ,˜K)˜ is defined by ˜
f (x)˙ =f(x),˙ ∀x∈X,
˜
f ([F])=
[f (y,˙ F)], ififf (f (FF)qk) qk→non-convergenty. ,
(3.7)
Since f is an s-map, it follow that ˜f is a well-defined Cauchy map for which the above diagram commutes. To prove that ˜fis ans-map it suffices to show that ˜f (A)q˜k
converges to only one element in ˜Y, wheneverA∈Cq˜˜ cconverges to only one element
in ˜X. If A≥jX(F) for someF∈C, thenjX(F)q˜c converges to only one point in ˜X,
which in turn implies thatFqc converges to only one point inX. Sincef andjY are
s-maps, ˜f (A)≥f˜◦jX(F)=jY◦f (F)q˜kconverges to only one element. On the other
hand, ifA≥jX(g)∩[g˙], whereg∈Cisqcnon-convergent, then it suffices to show that ˜
f◦jX(g)∩f ([˜ ˙g])q˜kconverges to only one point. Note that
˜
f◦jX(g)∩f˜[˙g]=
jY◦f (g)∩jYy˙, iff (g) qk→y,
jY◦f (g)∩f (˙F) , iff (g)isqknon-convergent. (3.8)
SincefandjYares-maps, it follows thatjY◦f (F)can converge to at most one point. This shows that ˜fis ans-map.
If ¯f:(X,˜C)˜ →(Y ,˜ K)˜ be anothers-map which makes the above diagram commute, then obviously ¯f◦jX(x)=f˜◦jX(x), ∀x∈X. So it remains to show that ¯f ([F])=
˜
f ([F]), for eachqc non-convergent filterF∈C. SincejX(F) q→c˜ [F], andboth ¯f ,f˜are s-map it follows that ¯f◦jX(F) q˜→k f ([¯ F])and ˜f◦jX(F) q˜→k f ([˜ F]). HencejY◦f (F) q˜→k
¯
f ([F]),f ([F])˜ . ButF∈Cisqcnon-convergent andf ,jYares-maps imply thatjY◦f (F)
Now we can define a functor on the category CHYexactly the same as theT2Wyler completion functor defined in [8]. Let C˜HYbe the subcategory of CHYconsisting of
all complete objects in CHY. We define ˜W: CHY→C˜HYas follows:
(1) ˜W (X,C)=(X,˜ C)˜, for all objects(X,C)in CHY.
(2) ˜W (f )=f˜, for all morphismsfin CHY, where ˜fis the same as in Proposition 3.10. Proposition3.11. W˜ defined as above is a covariant functor.
Proof. It follows from Proposition 3.10 that ˜W (f )=f˜is a morphism in C˜HY,
wheneverf is a morphism in CHY. Also, since ˜IX(x)˙ =IX(x)˙ =x˙=IX˜(x),˙ ∀x˙∈X˜
and ˜IX(F)=[IX(F)]=[F]=IX˜([F]),∀[F]∈X˜. This shows that ˜W (IX)=Iw˜(X).
Next we show that ˜W preserves the composition ofs-maps. Letf:(X,C)→(Y ,K)
and g :(Y ,K)→(Z,S). It is easy to see that ˜W (f◦g)(jX(X))= (f˜◦g)(jX˜ (X))=
(W (f )˜ ◦W (g))(jX˜ (X)). We show thatf˜◦g([F])=f˜◦g([˜ F]), whenever[F]∈X˜\jX(X). Note that
f˜◦g([F])=
f◦g(F) , iff◦g(F)isqs non-convergent,
˙
z, iff◦g(F) q→s z. (3.9)
SinceFisqcnon-convergent andgis ans-map,g(F)qkconverges to at most one point inY. So
˜
g([F])=
g(F) , ifg(F)isqknon-convergent,
˙
y, ifg(F) q→k y. (3.10)
Therefore, it follows that
˜
f◦g([˜ F])=
˜
fg(F), ifg(F)isqknon-convergent,
˜
fy˙, ifg(F) qk→y. (3.11)
Ifg(F) q→k y, thenf◦g(F)q→s f (y). SinceFisqcnon-convergent andf◦gis ans-map, f (y)=z, which shows thatf˜◦g(F)=f˜◦g(˜ F)wheneverf◦g(F)isqs convergent. On the other hand, ifg(F)isqknon-convergent, thenf◦g(F)converges to at most one point. Observe that
˜
f[g(F)]=
f◦g(F) , iff◦g(F)isqsnon-convergent,
˙
t, iff◦g(F) q→s t. (3.12)
Sincef◦g(F)converges to at most one point,t=z. Therefore, ˜f◦g([˜ F])=f˜◦g([F]), i.e., ˜W (f◦g)=f˜◦g˜. This proves Proposition 3.11.
Lemma3.12. Let(X,C)and(Y ,K)be inCHY. Amorphismf :(X,C)→(Y ,K)is an epimorphism if there exists aqc non-convergent filterF∈C such thatf (F) q→k y, whenevery∈Y\f (X).
Proof. Letα:(Y ,K)→(Z,S)andβ:(Y ,K)→(T ,U)be twos-maps such thatα◦
f=β◦f. For eachy∈Y\f (X), there exists aqcnon-convergent filterF∈Csuch that
f (F) q→k y.αandβares-maps imply thatα◦f (F)q→s α(y)only andβ◦f (F)q→u β(y) only. But, sinceα◦f (F)=β◦f (F)andα◦fis ans-map, it follows thatα(y)=β(y). Therefore,α=β, which implies thatf is an epimorphism.
Note that the embedding mapjin the Wyler completion((X,˜C),j)˜ of a Cauchy space
(X,C)is an epimorphism.
Proposition3.13. In the Cauchy categoryCHYWyler completion is the finest com-pletion of a Cauchy space.
Proof. Let(X,C)be a Cauchy space andκ=((Y ,K),ψ)be a completion of(X,C)
in the category CHY, i.e.,ψandψ−1ares-maps. We show that the Wyler completion
is finer thanκ.
Define a maph:(X,˜C)˜ →(Y ,K)ash(x)˙ =ψ(x), ∀x ∈X andfor eachqc non-convergent filterF∈C,h([F])=y, whereψ(F) q→k y. Sinceψis ans-map,his well defined. Also the following diagram commutes:
(X,C) ψ
/
/
j
(Y ,K)
˜ X,C˜.
h
:
:
u
u
u
u
u
u
u
u
u
(3.13)Next we show thathis an s-map. It is routine to show that his a Cauchy map. Let
A∈C˜converge to only one point. IfA≥j(F), thenFqc converges to only one point. Sinceψis ans-map,ψ(F) qkconverges to only one point. Therefore, h(A)converges to only one point. IfA≥j(g)∩[˙g], wheregisqcnon-convergent, thenψ(g)converges to at most one point, sayy. But, sinceh([g])=y, h(A)≥h◦j(g)∩h([˙g])converges only toy. This shows thathis ans-map. Therefore, Wyler completion is finer thank
in CHY. This completes the proof of Proposition 3.13.
From Lemma 3.12 andProposition 3.13, we obtain the following property of the subcategory C˜HY.
Proposition3.14. C˜HYis an epireflective subcategory ofCHY.
For a Cauchy categoryA, let ˜Adenote the full subcategory of all complete objects in
A. A Cauchy categoryAis saidto be aCauchy completion category, if there is a reflector [9]R:A→A˜such that for each object(X,C)inA,(R(X,C),ϕx)is a completion of
completion functor. Any subcategory of CHYwhich admits a completion functor is calledacompletion subcategoryof CHY. Note that by takings-maps as morphisms
a completion functor couldbe definedon the Cauchy space unlike the completion functors in [3, 8], which were restrictedtoT2CHY andits subcategories.
Kent andRichardson [8], have constructedcompletion forT3Cauchy space. An at-tempt has been made to construct a regular completion of a Cauchy space. In fact, in the next proposition we show that the following condition is a necessary and sufficient condition for a Cauchy space(X,C)to have a regular completion:
(∗)F∈C implies that there exists a complete regular Cauchy space(Y ,K)andan
s-mapf:(X,C)→(Y ,K)such thatf (F)∈K.
Note that any complete regular Cauchy space preserves this property.
Lemma3.15. ACauchy space(X,C)has a regular, stable completion if and only if ((X,r˜ C),j)˜ is a regular completion of(X,C).
Proof. (⇒) Let ((Y ,D),ϕ) be a regular, stable completion of (X,C). Then by Proposition 3.8,((Y ,D),ϕ)is equivalent to a regular, stable completion((X,D˜ ),j)
in standard form. By Proposition 3.6, D≤C˜andhenceD≤rC˜. Since (X,D˜ )and (X,˜C)˜ are both completion of(X,C), so is((X,r˜ C),j)˜ .
(⇐)Let((X,r˜ C),j)˜ be a regular completion of (X,C). We show that this is also a stable completion. Letz=[g]∈X˜\j(X)andassumej(F) qr→c˜ z, whereF∈C. IfFqc
converges, thenj(F∩g)=j(F)∩j(g)∈rC˜, which implies thatF∩g∈C, a contradic-tion sincegis notqcconvergent. IfFis notqcconvergent and[F]=[g]thenj(F)qrc˜
converges to[F]and[g], whencej(F∩g)∈rC˜. This implies thatF∩g∈C, which is again a contradiction. Thus,[F]=[g]is the unique limit ofj(F)in(X,r˜ C)˜, andthe completion((X,r˜ C),j)˜ is stable.
Proposition3.16. ACauchy space(X,C)has a regular stable completion if and only if(X,C)satisfies the condition(∗).
Proof. (⇐) Assume the condition (∗). By Lemma 3.15, we needonly to show that ((X,r˜ C),j)˜ is a completion of (X,C). It is routine to show that j is an injec-tive Cauchy map for which ˜X=clqr˜cj(X)and (X,r˜ C)˜ is complete [6]. So it remains only to show thatj−1is a Cauchy map. If not,∃H∈rC˜such thatj−1H∈C. By(∗)there
exists a complete regular Cauchy space(Y ,K)ands-mapf:(X,C)→(Y ,K)such that
f (j−1H)∈K. Let ˜f:(X,˜C)˜ →(Y ,K)be thes-extension off, to the Wyler completion,
as in Proposition 3.10. Since(Y ,K)is regular, ˜f :(X,r˜ C)˜ →(Y ,K)is also a Cauchy map. SoH∈rC˜implies that ˜f (H)∈K. But sincef (j−1(H))≥f (˜H), f (j−1(H))∈K,
a contradiction. Thus j−1 is a Cauchy map, so ((X,r˜ C),j)˜ is a regular completion
of(X,C).
(⇒)By Lemma 3.15,((X,r˜ C),j)˜ is a regular stable completion of(X,C). IfF∈C, thenj(F)∈rC˜, sincej−1is a Cauchy map. Also, since this completion is stable,jis
ans-map. So(X,C)satisfies(∗). This completes the proof of Proposition 3.16. Note that if the inverse of an injective Cauchy map is a Cauchy map then it is also an
s-map. Soj−1in the completion((X,r˜ C),j)˜ is ans-map. Hence we have the following
Corollary3.17. If(X,C)and (Y ,K)are two Cauchy spaces satisfying(∗) and f:(X,C)→(Y ,K)is ans-map, thenf˜:(X,r˜ C)˜ →(Y ,r˜ K)˜ is also ans-map, wheref˜is as in Proposition 3.10.
Let SCHYandRCHYbe the full subcategories of CHYconsisting of Cauchy spaces
satisfying(∗)andregular Cauchy spaces, respectively. Since every complete regular Cauchy space is in SCHY, SCHY=RCHY. Define a functor S : SCHY→SCHY as
follows:S(X,C)=(X,r˜ C)˜ and S(f )=f˜. Proof of the following proposition is now immediate.
Proposition3.18. We have thatSis a completion functor andSCHYis a comple-tion subcategory ofCHY.
There can be many applications of this theory in the completion of linear Cauchy spaces andCauchy groups. Fric, Kent andRichardson have studiedthese spaces with theT2restriction on the underlying spaces. However, if we develop this theory with-out theT2restriction, we can generalize several problems in functional analysis. Of course, in those cases we have to look at linears-maps andstable completions with compatible algebraic structures. The categorical properties like Cartesian closedness of the subcategory CHYof CHY also remain to be investigated.
References
[1] N. Bourbaki, Elements of Mathematics. General Topology. Part 1, Hermann, Addi-son Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., Paris, 1966. MR 34#5044a. Zbl 301.54001.
[2] A. Csaszar, λ-complete filter spaces, Acta Math. Hungar. 70 (1996), no. 1-2, 75–87. MR 97d:54005. Zbl 855.54038.
[3] R. Fric andD. C. Kent,Completion functors for Cauchy spaces, Int. J. Math. Math. Sci.2 (1979), no. 4, 589–604. MR 80m:54042. Zbl 428.54018.
[4] H. Herrlich andG. E. Strecker,Category Theory, An introduction, 2nd ed., Sigma Se-ries in Pure Mathematics, vol. 1, Heldermann Verlag, Berlin, 1979. MR 81e:18001. Zbl 437.18001.
[5] H. H. Keller,Die Limes-Uniformisierbarkeit der Limesraume., Math. Ann.176(1968), 334– 341. MR 37#874. Zbl 155.50302.
[6] D. C. Kent,The regularity series of a Cauchy space, Int. J. Math. Math. Sci.7(1984), no. 1, 1–13. MR 85d:54034. Zbl 564.54004.
[7] D. C. Kent andG. D. Richardson,Regular completions of Cauchy spaces, Pacific J. Math. 51(1974), 483–490. MR 52#11811. Zbl 291.54024.
[8] , Cauchy completion categories, Canad. Math. Bull. 32 (1989), no. 1, 78–84. MR 90a:54007. Zbl 675.54004.
[9] G. Preuss,Theory of Topological Structures, Mathematics andits Applications, vol. 39, D. Reidel Publishing Co., Dordrecht, Boston, 1988, An approach to categorical topol-ogy. MR 89m:54014. Zbl 649.54001.
[10] E. E. Reed,Completions of uniform convergence spaces, Math. Ann.194(1971), 83–108. MR 45#1109. Zbl 217.19603.
[11] S. Willard,General Topology, Addison Wesley Publishing Co., Reading, Mass. London Don Mills, Ont., 1970. MR 41#9173. Zbl 205.26601.
[12] O. Wyler,Ein Komplettierungsfunktor fur uniforme Limesraume, Math. Nachr.46(1970), 1–12. MR 44#985. Zbl 207.52603.
Nandita Rath: Department of Mathematics, University of Western Australia, Ned-lands, WA6009, Australia