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NEIGHBORHOOD SPACES
D. C. KENT and WON KEUN MIN
Received 7 February 2002
Neighborhood spaces, pretopological spaces, and closure spaces are topological space generalizations which can be characterized by means of their associated interior (or clo-sure) operators. The category NBD of neighborhood spaces and continuous maps contains PRTOP as a bicoreflective subcategory and CLS as a bireflective subcategory, whereas TOP is bireflectively embedded in PRTOP and bicoreflectively embedded in CLS. Initial and final structures are described in these categories, and it is shown that the Tychonov theorem holds in all of them. In order to describe a successful convergence theory in NBD, it is necessary to replace filters by more generalp-stacks.
2000 Mathematics Subject Classification: 54A05, 54A10.
1. Introduction. In his original definition of topology, Hausdorff [7] assigned to each point x in a setX a system of neighborhoods subject to certain axioms. We will define a neighborhood structureν onXin the same way, using axioms weaker than those of Hausdorff, but just strong enough to include both pretopological and closure spaces as special cases of neighborhood spaces. Of particular significance is the fact that pretopologies, supratopologies, and neighborhood structures onXare all uniquely determined by their associated interior (or closure) operators, which, in each case, can be axiomatically described by generalizing in different ways the Kuratowski interior operator that characterizes a topology.
To better understand the preceding statements, we recall that a topological space
(X,τ) has an interior operator I which satisfies the following Kuratowski interior axioms:
(i) I(X)=X;
(ii) I(A)⊆A, for allA⊆X;
(iii) I(A∩B)=I(A)∩I(B), for allA,B∈2X; (iv) I(I(A))=I(A), for allA⊆X.
Furthermore, ifI: 2X→2X isany set function satisfying these axioms, thenI de-termines a unique topology onXwhose interior operator isI.
Pretopological spaceswere introduced by Choquet [4] in 1948. A pretopologypon
During the past forty years, there has been a growing interest in what Nj˙astad [10] calls nearly open sets, which include semi-open sets [8,10], as well as preopen and semi-preopen sets [1]. Asupratopology, [9], is defined to be a collection of subsets of a setX(calledsupraopen sets) which containsXand is closed under arbitrary unions, but, unlike a topology, is not required to be closed under finite intersections. If(X,τ)
is a topological space, the collections of semi-open, preopen, and semi-preopen sets relative toτeach form supratopologies onXderived fromτ. Given a supratopologyσ
onX, the interior of any set is its largest supraopen subset, and the interior operator thus obtained satisfies axioms (i), (ii), and (iv), along with an additional axiom: (v) if
A⊆B, thenI(A)⊆I(B)which is, in fact, a generalization of (iii).
Furthermore, ifI: 2X→2X is any set function satisfying (i), (ii), (iv), and (v), then the collection{A⊆X:I(A)=A}is a supratopology onXwhose associated interior operator isI.
A pair(X,σ )consisting of a setXwith a supratopologyσ will be called aclosure space[5,6]. Closure spaces are typically characterized by the closure axioms corre-sponding to axioms (i), (ii), (iv), and (v) of the preceding paragraph. Their study is partially motivated by Birkhoff’s study of the relation between closures and certain Galois connections [2]. A brief discussion of motivations and applications for closure spaces is found in [5]; these are quite independent of those mentioned in the preceding paragraph, although the resulting structures are equivalent.
Throughout this paper, we use the terminterior operator onXto mean a set func-tionI: 2X→2X which satisfies (i), (ii), and (v); the interior operators for topological, pretopological, and closure spaces are special cases. Given any interior operatorIon
Xandx∈X, a subsetVis called aneighborhood of xifx∈I(V ). In this way, each interior operatorI determines a collection of subsets ofXfor eachx∈X, and these neighborhood collections, when appropriately axiomatized inSection 2, will define a neighborhood structureνonX, and the pair(X,ν)is called aneighborhood space. The natural morphisms between neighborhood spaces are calledcontinuous maps.
The category NBD of neighborhood spaces and continuous maps contains, as full subcategories, the category PRTOP of pretopological spaces, CLS of closure spaces, and TOP of topological spaces. By studying initial and final structures in these cate-gories, we show that PRTOP is bicoreflectively embedded in NBD, CLS is bireflectively embedded in NBD, and TOP is bireflectively embedded in PRTOP and bicoreflectively embedded in CLS. Some properties appropriate to the category NBD, including com-pactness, are investigated inSection 5, where it is shown that the Tychonov theorem holds in all of the categories cited above. It is also shown that a satisfactory conver-gence theory can be obtained for neighborhood and closure spaces if filter is replaced by the more general notion of ap-stack. A new notion of “p-compactness” based on ultrap-stacks is introduced and shown to be preserved under arbitrary products in NBD and CLS.
called afilter. The concept we need is intermediate in generality between a stack and a filter.
Definition2.1. A stackᏴon a setXis called ap-stackif it satisfies the following condition:
(p)A,B∈ᏴimpliesA∩B≠∅.
Condition (p) is called thepairwise intersection property(PIP) which is strictly weaker than the well-knownfinite intersection property (FIP). A collectionᏮof subsets ofX
with the PIP is called ap-stack base. For any collectionᏮ, we denote byᏮ = {A⊆X: ∃B∈Ꮾsuch thatB⊆A}the stack generated byᏮ, and ifᏮisp-stack base, thenᏮ is ap-stack. IfᏮis ap-stack base with the FIP, thenᏮis afilter subbase, and in this caseᏮgenerates the filter[Ꮾ]= {A⊆X:∃B1,...,Bn∈Ꮾsuch thatni=1Bi⊆A}.
Let pS(X)(resp.,F(X)) denote the set of allp-stacks (resp., filters) onX, partially ordered by inclusion. The maximal elements in pS(X)(resp.,F(X)) are calledultrap -stacks(resp.,ultrafilters). We may easily verify that every ultrafilter is an ultrapstack, and (via Zorn’s lemma) that everyp-stack is contained in an ultrapstack.
Lemma2.2. ForᏴ∈pS(X), the following are equivalent: (a) Ᏼis an ultrapstack;
(b) ifA∩H≠∅, for allH∈Ᏼ, thenA∈Ᏼ; (c) B ∈ᏴimpliesX\B∈Ᏼ.
Proof. (a)⇒(b). IfA∩H≠∅, for allH∈Ᏼ, andA ∈Ᏼ, thenᏴ∪ Awould be a
p-stack strictly larger thanᏴ.
(b)⇒(c). IfB ∈Ᏼ, then, by (b), there isH∈Ᏼsuch thatB∩H= ∅. ThusH⊆X\B, which impliesX\B∈Ᏼ.
(c)⇒(a). LetᏴ⊆, where∈pS(X)andK∈. IfK ∈Ᏼ, then by (c)X\K∈Ᏼ, and henceX\K∈, a contradiction.
IfXcontains at least three distinct elementsa,b,c, thep-stack base{{a,b},{a,c}, {b,c}}generates an ultrapstack which is not an ultrafilter. IfA⊆Xis nonempty,A denotes the filter of all oversets ofA, and ˙x= {x}denotes the ultrafilter generated by{x}.
For a function f:X→Y and Ᏼ∈pS(X), the image stackf (Ᏼ)in pS(Y ) hasp -stack base{f (H):H∈Ᏼ}. Likewise, ifᏳ∈pS(Y ),f−1(Ᏻ)denotes thep-stack onX
generated by{f−1(G):G∈Ᏻ}.
Proposition2.3. Letf:X→Y be a function,Ᏼ∈pS(X). Then (a)ifᏴis a filter, so isf (Ᏼ);
(b)ifᏴis an ultrafilter, so isf (Ᏼ); (c)ifᏴis an ultrapstack, so isf (Ᏼ).
Proof. (c) IfB∩f (H)≠∅, for allH∈Ᏼ, thenf−1(B)∩H≠∅, for allH∈Ᏼ, and
byLemma 2.2,f−1(B)∈Ᏼ, which impliesB∈f (Ᏼ), and thusf (Ᏼ)is an ultrapstack.
Definition2.4. LetXbe a set, and letν⊆pS(X)be given byν= {ν(x):x∈X}, whereν(x)⊆x˙, for allx∈X. Thenνis called aneighborhood structureonX,ν(x)
is called theν-neighborhood stack atx, and(X,ν)is called aneighborhood space. For convenience, neighborhood will be henceforth abbreviated by nbd.
Let N(X)be the set of all nbd structures onX, partially ordered as follows:ν≤µif and only ifν(x)≤µ(x), for allx∈X(in which caseνiscoarserthanµandµis finer thanν). Ap-stackᏴonX ν-converges tox(writtenᏴ→ν x) ifν(x)⊆Ᏼ.
Definition 2.5. If (X,ν) and (Y ,µ) are nbd spaces, then f :(X,ν)→(Y ,µ) is continuousifµ(f (x))⊆f (ν(x)), for allx∈X, andfisinterioriff (ν(x))⊆µ(f (x)), for allx∈X. A continuous, interior bijection is called ahomeomorphism, and in this case(X,ν)and(Y ,µ)arehomeomorphic.
Proposition2.6. The functionf:(X,ν)→(Y ,µ)is continuous if and only if, when-everᏴ∈pS(X)andᏴ→ν x,f (Ᏼ)→µ f (x).
The category having neighborhood spaces as objects and continuous maps as mor-phisms is denoted by NBD.
If(X,ν)is an nbd space andA⊆X, let
Iν(A)=x∈A:A∈ν(x);
Clν(A)=x∈X:A∩V≠∅,∀V∈ν(x). (2.1)
Proposition2.7. If(X,ν)is a neighborhood space,A⊆X, then (a) Iνsatisfies axiom (i), (ii), (v);
(b) Iν(A)= {x∈A:A∈Ᏼ, for everyp-stackᏴ→ν x};
(c) Clν(A)=X\Iν(X\A);
(d) Clν(A)= {x∈X:∃Ᏼ∈pS(X)such thatᏴ→ν xandA∈Ᏼ}.
Proof. (d) Ifx∈Clν(A), let Ᏼ=ν(x)∪ A. Ifx ∈Clν(A), someV ∈ν(x)has
an empty intersection withA, and so nop-stack containingAcanν-converge tox.
Theorem2.8. LetI(X)be the set of all interior operators onX, partially ordered byI1≤I2if and only ifI1(A)⊆I2(A), for allA⊆X. Ifϕ: N(X)→I(X)is defined by
ϕ(ν)=Iν, thenϕis an order isomorphism.
Proof. If ν ∈N(X), thenϕ(ν)=Iν ∈I(X) by Proposition 2.7. If I ∈I(X), let ψ(I)=νI, whereνI(x)= {A⊆X:x∈I(A)}, for eachx∈X. Thenψ:I(X)→N(X)
follows easily. To showIνI=I, note that for allA⊆X,x∈I(A)if and only ifA∈νI(x)
if and only ifx∈Iν(A)if and only ifA∈νIν(x). Thusϕ◦ψis the identity map on I(X)and ψ◦ϕis the identity map on N(X), where ϕis a bijection. Finally, ν≤µ
in N(X)if and only ifν(x)⊆µ(x), for all x ∈X if and only if A∈ν(x) implies
A∈µ(x) if and only ifIν(A)⊆ Iµ(A), for all A⊆X if and only if ϕ(ν)≤ϕ(µ).
(1)fis continuous; (2)f−1(I
µ(A))⊆Iν(f−1(A)), for allA⊆Y;
(3)f (Clν(B))⊆Clµf (B), for allB⊆X;
(4)fis interior;
(5)Iν(f−1(A))⊆f−1(Iµ(A)), for allA⊆Y;
(6)f−1(Cl
µ(B))⊆Clνf−1(B), for allB⊆Y.
If (X,ν) is a neighborhood space andᏴ ∈pS(X), the associated closurep-stack Clν(Ᏼ)and neighborhoodp-stackᐂν(Ᏼ)are generated by thep-stack bases{ClνH: H∈Ᏼ}and{A⊆X:IνA∈Ᏼ}, respectively. Note thatᐂν(x)˙ =ν(x)is theν-nbd stack
atx.
Proposition 2.10. The interior (or closure) operator for an nbd space (X,ν)is idempotent if and only if ᐂν(ν(x))=ν(x), for allx∈X.
Proof. Letᐂν(ν(x))=ν(x), for all x∈X. IfA⊆X and x∈Iν(A), then A∈
ᐂν(ν(x))impliesIν(A)∈ν(x), and sox∈Iν(Iν(A)). ThusIν=Iν·Iν. The converse
is obvious.
Definition2.11. A neighborhood space(X,ν)is defined to be (a) pretopologicalifν(x)is a filter, for allx∈X;
(b) regular if Clν(ν(x))=ν(x), for allx∈X;
(c) supratopologicalifᐂν(ν(x))=ν(x), for allx∈X;
(d) topologicalif pretopological and supratopological.
The termspretopology, supratopology, andtopology will be used for an nbd struc-tureνwhich is pretopological, supratopological, or topological. Also, a pretopological nbd space(X,ν)will be called apretopological space; likewise fortopological space. A supratopological nbd space will be called aclosure space.
Theorem2.12. A neighborhood space(X,ν)is
(a) pretopological if and only ifIνsatisfies (i), (ii), and (iii);
(b) supratopological if and only ifIνsatisfies (i), (ii), (iv), and (v);
(c) topological if and only ifIνsatisfies (i), (ii), (iii), and (iv).
Proof. (c) is well known and (b) follows byProposition 2.10.
(a) Ifν is pretopological andA,Bare subsets ofX, thenIν(A∩B)⊆Iν(A)∩Iν(B)
follows by (v). Ifx∈Iν(A)∩Iν(B), thenA,B∈ν(x)impliesA∩B∈ν(x), sinceν(x)
is a filter, and henceIν(A)∩Iν(B)⊆Iν(A∩B). Conversely, ifA,B∈ν(x), thenx∈ Iν(A)∩Iν(B)=Iν(A∩B), and henceA∩B∈ν(x), soν(x)is a filter.
Whereas pretopologies are well known, additional discussion of supratopologies is appropriate. In 1983, Mashhour et al. [9] defined asupratopology σ onXto be a subcollection of 2XcontainingXand closed under arbitrary unions. The members of
σ are calledsupraopen sets. For example, if(X,τ)is a topological space, the collec-tions SO(X)= {A⊆X:A⊆ClτIτA}ofsemi-open sets, [8], and PO(X)= {A⊆X:A⊆ IτClτA}ofpreopen sets, [1], are supratopologies on Xderived fromτ. It is easy to
verify that ourDefinition 2.11(c) is equivalent to that given in [9]. Indeed, if (X,σ )
νσ(x)= {U∈σ:x∈U}, thenνσ has an idempotent interior operator and is thus
a supratopological nbd space. On the other hand, if(X,ν)is a supratopological nbd space, thenσν= {U⊆X:IνU=U}is a supratopology in the sense of [9], andνσν=ν.
The remaining details are left to the reader. The termclosure spaceis used rather than supratopological nbd space for brevity and consistency with current usage (see [5,6]). The full subcategories of NBD determined by the pretopological, regular, supratopo-logical, and topological nbd spaces are denoted, respectively, by PRTOP, REGNBD, CLS, and TOP. In the next two sections we study initial and final properties in these cate-gories to determine how the smaller catecate-gories are embedded in the larger ones.
3. The latticeN(X). For any nonempty setX, let PR(X), ST(X), T(X), and RN(X)
denote the sets of pretopologies, supratopologies, topologies, and regular nbd struc-tures onX, respectively, all partially ordered as subsets of N(X). In each case the resulting partially ordered set is a complete lattice, with the discrete and indiscrete topologies as greatest and least elements.
Lemma3.1. IfᏭ= {νi:i∈J} ⊆N(X),ν=infN(X)Ꮽ, andµ=supN(X)Ꮽ, thenν(x)= ∩{νi(x):i∈J}andµ(x)= ∪{νi(x):i∈J}.
Proposition3.2. LetᏭ= {νi:i∈J} ⊆N(X). Then
(1)if eachνi∈Ꮽis a supratopology, so issupN(X)Ꮽ;
(2)if eachνi∈Ꮽis a regular, so issupN(X)Ꮽ;
(3)if eachνi∈Ꮽis a pretopology, so isinfN(X)Ꮽ;
(4)if eachνi∈Ꮽis a topology, so aresupPR(X)ᏭandinfST(X)Ꮽ.
Proof. (1) Letµ=supN(X)Ꮽ; to showµis a supratopology, it suffices to show that V∈µ(x)impliesIµ(V )∈µ(x). ByLemma 3.1,V∈µ(x)impliesV∈ν(x)for some ∈J, and since νis a supratopology, Iν(V )∈ν(x). Butν≤µ implies Iν(V )⊆ Iµ(V ), soIµ(V )∈µ(x).
(2) Letµ=supN(X)Ꮽ,x∈X, andV ∈µ(x). ThenV∈ν(x)for some∈J, and
sinceνis regular, there isW∈ν(x)such that ClνW⊆V. Sinceν≤µ, ClµW⊆V
andW∈ν(x)impliesW∈µ(x). Thusµis regular.
(3) Letν=infN(T )Ꮽ; to showνis a pretopology, it suffices to show that eachν(x)
is a filter for arbitraryx∈X. ByLemma 3.1,ν(x)=i∈Jνi(x), andν(x)is a filter
since eachνi(x)is a filter.
(4) The first part is well known, and the second follows because any intersection of topologies onXis again a topology.
IfAis a complete lattice andB⊆Ahas the inherited order, we say thatB is sup-dense (resp.,inf-dense) in Aif every a∈A is the supremum (resp., infimum) inA
of members ofB. Furthermore,B is said to beorder denseinAifAis the smallest sub-complete lattice of itself containingB.
Theorem3.3. (1) ST(X)is inf-dense inN(X). (2) PR(X)is sup-dense inN(X).
Proof. (1) Letν∈N(X)and, for eachy∈X, letµy be defined by
µy(x)=
ν(y), y=x; ˙
x, y≠x. (3.1)
We may easily verify thatµy∈ST(X)andν=infN(X){µy:y∈X}.
(2) Assumex∈A⊆X, and letµA,x be the pretopology defined by
µA,x(y)=
[A], y=x;
{X}, y≠x. (3.2)
Thenν(x)={µA,x(x):A∈ν(x)}, and byLemma 3.1,ν=supN(X){µA,x:A∈ν(x), x∈X}.
(3) Ifσ is a supratopology expressed as a collection of supraopen sets andU∈σ, letτU be the topology{∅,U,X}. Obviouslyσ=supN(X){τU:U∈σ}.
(4) Ifνis a pretopology, thenµy (as defined in the proof of (1)) is a topology, and ν=infN(X){µy:y∈X}.
Corollary3.4. PR(X),ST(X), andT(X)are order dense inN(X).
Proposition3.5. Let(X,ν)be an nbd space. There is a finest supratopologyσ ν
coarser thanν, a finest regular nbd structureρν coarser thanν, and a coarsest pre-topologyπνfiner thanν.
Proof. Since the setS(ν)of all supratopologies onXcoarser thanνis nonempty (it contains the indiscrete topology),σ ν=supN(X)S(ν). Similarly, the existence ofρν
andπν is obtained with the help ofProposition 3.2.
The structuresσ ν,ρν, andπνare called, respectively, thesupratopological, regu-lar, andpretopological modificationsofν. In the case ofσ νandπν, there are simple characterizations:
σ ν=U⊆X:IνU=U;
πν(x)=ν(x), ∀x∈X. (3.3)
Theorem3.6. (1)If νis a pretopology,σ ν is a topology. (2)If νis a supratopology,πνis a topology.
Proof. (1) LetU,V ∈σ ν. Sinceν is a pretopology,Iν(U∩V )=Iν(U)∩Iν(V )= U∩V. ThusU∩V∈σ ν, soσ ν is a topology.
(2) It suffices to show thatIπν is idempotent. LetA⊆X and x ∈IπνA. Sinceν
is a supratopology, there areV1,...,Vn inν(x)such thatIνV=V,=1,...,n, and n
i=1Vi⊆A. Sinceπνis a pretopology andν≤πν,x∈ n
i=1Iν(Iν(Vi))⊆ n
i=1Iπν(Vi) =Iπν(Iπν(ni=1Vi))⊆IπνIπν(A).
(2)If(X,ν)is a closure space, then the topological modificationπν is the coarsest topology onXfiner thanν.
The terminology of Corollary 3.7 is not ambiguous since the only structures which are both pretopological and supratopological are topologies, for which the topological modification is the structure itself.
4. Initial and final structures. LetXbe a set,{(Xi,νi):i∈J}a collection of nbd
spaces,{fi:i∈J}a corresponding collection of functionsfi:X→Xi. If there is a
coarsest nbd structureνonX, making eachfi:(X,ν)→(Xi,νi)continuous, thenν
is called theinitial nbd structureonXinduced by the given collections of spaces and functions. Initial structures in the categories PRTOP and CLS are defined analogously. Final structureis the dual notion obtained by reversing all arrows and replacing coars-est by fincoars-est. Aninitial propertyP is one preserved under formation of initial structures (i.e., if each(Xi,νi)has property P, so does(X,ν));final property is defined dually.
Proposition4.1. LetXbe a set,{(Xi,νi):i∈J}a collection of nbd spaces,{fi: i∈J}a collection of functions fi:X→Xi. The initial structureν on Xexists and is
specified byν(x)=i∈Jfi−1(νi(fi(x))), for allx∈X.
Proof. For each∈J, the nbd structureµonXdefined byµy(x)=f−1(ν(f(x)))
is the coarsest makingf:(X,µ)→(X,ν)continuous. Thus,ν=supN(X){µi:i∈J}
is the coarsest nbd structure makingfi:(X,ν)→(Xi,νi)continuous for alli∈J, and
the asserted characterization forν(x), for anyx∈X, follows byLemma 3.1.
Proposition4.2. Letνbe the initial nbd structure onXinduced by{(Xi,νi):i∈J}
and{fi:i∈J}, wherefi:X→Xi. LetᏴbe ap-stack onXandx∈X. ThenᏴ→ν xif
and only iffi(Ᏼ) ν→i fi(x), for alli∈J.
Proof. If fi(Ᏼ) ν→i fi(x), for all i ∈ J, then νi(fi(xi)) ⊆ fi(Ᏼ), and thus f−1
i (νi(fi(x)))⊆fi−1(fi(Ᏼ)), for alli∈J. Therefore,ν(x)⊆i∈Jfi−1(fi(Ᏼ))follows
byProposition 4.1. IfA∈ν(x), thenA∈f−1
(f(Ᏼ))for some∈J. Consequently,
there isH∈Ᏼ such thatf−1
(f(H))⊆A, implyingH⊆A, and henceA∈Ᏼ. Thus ν(x)⊆ᏴandᏴ→ν x.
Proposition 4.3. Let X be a set, {(Xi,νi) : i∈ J} a collection of nbd spaces, {fi:i∈J}a corresponding collection of functionsf :Xi→X. The final structureµ
onXexists and is specified by
(1) ifx∈i∈Jfi(Xi),µ(x)= {A⊆X:fi−1(A)∈νi(a), for alla∈fi−1(x)and for
alli∈Jsuch thatx∈fi(Xi)};
(2) otherwise,µ(x)=x˙.
Proof. Clearly,µ∈N(X). Iffi:(Xi,νi)→(X,µ)and a∈Xi, thenA∈µ(fi(a)),
sof−1
i (A)∈νi(a); thusµ(fi(a))⊆fi(νi(a)), which implies thatfiis continuous. It
is routine to verify thatµas specified is the finest nbd structure onX, making each
fi:(Xi,νi)→(X,µ)continuous.
Proof. In the notation and terminology ofProposition 4.1, assume each(Xi,νi)
is regular, and, for any ∈J, letµbe the nbd structure onX defined by µ(x)= f−1
(ν(f(x))). To show thatµis regular, letA∈µ(x). Then, there isB∈ν(f(x))
such thatf−1
(B)⊆A. Sinceνis regular, there isC∈ν(f(x))such that ClνC⊆B.
ByProposition 2.9, f(Clµ(f−1(C)))⊆Clνf(f−1C)⊆ClνC, and so Clµf−1(C)⊆ f−1
(ClνC)⊆A. Since f−1(C)∈µ(x),(X,µ)is regular. As shown in the proof of Proposition 4.1,ν=supN(X){µ:∈J}, and thus(X,ν)is regular byProposition 3.2(2).
If each(X,ν)is supratopological, a simple argument shows thatµis a
supratopol-ogy, and thereforeνis a supratopology byProposition 3.2(1).
In the notation and terminology of Proposition 4.3, assume each (Xi,νi) is
pre-topological. To show that the final structureµ is pretopological, it suffices to show that µ(x)is a filter, for allx ∈X. If x ∈i∈Jfi(Xi), µ(x)=x˙ is a filter. Letx∈
i∈Jfi(Xi)andA,B∈µ(x). Iffi(a)=x, thenf−1(A)andf−1(B)are inνi(a)and
hencef−1(A∩B)∈ν
i(a). ThusA∩B∈µ(x)andµ(x)is a filter.
Recalling the notationsσ ν,ρν, andπν for the supratopological, regular, and pre-topological modifications of a given nbd structureν, we obtain the following corollary toTheorem 4.4.
Corollary4.5. Letf:(X,ν)→(Y ,µ)be continuous. Then so are the following: (1) f:(X,σ ν)→(Y ,σ µ);
(2) f:(X,ρν)→(Y ,ρµ); (3) f:(X,πν)→(Y ,πµ).
Proof. (1) Sinceσ µ≤µ,f:(X,ν)→(X,σ ν)is continuous, and so isf:(X,ν)→
(X,σ µ), whereνis the initial structure onXinduced by(X,σ µ)andf. Butνis a
supratopology byTheorem 4.4, and sinceν≤ν,ν≤σ ν.
(2) Similar to (1).
(3) Sinceν≤πν,f:(X,πν)→(X,µ)is continuous, and so isf:(X,πν)→(X,µ),
whereµis the final structure induced onXby(X,πν)andf. Butµis a pretopology byTheorem 4.4, and sinceµ≤µ,πµ≤µ.
The next lemma is a rather obvious result; we omit the straightforward proof.
Lemma4.6. (1)Let{(Xi,νi):i∈J}be a collection of closure spaces and{fi:i∈J}
a collection of maps fi:Xi→X. If ν is the final structure onX in NBD induced by
these collections, thenσ ν is the final structure onXin SUPTOP induced by the same collections.
(2)Let{(Xi,µi):i∈J}be a collection of pretopological spaces and {gi:i∈J}a
collection of mapsgi:X→Xi. Ifµis the initial structure onXin NBD induced by these
collections, thenπµis the initial structure in PRTOP induced by these same collections.
Theorem4.7. Topological is an initial property in PRTOP and a final property in CLS.
Proof. Using the notation and assumptions ofLemma 4.6(2), assume that each
With the help ofTheorem 4.4andCorollary 4.5, we easily show that the modifica-tionsσ,ρ, andπ can be used to define functors mapping NBD onto CLS, RNBD, and PRTOP, respectively. The functors corresponding toσ and ρare reflectors, the one corresponding toπ is a coreflector. However, whenσ is restricted to PRTOP, it fol-lows byTheorem 4.7that the functor determined byσ is a reflector from PRTOP onto TOP, and, likewise,πdetermines a coreflector from CLS onto TOP. These categorical results can be summarized as follows.
Theorem4.8. The categories NBD, PRTOP, CLS, TOP, and RNBD are topological. CLS and RNBD are bireflective subcategories of NBD. PRTOP is a bicoreflective subcat-egory of NBD. TOP is a bireflective subcatsubcat-egory of PRTOP and a bicoreflective subcate-gory of CLS.
Subspaces and products are defined in the usual way as special cases of initial structures, coproducts and quotients as special cases for final structures. For future reference, we mention the following as consequences ofProposition 4.1.
Proposition4.9. (1)If(X,ν)is an nbd space andA⊆X, the subspace(A,νA)is
described at eachx∈AbyνA(x)= ∩{A∩V:V∈ν(x)}.
(2)Let{(Xi,νi):i∈J}be a set of nbd spaces andX=Πi∈JXithe Cartesian product
set. The product nbd space(X,ν)=Πi∈J(Xi,νi)is characterized for eachx∈X by: ν(x)=i∈JPi−1(νi(Pi(x)))for allx∈X, wherePiis theith projection map.
ByTheorem 4.4,Proposition 4.9also describes subspaces and products of closure spaces. Alternate characterizations for these in terms of supraopen sets are given in the next proposition.
Proposition 4.10. (1) If (A,νA) is a subspace of the closure space (X,ν), then νA= {V∩A:V∈ν}.
(2) If (X,ν)=Πi∈J{(Xi,νi)}where each (Xi,νi) is closure, thenν consists of all
unions of sets of the formP−1
i (Vi),Vi∈νi,i∈J.
Remark4.11. The description of products in CLS contrasts with that in TOP, where sets of the formP−1
i (Vi),Vi∈νi,i∈Jconstitute a subbase rather than a base for the
product topology.
5. Some properties of neighborhood spaces. Various topological properties have been studied in the setting of pretopological spaces (e.g., see [2]), and more recently for closure spaces [5]. A more unified approach is to study these and other properties in the setting of nbd. spaces. Among the properties we consider is compactness, and it is interesting to note that the Tychonov theorem holds in NBD, whereas some other results familiar from topology do not.
Definition5.1. Let(X,ν)be an nbd space,A⊆X. A collectionᏯof subsets ofX
is aν-neighborhood cover ofAif, for eachx∈A, there isV∈ν(x)such thatV∈Ꮿ.
Definition5.2. An nbd space(X,ν)is defined to be (1) T1if∩ν(x)= {x}, for allx∈X;
(3) T3if regular andT1;
(4) compactif eachν-neighborhood cover ofXhas a finite subcover;
(5) locally compactif eachν(x)contains a compact set (i.e., a set which is compact when considered as a subspace of(X,ν)).
Theorem5.3. (1)AT3nbd space isT2.
(2)An nbd space(X,ν)isT2if and only if eachν-convergentp-stack has a unique limit.
Proof. (1) Assume(X,ν)is T3andx≠y. UsingT1, chooseU∈ν(x)such that
y ∈U; then there isV∈ν(x)such that ClνV⊆U, andX\V∈ν(y)is disjoint fromV.
(2) If Ᏼ →ν x,y, then ν(x)⊆Ᏼ and ν(y)∈Ᏼ imply that x and y do not have disjoint neighborhoods. Conversely, if(X,ν)is notT2, then there arex,yinXsuch thatV∈ν(x)andW∈ν(y)imply thatW∩V≠∅. ThusᏴ=ν(x)∪ν(y)is ap-stack
ν-converging to bothxandy.
Theorem 5.4. A subsetA of an nbd space(X,ν)is compact if and only if each ultrafilterᏲonXcontainingAisν-convergent to a point inA.
Proof. Let(X,ν)be compact, and assume there is an ultrafilterᏲcontainingA
which fails to converge to a point inA. Thus, for eachx∈X, there isVx∈ν(x)such
thatVx ∈Ᏺ, andᐁ= {Vx:x∈X}is an nbd cover ofAwith no finite subcover. Thus,
Ᏺmust converge to a point inA.
Conversely, ifᐁis aν-nbd cover of Awith no finite subcover, thenᐃ= {A\V:
V∈ᐁ}is a filter subbase, and ifᏳis an ultrafilter containing[ᐃ], thenA∈ᏳbutᏳ cannot converge to any point inA.
Theorem5.5. The product(X,ν)=Πi∈J(Xi,νi)of a set of nbd spaces is compact if
and only if each component space(Xi,νi)is compact.
Proof. We easily verify that a continuous image of a compact nbd space is com-pact, and hence each(Xi,νi)is compact if(X,ν)is compact. Conversely, assume each (Xi,νi)is compact and letᏲbe an ultrafilter onX. ThenPi(Ᏺ)is an ultrafilter onXi,
whichνi-converges to somexi∈XibyTheorem 5.4. Ifx∈Xis defined byPi(x)=xi,
for alli∈J, thenᏲ→ν xbyProposition 4.2, and therefore, byTheorem 5.4,(X,ν)is compact.
For closure spaces, compactness and regularity have more conventional character-izations. We omit these straightforward proofs.
Theorem5.6. Let(X,ν)be a closure space. Then
(1) (X,ν)is regular if and only if, wheneverAis supraclosed andx∈X\A, there are disjoint supraopen setsU,Vsuch thatx∈UandA⊆V;
(2) (X,ν)is compact if and only if each cover ofXby supraopen sets has a finite subcover.
(a) a closed subspace of a compact,T2nbd space need not be compact; (b) a compactT2nbd space need not be regular;
(c) a product (in NBD or CLS) of two locally compact supratopological spaces need not be locally compact.
For (c), we can indeed choose each locally compact space to be the real line with its usual topology; the conclusion follows from the remark followingProposition 4.10.
Since ultrap-stacks play a role in nbd spaces analogous to that of ultrafilters in topological spaces, it is of some interest, in view of Theorem 5.4, to consider the following stronger version of compactness in which ultrafilters are replaced by ultra
p-stacks.
Definition5.7. A subsetAof an nbd space(X,ν)isp-compact if every ultrap-stack containingA ν-converges to a point inA. The space(X,ν)isp-compactifXis ap-compact set.
Proposition5.8. The continuous image of ap-compact set isp-compact.
Proof. Letf:(X,ν)→(Y ,µ)be continuous, letA⊆X bep-compact, and letᏴ be an ultrapstack containingf (A). IfᏳis an ultrapstack containing thep-stack base {f−1(H):H∈Ᏼ} ∪ A, thenᏳ→ν xfor somex∈A, andᏴ=f (Ᏻ)→µ f (x)∈f (A),
thus proving thatf (A)isp-compact.
Theorem5.9. A subsetAof an nbd space(X,ν)isp-compact if and only if each
ν-neighborhood cover ofAhas a two-member subcover.
Proof. SupposeᏴis an ultrapstack containingAwhich fails toν-converge to any point inA. Then for eachx∈A, there isUx∈ν(x)such thatUx ∈Ᏼ. ByLemma 2.2(c), A\Ux∈Ᏼ, for allx∈A. Thusᐁ= {Ux:x∈A}is aν-neighborhood cover ofA. Butᐁ
has no two-element subcover ofA, for ifU,V∈ᐁandA⊆U∪V, then(A\U)∩(A\V )= A\(U∪V )= ∅, contradicting the assumption thatᏴis ap-stack.
Conversely, letᐁbe aν-neighborhood cover ofAwith no two-element subcover of
A. ThenᏮ= {A\U:U∈ᐁ}isp-stack base, and any ultrapstack containingᏮcannot converge to any point inA.
Theorem5.10. The product(X,ν)=Πi∈J(Xi,νi)of a set of nbd space isp-compact
if and only if each component space(Xi,νi)isp-compact.
Proof. Essentially the same as that ofTheorem 5.5.
It is well known thatTheorem 5.5(the Tychonov theorem) holds for products in TOP and PRTOP, but this is not true forTheorem 5.10. Indeed, a two-element discrete topological space isp-compact and the product of this space with itself isp-compact relative to NBD and CLS, but not relative to TOP or PRTOP.
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D. C. Kent: Department of Mathematics, Washington State University, Pullman, WA 99164-3113, USA
E-mail address:[email protected]
Won Keun Min: Department of Mathematics, Kangwon National University, Chun-cheon,200-701, Korea
