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p-TOPOLOGICAL AND p-REGULAR: DUAL NOTIONS IN CONVERGENCE THEORY
SCOTT A. WILDE and D. C. KENT
(Received 3 July 1997 and in revised form 20 August 1997)
Abstract.The natural duality between “topological” and “regular,” both considered as convergence space properties, extends naturally top-regular convergence spaces, resulting in the new concept of ap-topological convergence space. Taking advantage of this duality, the behavior ofp-topological andp-regular convergence spaces is explored, with particular emphasis on the former, since they have not been previously studied. Their study leads to the new notion of a neighborhood operator for filters, which in turn leads to an especially simple characterization of a topology in terms of convergence criteria. Applications include the topological and regularity series of a convergence space.
Keywords and phrases. Convergence space, interior map, closure map,p-topological,p- regular, initial structure, final structure, topological series, regularity series.
1991 Mathematics Subject Classification. 54A20, 54A10, 54D10.
Introduction. In 1990, G. Richardson and one of the authors introduced the notion ofp-regular convergence space, [6], defined as follows: Ifq andp are convergence structures on a setX, then the space(X,q)isp-regular if clpᏲ →q xwheneverᏲ →q x, where “clp” is thep-closure operator. Clearlyp-regularity is equivalent to regularity when p=q. By varyingp, one can characterize various convergence properties in terms ofp-regularity (see [6, 7]).
More recently, Kent and Richardson [7] developed some ideas and results due to Kowalsky [8], Cook and Fischer [1], and Biesterfeldt [2] to give convergence character- izations of the properties “topological” and “regular” so as to reveal a fundamental duality between these notions. These characterizations made use of “diagonal” ax- iomsFandRwhich are in a natural way dual to each other. (It should be noted that the axiom calledRin this paper was calledDFin [7].)
In this paper, we begin by proving thep-regularity of a convergence space(X,q)also has a “diagonal” characterization in terms of an axiom we callRp,q, which is obtained by making a minor alteration in the axiomR. We then use the dual axiomFp,qto define (and introduce) the dual notion of a “p-topological convergence space.”
Our goal is two-fold. We wish to study and develop this new concept of a p- topological convergence space, while simultaneously exploring the duality alluded to in the title of the paper. The approach based on duality is most useful in examining the structural behavior ofp-topological andp-regular spaces as well as their upper and lower modifications. This approach is adopted in Sections 1 and 4. In Section 2, we study some aspects ofp-topological spaces which do not have obvious analogues in the setting ofp-regular spaces. Section 3 introduces the “neighborhood operator
for filters” which seems to be “tailor-made” for the study ofp-topological spaces and is used extensively in Section 4. The characterization ofp-topological spaces, given in Theorem 3.2, yields a corollary which gives a simple and elegant characterization of a topology in terms of convergence criteria.
As is shown in Section 2 of this paper and also in [6, 7], both of the properties
“p-topological” and “p-regular” can be adapted to characterize various convergence and topological concepts and, thereby, reveal underlying relationships between them.
Other applications of these notions include theregularityandtopological seriesof a convergence space which are discussed briefly in Section 5.
1. The Axioms Fp,qand Rp,q. For standard notation and terminology pertaining to convergence spaces, the reader is referred to [7]. In particular,F(X)denotes the set of all filters on a setX,U(X)the set of all ultrafilters onX, andC(X)the complete lattice of all convergence structures onX(with the discrete topology as the greatest element). Let ˙xdenote the fixed ultrafilter onXgenerated byx∈X.
If(X,q)is a convergence space andJan arbitrary set, letᏲ∈F(J)and letσ :J → F(X)be an arbitrary “selection function.” We defineκσᏲ to be the filter∪F∈Ᏺ∩x∈F
σ (x)inF(X); κσᏲis called thecompression of Ᏺrelative toσ.
We, next, define two axioms pertaining to two convergence structuresp,qon a setX.
Fp,q: LetJbe any set,ψ:J →X, and letσ:J →F(X)have the property thatσ (y)→p ψ(y)for ally∈J. IfᏲ∈F(J)is such thatψ(Ᏺ)→q x, thenκσᏲ →q x.
Rp,q: LetJbe any set,ψ:J →Xand letσ :J →F(X)have the property thatσ (y)→p ψ(y), for ally∈J. IfᏲ∈F(J)is such thatκσᏲ →q x, thenψ(Ᏺ)→q x.
Theorem1.1. Let(X,q)be a convergence space andp∈C(X). Then(X,q)isp- regular if and only ifpandqsatisfyRp,q.
Proof. Recall that(X,q)isp-regular if clp(Ᏺ) →q xwheneverᏲ →q x.
(⇐)Assume that(X,q)andpsatisfyRp,q. LetJ= {(Ᏻ,y):Ᏻ∈U(X),y∈X,Ᏻ →p y}. Defineψ:J →X byφ(Ᏻ,y)=y and σ: J →F(X)byσ (Ᏻ,y)=Ᏻ. Note that σ (z) →p ψ(z), for allz=(Ᏻ,y)∈J.
Assume thatᏲ →q x. We define a filter Ᏼ∈F(J)as follows: for each F ∈Ᏺ, let HF = {(Ᏻ,y)∈J: F ∈Ᏻ}, and letᏴ be the filter on J generated by {HF: F ∈Ᏺ}.
SinceF ∈σ (Ᏻ,y)for every (Ᏻ,y)∈HF, F ∈κσᏴ, soκσᏴ≥Ᏺ. Thus,κσᏴ →q x.
But observe thatψ(HF)=clp(F), so clp(Ᏺ)≥ψ(Ᏼ). ByRp,q,ψ(Ᏼ) →q x, which then implies that clp(Ᏺ)→q x. Thus,(X,q)isp-regular.
(⇒)Assume that(X,q)isp-regular. LetJ,σ,φbe as inRp,qand letᏲ∈F(J)such thatκσᏲ →q x. We claim that clp(κσᏲ)≤ψ(Ᏺ). LetF∈Ᏺand chooseAy∈σ (y), for everyy ∈F. Thenσ (y) →p ψ(y), for everyy∈F, which implies that ψ(y)∈ clp(
y∈FAy)holds for everyy∈F, and soψ(F)⊆clp(
y∈FAy). Since
y∈FAy is a basic set inκσᏲ, the claim is verified. Byp-regularity, clp(κσᏲ) →q x, which implies thatψ(Ᏺ) →q x.
If (X,q) is a convergence space and p ∈C(X), then (X,q) is defined to be p-
topological if(X,q) andp satisfy the axiomFp,q. Note that, by Theorem 1.1,(X,q) isp-regular if and only if(X,q)andpsatisfyRp,q. SinceFp,qandRp,qare dual to each other, “p-topological” and “p-regular” are likewise dual properties. In the special case wherep=q,Fq,qandRq,qare denoted byFandR, respectively.
Theorem1.2. Let(X,q)be a convergence space.
(i) (X,q)is topological if and only if(X,q)satisfiesF.
(ii) (X,q)is regular if and only if(X,q)satisfiesR.
Proof. The first assertion is proved in [6], the second by combining results from [2, 1].
It follows from Theorem 1.2 that “p-topological” generalizes “topological” in the same way that “p-regular” generalizes “regular”. In the next theorem,Fp,q andRp,q
are applied directly to determine the behavior of these properties relative to initial constructs.
Theorem1.3. Initial structures.
(i) Let{(Xi,qi):i∈I}be a set of spaces together with a set of convergence structures piwhich satisfiesFpi,qi, for alli∈I. Let X be a set and letfi:X →Xibe a mapping, for eachi∈I. Ifqis the initial structure onXrelative to the families{(Xi,qi):i∈I}
and{fi:i∈I}, andpis the initial structure onXrelative to{pi:i∈I}and{fi:i∈I}, then(X,q)andpsatisfyFp,q.
(ii) Statement (i) remains valid if Fpi,qi is replaced by Rpi,qi andFp,q is replaced byRp,q.
Proof. (i) It is well known thatq-convergence is characterized by:Ᏺ →q xif and only iffi(Ᏺ) →qi fi(x), for alli∈I. LetJ be a set andψ:J →X andσ: J →F(X) have the property thatσ (j) →p ψ(j) for allj ∈J. Defineσi(j) and ψi(j)so that σi(j)=fi(σ (j)) and ψi(j)=fi(ψ(j)) for all j∈J and i∈I, respectively. Thus, σi(j) →pi ψi(j)for allj∈J. Also,fi(κσᏲ)=κ(fi◦σ )Ᏺ=κσiᏲ. Now, let Ᏺ∈F(J) have the property thatψ(Ᏺ) →q xwhich then implies thatfi(ψ(Ᏺ)) →qi fi(x)for all i∈I, by the property ofq being the initial structure of all theqi. Thus,fi(κσᏲ)= κσiᏲ →qi fi(x)for alli∈Iby the propertyFpi,qi.Hence,κσᏲ →q xby the definition ofqand this implies that(X,q)andpsatisfyFp,q.
(ii) This proof is essentially the same as that of (i).
Corollary1.4. Asubspace of ap-topological (respectively,p-regular) space isp- topological (respectively,p-regsular), wherepdenotes the restriction ofpto the sub- space.
Corollary1.5. Let(X,q)=Πi∈I(Xi,qi)and(X,p)=Πi∈I(Xi,pi)be product con- vergence spaces. If each(Xi,qi)ispi-topological (respectively,pi-regular), then(X,q) isp-topological (respectively,p-regular).
Corollary1.6. LetXbe a set and letΛ= {qi:i∈I}andΓ= {pi:i∈I}be subsets of C(X). Letq=supΛandp=supΓ. If(Xi,qi)ispi-topological (respectively,pi-regular) for eachi∈I, then(X,q)isp-topological (respectively,p-regular).
Before proving the analogue of Theorem 1.3 for final structures, we give a simpler characterization forp-topological spaces which makes use of thep-interior operator Ᏽp.
Theorem1.7. Let(X,q)be a convergence space andp∈C(X). Then(X,q)isp- topological if and only if, wheneverᏲ →q x, there existsᏳ →q xsuch thatᏲ≥ᏵpᏳ.
Proof. Assume that(X,q)andp satisfyFp,q. Let Ᏺ →q x andJ= {(Ᏻ,x),:Ᏻ∈ U(X),Ᏻ →p x}. Let ψ: J → X be defined by ψ((Ᏻ,x))=x and σ: J →F(X)by σ ((Ᏻ,x))=Ᏻ. Note thatψis ontoX (sincex∈X⇒(x,x)˙ ∈J). IfᏴ=ψ−1(Ᏺ), then Ᏼ∈F(J)andψ(Ᏼ)=Ᏺ →q x. ByFp,q,κσᏴ →q x. Thus, to show that(X,q)satisfies the given condition, it suffices to show thatᏲ≥Ᏽp(κσᏴ).
LetF ∈Ᏺ; thenψ−1(F)is a basic set inᏴ. Note thatᏳ∈U(X)and Ᏻ →p x∈F imply that(Ᏻ,x)∈ψ−1(F). Sinceσ ((Ᏻ,x))=Ᏻ, for each pair(Ᏻ,x)∈ψ−1(F), choose G(Ᏻ,x)∈Ᏻ. ThenA= ∪{G(Ᏻ,x):(Ᏻ,x)∈ψ−1(F)} is a basic set inκσᏴ. For a given y∈F,Ay= ∪{G(Ᏻ,y):Ᏻ∈U(X),Ᏻ →p y} ∈ᐂp(y)(sinceᐂp(y)is the intersection of all ultrafilters whichp-converge to y). SinceAx⊆A,for allx∈F,A∈ᐂp(x),for all x∈F. Thus,F⊆Ᏽp(A)and we obtain the desired conclusion thatᏲ≥Ᏽp(κσᏴ).
Conversely, letJ,ψ,σ ,andᏲbe as inFp,qand letψ(Ᏺ) →q x. Since(X,q)satisfies the specified condition, there exists a filterᏳ →q x such thatψ(Ᏺ)≥ᏵpᏳ. To show thatκσᏲ →q x, it suffices to show thatκσᏲ≥Ᏻ. LetG∈Ᏻand chooseF∈Ᏺsuch that ψ(F)⊆Ᏽp(G). For eachy∈F,ψ(y)∈Ᏽp(G)implies thatG∈σ (y). Thus,G∈κσᏲ, which yields the desired result thatκσᏲ≥Ᏻ.
Letf: (X,q) →(Y ,p)be a function between convergence spaces. We definef to be aninterior map iff (Ᏽp(A))⊆Ᏽp(f (A))holds for allA⊆X, and aclosure mapif clp(f (A))⊆f (clq(A))holds for allA⊆X. Closure maps were introduced in [6], where they were found to be useful in the study ofp-regularity.
Theorem1.8. LetXbe a set,{(Xi,qi):i∈I}a set of convergence spaces, and{fi: i∈I}a set of functions mappingXitoXsuch thatX= ∪i∈If (Xi). Letq be the final convergence structure onXinduced by{fi:i∈I}and{(Xi,qi):i∈I}.
(i) If each (Xi,qi)ispi-topological for somepi∈C(Xi)andp is a convergence structure onX such that eachfi:(Xi,pi) →(X,p)is an interior map, then(X,q)is p-topological.
(ii) If each(Xi,qi)ispi-regular for somepi∈C(Xi)andpis a convergence structure onXsuch that eachfi:(Xi,pi) →(X,p)is a closure map, then(X,q)isp-regular.
Proof. (i) LetᏲ →q x. Then there existsj∈I,xj∈Xj such thatfj(xj)=xand Ᏺj
qj
→xjsuch thatfj(Ᏺj)≤Ᏺ. Since(Xj,qj)ispj-topological, there existsᏳj qj
→xj
such thatᏲj ≥Ᏽpj(Ᏻj). Sincefj is an interior map, fj
Ᏽpj(Ᏻj)
≥Ᏽp fj(Ᏻj)
. By continuity offj:(Xj,qj) →(X,q),fj(Ᏻj)→q x, andᏲ≥fj(Ᏺj)≥Ᏽp
fj(Ᏻj)
, so(X,q) isp-topological by Theorem 1.7.
The proof of (ii) is similar.
Corollary1.9. Letf:(X,q)→(X,q)be a convergence quotient map.
(i) Iff:(X,p)→(X,p)is an interior map and(X,q)isp-topological, then(X,q)
isp-topological.
(ii) Iff:(X,p) →(X,p)is a closure map and(X,q)isp-regular, then(X,q)is p-regular.
Corollary1.10. Let(X,q)=
i∈I(Xi,qi)be a disjoint sum of convergence spaces andp∈C(X). Letgi:Xi →Xbe the canonical injection.
(i) If for eachi∈I,(Xi,qi)ispi-topological andgi:(Xi,pi) →(X,p)is an interior map, then(X,q)isp-topological.
(ii) If for eachi∈I,(Xi,qi)ispi-regular andgi:(Xi,pi)→(X,p)is a closure map, then(X,q)isp-regular.
Corollary1.11. LetΛ= {qi:i∈I} ⊆C(X), letp∈C(X), and assume that(X,qi) ispi-topological (respectively,pi-regular), for alli∈I. Ifpi≤pfor eachi∈Iandq= infΛ, then(X,q)isp-topological (respectively,p-regular).
The final result of this section, which follows immediately from Corollaries 1.6 and 1.11, asserts that for a fixed convergence structure pon X, both of the properties
“p-topological” and “p-regular” are preserved under arbitrary infima and suprema in the latticeC(X).
Corollary1.12. LetΛ= {qi:i∈I} ⊆C(X)and letp∈C(X)be such that(Xi,qi) isp-topological (respectively,p-regular), for alli∈I. Letq=infΛandr=supΛ. Then both(X,q)and(X,r )arep-topological (respectively,p-regular).
2. More onp-topological spaces. In Section 1, we observed thatp-topological and p-regular properties exhibit essentially the same structural behavior. Now, we gain some additional insight into the behavior ofp-topological spaces by making use of Theorem 1.7. The first result of this section gives a simple characterization of pre- topological spaces which arep-topological.
Theorem2.1. Let(X,q)be a pretopological space andp∈C(X).
(i) (X,q)isp-topological if and only ifᐂq(x)=Ᏽpᐂq(x).
(ii) If(X,q)isp-topological, thenq≤τp, whereτpdenotes the topological modifi- cation ofp.
Proof. (i) Assume that(X,q) is p-topological. Since ᐂq(x) →q x, it follows by Theorem 1.7 thatᐂq(x)≥Ᏽpᐂq(x), and, hence,ᐂq(x)=Ᏽpᐂq(x). Conversely, if the given equality holds, thenᏲ →q ximpliesᏲ≥ᐂq(x)=Ᏽpᐂq(x), and since(X,q)is pretopological,ᐂq(x)→q x, so(X,q)isp-topological by Theorem 1.7.
(ii) If(X,q)isp-topological, then by (i)ᐂq(x)=Ᏽpᐂq(x), and it follows thatᐂq(x) has a filter base ofp-open sets (which are the same asτp-open sets). Thus,ᐂq(x)≤ ᐂτp(x), and sinceqis a pretopology,q≤τp.
Example2.2. Converse of Theorem 2.1(ii) is generally false.
LetX=Rbe the set of real numbers, and letτdenote the usual topology onR. Note thatτ∪{0}is a base for a topologyponR, whereτ < pandᐂp(x)=ᐂτ(x), for all x≠0, whereasᐂp(0)=˙0. Letqbe the pretopology onRdefined byᐂq(x)=ᐂτ(x) forx≠0 andᐂq(0)=ᐂτ(0)∨Q˙ (where ˙Qis the filter of oversets of the setQof
rational numbers). Note thatτ < q < p. Thenᐂq(x) →q x, butᏵpᐂq(0)=˙0≠ᐂq(0), so by Theorem 2.1(i),(X,q)is notp-topological.
Corollary2.3. Ifpandqare topological, then(X,q)isp-topological if and only ifq≤p.
Proof. If(X,q) isp-topological, thenp≤q follows from Theorem 2.1(ii). Con- versely, ifq≤p, thenᏵpᐂq(x)=ᐂq(x)follows becauseqis a topology, and so, the conclusion follows from Theorem 2.1(i).
The preceding example shows that Corollary 2.3 does not hold under the weaker condition thatqis pretopological.
Note that if(X,q)isp-topological, then(X,q)is obviouslyp-topological for any p≥p. Clearly, every convergence space isδ-topological, whereδdenotes the discrete topology.
Corollary2.4. If(X,q)isp-topological, then(X,πq)and(X,τq)arep-topological, andτq≤πq≤τp(whereπqdenotes the pretopological modification ofq).
Proof. LetᏲ →q x; then, by Theorem 1.7, there existsᏳ →q xsuch thatᏲ≥ᏵpᏳ≥ Ᏽpᐂq(x). This holds for everyᏲ →q x, soᐂπq(x)=ᐂq(x)≥Ᏽpᐂπq(x). Thus,πqis p-topological from Theorem 2.1(i).τq≤πq≤τp follows from Theorem 2.1(ii), and (X,τq)isτp-topological from Corollary 2.3,(X,τq)isp-topological from the remark preceding the corollary, sinceτp≤p.
Corollary2.5. Letιdenote the indiscrete topology onX. Then(X,ι)isp-topological, for everyp∈C(X).
Proof. By Corollary 2.3,(X,ι)is ι-topological, and, hence, p-topological for all p∈C(X)by the remark preceding Corollary 2.4.
Given a convergence space(X,q), letρqdenote the finest completely regular topol- ogy onXcoarser thanq, and letωqbe the finest completely regular topology onX coarser thanq.
Theorem2.6. Aconvergence space (X,q) is a regular (respectively, completely regular) topological space if and only if (X,q) is ρq-topological (respectively, ωq- topological).
Proof. If (X,q) is a regular topological space, thenq =ρq and (X,q) is obvi- ouslyq-topological. Hence,ρq-topological. Conversely, if(X,q)isρq-topological, then (X,q)is clearlyq-topological, and, hence, topological. By Corollary 2.4,q≤ρq, and, hence,q=ρq. Thus,(X,q)is regular and topological.
Let (X,q) be a topological space, and let q be the topology on X generated by Ꮾq= {X}∪{U⊆X:U∈qandU⊆Kfor someq-compact subsetKofX}.
Theorem2.7. AT1topological space(X,q)is locally compact if and only if(X,q) isq-topological.
Proof. Let (X,q) be locally compact andx ∈X. Let U ∈ᐂq(x)be q-open. By
local compactness, there is a compact setA∈ᐂq(x). LetVbe aq-open set such that V⊆U∩A. ThenVisq-open. So,Ᏽqᐂq(x)=ᐂq(x), which implies, by Theorem 2.1(i), that(X,q)isq-topological.
Conversely, let(X,q) beq-topological and x∈X. Sinceq isT1, there existsU∈ ᐂq(x)such thatU≠X. SinceᏵqᐂq(x)=ᐂq(x), by Theorem 2.1(i),Ᏽq(U)∈ᐂq(x), andᏵq(U)=U⊆A, which implies thatA∈ᐂq(x). Thus,(X,q)is locally compact.
A related theorem characterizing local compactness in terms ofp-regularity is the following result, which is a direct corollary of [6, Thm. 3.1].
Theorem2.8. Let(X,q) be aT1convergence space. Let p be the topology on X having as a base of closed sets all the nonempty subsets ofq-compact sets. Then(X,q) is locally compact if and only if(X,q)isp-regular.
3. The neighborhood operator for a filter. In this section, we introduce a new filter notion which is essentially dual to the “closure of a filter,” thereby obtaining another characterization of “p-topological” which further illustrates its duality with
“p-regular”.
Let(X,q)be a convergence space,Ᏺ∈F(X). ThenᐂqᏲ= {A∈Ᏺ:Ᏽq(A)∈Ᏺ}is called theq-neighborhood filter ofᏲ.
Proposition3.1. If(X,q)is a convergence space andᏲ∈F(X), thenᐂqᏲis the finest filter onXsuch thatᏲ≥Ᏽq(ᐂqᏲ).
Proof. It is clear thatᐂqᏲis a filter onXsuch thatᏵqᐂqᏲ≤Ᏺ. IfᏳis any filter onXsuch thatᏵqᏳ≤Ᏺ, thenG∈ᏳimpliesᏵqG∈Ᏺ, and, hence,G∈ᐂqᏲ.
IfᏲ=x, it is obvious from the definition that˙ ᐂqx˙=ᐂq(x)is theq-neighborhood filter atx.
Recall that(X,q)isp-regular ifᏲ →q ximplies clpᏲ →q x. The corresponding dual characterization for ap-topological space is the following.
Theorem3.2. Aconvergence space (X,q)is p-topological if and only ifᏲ →q x impliesᐂpᏲ →q x.
Proof. Let(X,q)bep-topological andᏲ →q x. By Theorem 1.7, there isᏳ →q xsuch thatᏲ≥ᏵqᏳ. By Proposition 2.1, ᐂpᏲ≥Ᏻ, and soᐂpᏲ →q x. Conversely if the con- dition holds, we can setᏳ=ᐂpᏲin Theorem 1.7, and, thus,(X,q)is p-topological.
Corollary3.3. Aconvergence space (X,q)is topological if and only if Ᏺ →q x impliesᐂqᏲ →q x.
Theq-neighborhood filter of a filter can also be described by means of the compres- sion operator for filters defined in Section 1.
Proposition3.4. Let (X,p) be a convergence space and let σ: X →F(X) be defined byσ (x)=ᐂp(x)for allx∈X. Then for anyᏲ∈F(X),κσᏲ=ᐂp(Ᏺ).
Proof. LetA∈ᐂp(Ᏺ). ThenᏵp(A)∈Ᏺ. IfF =Ᏽp(A), then for eachx∈F, A∈
ᐂp(x), and soA=
x∈FVx, where each Vx=A, is a basic set inκσᏲ. Conversely, let A∈κσᏲ. Then A contains a basic set of the formB =
y∈FVy, whereF ∈Ᏺ and Vy ∈ᐂp(y), for ally ∈F. To show that A∈ᐂp(Ᏺ), it suffices to show that F⊆Ᏽp(B). x∈Fimplies thatx∈Vx⊆B. Thus,B∈ᐂp(x)and so,x∈Ᏽp(B).
Let(X,q)be a convergence space andᏲ∈F(X). For anyn∈N, the set of natural numbers, thenth iterations of the closure and neighborhood operators for a filterᏲ are given inductively by:
clnqᏲ=clq
cln−1q Ᏺ, ᐂnqᏲ=ᐂq
ᐂn−1q Ᏺ. (1)
The next two propositions summarize (without proof) some additional elementary properties of the neighborhood operator for filters.
Proposition3.5. Let(X,q)be a convergence space,n∈N, and{Ᏺi:i∈I} ⊆F(X).
Then:
(i) ᐂnp(∩i∈IᏲi)= ∩i∈IᐂnpᏲi;
(ii) If∨i∈IᏲiexists, thenᐂnp(∨i∈IᏲi)≥ ∨i∈IᐂnpᏲi;
(iii) Equality holds in(ii)under the additional assumption that{Ᏺi:i∈I}is an upward directed set of filters.
Proposition3.6. Letf:(X,q) →(Y ,p)be a function between convergence spaces.
LetᏲ∈F(X)andn∈N.
(i) Iff is continuous, thenf
ᐂnqᏲ≥ᐂnpf (Ᏺ).
(ii) Iffis an interior map, thenf
ᐂnqᏲ≤ᐂnpf (Ᏺ).
4. Lower and upper modifications. It was established in Corollary 1.12 that each of the properties p-topological and p-regular is preserved under both infima and suprema in the latticeC(X). Since an indiscrete space is bothp-topological and p- regular for any choice ofp, we immediately obtain the following.
Proposition4.1. Let(X,q)be a convergence space andp∈C(X).
(i) There is a finestp-topological convergence structureτpqonXcoarser thanq.
(ii) There is a finestp-regular convergence structurerpqonXcoarser thanq.
The structuresτpqandrpqare called thelower p-topologicalandlowerp-regular modificationsofq, respectively. The dual relationship between these concepts is evi- dent in the next theorem.
Theorem4.2. Let(X,q)be a convergence space andp∈C(X).
(i) Ᏺτ→pqxif and only if there existsᏳ →q xsuch thatᏲ≥ᐂnpᏳ, for somen∈N.
(ii) Ᏺr→pqxif and only if there existsᏳ →q xsuch thatᏲ≥clnpᏳ, for somen∈N.
Proof. (i) Letqbe defined byᏲ →q x if and only if there isᏳ →q x such that Ᏺ≥ᐂnp(Ᏻ), for somen∈N. One may easily verify thatqis a convergence structure.
IfᏲ →q x, thenᏲ≥ᐂnp(Ᏺ)for anyn∈N, and soᏲ →q x. Thus,q≤q. To show that
qisp-topological, letᏲ →q xand letᏳ →q xbe such thatᏲ≥ᐂnp(Ᏻ)for somen∈N.
Thenᐂp(Ᏺ)≥ᐂn+1p (Ᏻ), soᐂp(Ᏺ) →q x, and by Theorem 3.2,qisp-topological.
Finally, assume thatrisp-topological andr≤q. LetᏲ →q x. Then there isᏳ →q x such that Ᏺ≥ᐂnp(Ᏻ) for some n∈N. Ᏻ →q x implies Ᏻ →r x, and since r is p- topological, ᐂnp(Ᏻ) →r x, for alln∈N. ButᏲ≥ᐂnp(Ᏻ)for somen∈N, and, hence, Ᏺ →r x. Thus,r≤qand the proof is complete.
(ii) See [6, Thm. 2.2].
Since the discrete topologyδ on a setX is generally neitherp-topological norp- regular for an arbitraryp∈C(X), the existence of an upperp-topological (or upper p-regular) modification for someq∈C(X)depends on the existence of ap-topological (orp-regular) convergence structure onX finer thanq. Clearly,τpδis the finest p- topological structure inC(X)andrpδis the finestp-regular member ofC(X). Thus, a coarsestp-topological (respectively,p-regular) convergence structure onXfiner than qexists if and only ifq≤τpδ(respectively,q≤rpδ). Using Theorem 4.2, this result may be restated as follows.
Theorem4.3. Let(X,q)be a convergence space andp∈C(X).
(i) There is a coarsestp-topological convergence structureτpqonXfiner thanqif and only ifᐂnp(˙x)→q x, for allx∈Xand for alln∈N.
(ii) There is a coarsestp-regular convergence structurerpqonXfiner thanqif and only ifclnp(x)˙ →q x, for allx∈Xand for alln∈N.
When they exist,τpqandrpqare called theupperp-topologicalandupperp-regular modificationsofq, respectively. Note that forτpqto exist, it is necessary thatq≤p, and thatrpqwill exist wheneverpisT1.
Theorem4.4. Let(X,q)be a convergence space andp∈C(X).
(i) Ifτpqexists, thenᏲτ→pqxif and only ifᐂnp(Ᏺ∩x)˙ →q x, for alln∈N.
(ii) Ifrpqexists, thenᏲr→pqxif and only ifclnp(Ᏺ∩x)˙ →q x, for alln∈N.
Proof. (i) Letq∗be defined byᏲ q→∗ xif and only ifᐂnp(Ᏺ∩x)˙ →q x, for alln∈N.
It is easily shown thatq∗ is a convergence structure. IfᏲ q→∗ x, thenᏲ≥ᐂnp(Ᏺ∩x),˙ andᐂnp(Ᏺ∩x)˙ →q ximpliesᏲ →q x. Thus,q≤q∗. To show thatq∗isp-topological, Ᏺ q→∗ x impliesᐂn+1p (Ᏺ∩x)˙ =ᐂnp(ᐂp(Ᏺ∩x))˙ →q x for all n∈ N, which implies ᐂnp(ᐂp(x)∩x)˙ →q x, for alln∈N. Thus, ᐂp(x) q→∗ x and so, q∗ is p-topological by Theorem 3.2.
Finally, assume thatr isp-topological andq≤r. Then, by Theorem 3.2, Ᏺ →r x impliesᐂnp(Ᏺ∩x)˙ →r x, for eachn∈N, and, hence,ᐂnp(Ᏺ∩x)˙ →q x, for eachn∈N.
But this impliesq∗≤r. Thus,q∗=τpq.
The proof of (ii) is similar.
Theorem4.5. Let(X,q)and(X,q)be convergence spaces and let f: (X,q) → (X,q)be continuous. Assume thatp∈C(X)andp∈C(X).
(i) Iff:(X,p) →(X,p)is continuous, then both of the mappingsf:(X,τpq) → (X,τpq)andf:(X,rpq) →(X,rpq)are continuous.
(ii) If both ofτpqandτpqexist andf:(X,p) →(X,p)is an interior map, then f:(X,τpq) →(X,τpq)is continuous.
(iii) If both ofrpqandrpqexist andf:(X,p) →(X,p)is a closure map, thenf:
(X,rpq) →(X,rpq)is continuous.
Proof. Those results pertaining top-regular structures have been proved in [6].
Those pertaining top-topological structures can be proved analogously using Theo- rems 4.2(i) and 4.4(i), along with Proposition 3.6.
The next two theorems show that the lower modifications behave reasonably well relative to final structures, whereas the upper modifications exhibit comparable be- havior relative to initial structures.
Theorem4.6. LetXbe a set and let{(Xi,qi):i∈I}and{(Xi,pi):i∈I}be collec- tions of convergence spaces, and for alli∈I,fi:Xi →X. Letqbe the final structure onXinduced by{fi:i∈I}and{(Xi,qi):i∈I}and letp∈C(X). Furthermore, assume thatX= ∪i∈Ifi(Xi).
(i) If eachfi:(Xi,pi) →(X,p)is a continuous interior map, thenτpq is the final structure onXinduced by{fi:i∈I}and{(Xi,τpiqi):i∈I}.
(ii) If eachfi:(Xi,pi) →(X,p) is a continuous closure map, thenrpqis the final structure onXinduced by{fi:i∈I}and{(Xi,rpiqi):i∈I}.
Proof. (i) Letsdenote the final structure onXinduced by{fi:i∈I}and{(Xi,τpiqi) :i∈I}. LetᏲ →s x. Then there isi∈I,xi∈Xi, andᏳi
τpiqi
→ xisuch thatᏲ≥fi(Ᏻi).
Thus, there isᏴi qi
→xi and n∈N such thatᏳi ≥ᐂnpiᏴi by Theorem 4.2. Hence, fi(Ᏼi) →q xandᏲ≥fi(Ᏻi)≥fi(ᐂnpiᏴi)=ᐂnpfi(Ᏼi), where the last inequality follows by Proposition 3.6. Thus,Ᏺτ→pqx.
Conversely, letᏲτ→pqx. Then there isᏳ →q xandn∈Nsuch thatᏲ≥ᐂnpᏳ.Ᏻ →q x implies there isi∈I,xi∈Xi, andᏴi qi
→xisuch thatᏳ≥fi(Ᏼi). Note thatᏲ≥ᐂnpᏳ≥ ᐂnpfi(Ᏼi)=fi
ᐂnpiᏴi
. SinceᏴi →qi xi,ᐂnpi(Ᏼi)τpi→qixi, and, thus,Ᏺ →s x.
The proof of (ii) is the similar.
To avoid needless repetition, we state the next three corollaries to Theorem 4.6 only for the lowerp-topological modifications. Analogous results obviously hold for the lowerp-regular modifications as well.
Corollary4.7. Letf:(X,q) →(X,q)be a convergence quotient map and f:
(X,p) →(X,p)an interior-preserving map. Thenf:(X,τpq) →(X,τpq)is a con- vergence quotient map.
Corollary4.8. Let(X,q)=
i∈I(Xi,qi)be a disjoint sum of convergence spaces, and letp∈C(X)be such that eachgi:(Xi,pi) →(X,p)is an interior-preserving map, wheregi:Xi →Xis the canonical injection. Then(X,τpq)=
i∈I(Xi,τpiqi).
Corollary4.9. LetΛ= {qi: i∈I} ⊆C(X)and letp∈C(X). Ifq=inf Λ, then τpq=inf{τpqi:i∈I}.
Theorem4.10. LetXbe a set and let{(Xi,qi):i∈I}and{(Xi,pi):i∈I}be collec-
tions of convergence spaces and, for alli∈I, letfi:X →Xi. Letqbe the initial structure onXinduced by{fi:i∈I}and{(Xi.qi):i∈I}, and assume thatpis a structure such thatfi:(X,p) →(Xi,pi)is continuous, for alli∈I.
(i) Ifτpiqiexists for alli∈Iand eachfi:(X,p) →(Xi,pi)is an interior map, then τpqexists and is the initial structure onXinduced by{fi:i∈I}and{(Xi,τpiqi):i∈I}.
(ii) Ifrpiqiexists for alli∈Iand eachfi:(X,p) →(Xi,pi)is a closure map, then rpqexists and is the initial structure onXinduced by{fi:i∈I}and{(Xi,rpiqi):i∈I}.
Proof. (i) To showτpq exists, it suffices, by Theorem 4.3, to show that ifᏲ≥ ᐂnp(x)for somex∈Xandn∈N, thenᏲ →q x. Sincef:(X,p) →(Xi,pi)is continu- ous for alli∈I,fi
ᐂnp(x)
≥ᐂnpi(fi(x)), and since eachτpiqiexists by assumption, ᐂnpi(fi(x)) →qi fi(x)for eachi∈I. Sinceqis the initial structure,ᐂnp(x) →q x, and, hence,Ᏺ →q x. The remainder of the proof of (i) is straight-forward and is omitted.
The proof of (ii) exactly parallels that of (i).
The corollaries of Theorem 4.10, like those of Theorem 4.6, are stated only for the upperp-topological modifications. The corresponding results involving upperp- regular modifications can be supplied by the reader.
Corollary4.11. Let(X,q) be a subspace of(X,q)and letp∈C(X). Also let (X,p) be a subspace of (X,p)and assume that τpq exists. If X is p-open, then (X,τpq)is a subspace of(X,τpq).
Proof. SinceXisp-open in(X,p), the identity map from(X,p)into(X,p)is a continuous interior map, and so, the conclusion follows from Theorem 4.10(ii).
Corollary4.12. Let(X,q)=Πi∈I(Xi,qi), letpi∈C(Xi)be such thatτpiqiexists for each i∈I, and letp∈C(X)be such that the ithprojection map πi: (X,p) → (Xi,pi)is continuous interior map for alli∈I. Thenτpqexists, and(X,τpq)=Πi∈I(Xi, τpiqi).
Corollary4.13. LetXbe a set,Λ= {qi:i∈I} ⊆C(X), and letp∈C(X)be such thatτpqiexists for alli∈I. Ifq=supΛ, thenτpqexists andτpq=sup{τpqi:i∈I}.
5. The topological series of a convergence space. If(X,q)is a convergence space, it is well known that there is a finest topologyτqcoarser thanqand a finest regular convergence structurer qcoarser thanq. These are thetopologicalandregular modi- ficationsofq. However, neitherτq-convergence norr q-convergence can be described directly in terms ofq-convergence. Consequently, descending ordinal series have been devised to “bridge the gap” betweenqand these two lower modifications.
Theregularity series(rαq), introduced in [4] and studied also in [5], can be easily characterized by means of the lowerp-regular modification for an arbitrary ordinal numberαas follows:rαq=rpαq, wherep0=δ,p1=q,pα=rα−1qifα−1 exists, and pα=inf{rβq:β < α}ifαis a limit ordinal. The least ordinalαfor whichrαq=rα+1q is called thelength of the regularity series and is denoted by/Rq. It is easy to verify thatrαq=r qif and only ifα≥/Rq.
Thedecomposition series(παq), introduced in [3], is a descending ordinal sequence
of pretopologies terminating in τq. Just as the regularity series gives an ordinal measure of how “non-regular” a given convergence space is, so likewise does the decomposition series measure how “non-topological” the given space is. However, the construction of the regularity and decomposition series are fundamentally so dif- ferent that interactions or comparisons between them are difficult to find or interpret.
The existence of the lowerp-topological modification and its dual relationship to the lowerp-regular modification provide means for constructing a new descending ordinal sequence called thetopological series (ταq)of(X,q)which, like the decom- position series, stretches betweenqandτq. Following the preceding description of the regularity series, we define:ταq=τpαq, wherep0=δ,p1=q,pα=τα−1qifα−1 exists, andpα= inf{τβq:β < α}ifαis a limit ordinal.The resultingtopological se- ries is the exact dual of the regularity series. It can be shown that the length of the topological series cannot exceed that of the decomposition series. Additional results pertaining to these and other related ordinal series will be published later.
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Wilde and Kent: Department of Mathematics, Washington State University, Pull- man, WA99164-3113, USA
