ON THE COMPLETENESS OF NON-SYMMETRICAL UNIFORM CONVERGENCE …ma.fme.vutbr.cz › archiv › 8_1 › ma_8_1_leseberg_vaziry_final.pdf · Here, we should mention that now classical

Math. Appl. 8 (2019), 37–57DOI: 10.13164/ma.2019.04

ON THE COMPLETENESS OF NON-SYMMETRICAL UNIFORM

CONVERGENCE WITH SOME LINKS TO APPROACH SPACES

DIETER LESEBERG and ZOHREH VAZIRY

Abstract. The quasitopos b-UFIL of b-uniform filter spaces [16] are an appropri-

ate tool for studying convergence from a higher point of view as demonstrated in

recent papers by the above mentioned authors. In addition BORN, the category ofbornological spaces and bounded maps, can be integrated as bicoreflective subcate-

gory of b-UFIL. As already shown symmetric b-uniform filter spaces have “Cauchy

completions” which generalize some important ones as for example those which wereconsidered by Wyler, Preuss, Czaszar and Hausdorff, respectively. In the present

paper we will construct a completion, called ultracompletion, for a suitable not

necessarily symmetric b-uniform filter space and compare this one with a constructpresented for quasi-uniform spaces by Carlson and Hicks in the past. Furthermore,

among others, we get the result that every quasiuniform limit space in the sense of

Behling has an ultracompletion. At the end of this article, we consider some impor-tant links to generalized approach spaces, those which were introduced by Lowen.

So it is shown that b-topological closure operators can be completely described byso-called approach-bornologies, which represent a common generalization of both

approach spaces and bornological spaces, respectively. Thus, as interesting corol-

lary we obtain the result that APB the category of approach-bornological spacesand contracted maps intersects b-URING, the full subcategory of b-UFIL, whose

objects have ultracompletions.

1. Introduction

In our last paper “The Cauchy-completion of a symmetric b-uniform filter space”we announce the question whether there exist completions for given not necessar-ily symmetric b-uniform filter spaces. Here, a b-uniform filter space is a tripleconsisting of a non-empty subset BX ⊂ PX and a non-empty subset

µ ⊂ FIL(X ×X) := {U : U is filter on X ×X}

such that the following conditions hold

(buf1) B1 ⊂ B ∈ BX implies B1 ∈ BX ;(buf2) x ∈ X implies {x} ∈ BX ;

(buf3) B ∈ BX\{∅} implies•B ×

•B∈ µ, where

•B:= {A ⊂ X : A ⊃ B};

(buf4) U ∈ µ and U ⊂ U1 ∈ FIL(X ×X) imply U1 ∈ µ.

MSC (2010): primary 54D35, 54A05, 54A20; secondary 54B30, 54E15.Keywords: b-uniform filter space, b-filter space, bornological space, ultracompletion, pre-

Cauchy filter, set-convergence, approach space, preuniform convergence space, quasi-uniformspace, bounded set, bounded topology.

37

38 D. LESEBERG and Z. VAZIRY

In general, for filters F and G, their cross product is defined by F × G := {R ⊂X ×X : ∃ F ∈ F ∃ G ∈ G s.t. R ⊃ F ×G}.For b-uniform filter spaces (X,BX , µX), (Y,BY , µY ) a function f : X −→ Y iscalled b-uniformly continuous map, provided that the following conditions aresatisfied,

(buc1) B ∈ BX implies f [B] ∈ BY ;(buc2) U ∈ µX implies (f × f)(U) ∈ µY .

By b-UFIL we denote the category of b-uniform filter spaces and b-uniformlycontinuous maps.

In this context we point out that BX defines a B-set in the sense of Wyler [25].On the other hand if BX is discrete, meaning that BX = DX := {∅} ∪ {{x} :x ∈ X} holds, then b-uniform filter spaces and preuniform convergence spaces inthe sense of Preuss [22] are essentially the same (up to isomorphism); meaningthat DISb-UFIL the full subcategory of b-UFIL, whose objects are discreteis isomorphic to PUCONV, the category of preuniform convergence spaces anduniformly continuous maps.

Furthermore the quasi-topos BOUND of bound spaces and bounded maps [23]can also be fully embedded into b-UFIL.

Here, a bound space is a pair (X,BX), where BX denotes a B-set, and themorphisms between them are the bounded maps, compare with (buc1). Then, foran arbitrary B-set BX we consider the pair (BX , µb), where µb is defined by setting

µb := {U ∈ FIL(X ×X) : ∃ B ∈ BX s.t.•B ×

•B⊂ U}.

Conversely, for a bounded b-uniform filter space (X,BX , η), meaning that (BX , η)is satisfying the following condition,

(B) V ∈ η implies the existence of an B ∈ BX with B ×B ∈ V,

the so defined assignments deliver an isomorphism between the category BOUNDand B-UFIL, the full subcategory of b-UFIL whose objects are bounded. Fur-thermore B-UFIL is bicoreflective in b-UFIL. In fact let (X,BX , µ) be a b-uniform filter space, then by setting µB := {U ∈ µ : ∃ B ∈ BX B × B ∈ U} weobtain a bounded b-uniform filter space such that 1X : (X,BX , µB) −→ (X,BX , µ)is the bicoreflection of (X,BX , µ) with respect to B-UFIL. Now, in addition letus call a b-uniform filter space (X,BX , µ) bornological provided that BX formsa bornology, meaning that BX in addition satisfies B1, B2 ∈ BX imply B1 ∪B2 ∈BX . By BONb-UFIL we denote the full subcategory of b-UFIL, whose objectsare bornological. Then, the category BORN of bornological spaces and boundedmaps, [9] can be regarded as full subcategory of the intersection of BONb-UFILand B-UFIL, respectively. Finally we still mention the concept of final b-uniformfilter spaces, regarded as full subcategory of b-UFIL, denoted by FINb-UFILand whose objects are final, meaning that the corresponding B-sets are satisfyingthe following condition,

(f) B ∈ BX implies B is finite.

Here we note that every finite b-uniform filter space is final, and furthermore westate that each discrete b-uniform filter space is final, too.

ON THE COMPLETENESS OF NON-SYMMETRICAL... 39

Thus final b-uniform filter spaces represent roughly spoken a common gener-alization of uniform convergence and the bornologies of finite sets. Later (seeup to 5.11) we will see which important role they are playing in the context ofultracompleteness.

Finally, we infer that FINb-UFIL is bicoreflective in b-UFIL. To these facts,we note that in our view filters may contain the empty set. This is of importance,too, if one considers set-convergence spaces in the sense of Wyler [25].

A set-convergence is a pair (BX , q), where BX is B-set and q ⊂ BX × FIL(X),such that the following conditions are satisfied,

(SC1) B ∈ BX implies (B,•B) ∈ q;

(SC2) (∅,F) ∈ q implies F = PX;(SC3) (B,F) ∈ q and F ⊂ F1 ∈ FIL(X) imply (B,F1) ∈ q.

Then, the triple (X,BX , q) is called a set-convergence space. Often we write F q Biff (B,F) ∈ q is valid.

A function f : X → Y between set-convergence spaces (X,BX , qX), (Y,BY , qY )is called b-continuous provided that f is bounded, and in addition, f transfers con-vergent filters. By SETCONV we are denoting the corresponding category. Animportant subcategory of SETCONV is the bireflective full subcategory RO-SETCONV, whose objects are reordered. Here, a set-convergence (BX , q) iscalled reordered, and the triple (X,BX , q) a reordered set-convergence space, pro-vided it satisfies the following conditions,

(RO) ∅ 6= B1 ⊂ B ∈ BX and F q B imply F q B1.

Here, we should mention that now classical point-convergence spaces, such asKENT-convergence spaces, limit spaces, pseudo-topological spaces, pre-topologicalspaces or topological spaces as well can be regarded as special reordered set-convergence spaces by restricting BX to DX [6] (see Section 3).

Note also that RO-SETCONV can be fully embedded into b-UFIL as follows:For reordered set-convergence space (X,BX , q) we consider the triple (X,BX , µq),where µq is defined by setting:

µq := {U ∈ FIL(X ×X) : ∃ F ∈ FIL(X) ∃ B ∈ BX\{∅}

(F q B and•B ×F ⊂ U)} ∪ {P (X ×X)}.

Then, (BX , µq) forms a so-called b-uniform convergence, and the triple (X,BX , µq)a b-uniform convergence space.

Here, a b-uniform filter space (X,BX , η) is called b-uniform convergence space,provided that (BX , η) satisfies the following condition,

(cv) U ∈ η implies the existence of F ∈ FIL(X) and B ∈ BX with U ⊃•B

×F ∈ η.

In this context F pη B is defined by setting:

F pη ∅ iff F = PX and

F pη B iff•B ×F ∈ η for every B ∈ BX\{∅}.

The so-defined assignments deliver an isomorphism between the categories

RO-SETCONV and b-UCONV,


whose objects are the b-uniform convergence spaces. Thus our new developed con-cept seems to be an appropriate tool for a common study of all former investigatedcategories.

In giving some intrinsic examples let us consider a quasi-uniform space (X,U),[12] and let U−1 := {U−1 : U ∈ U} be the conjugate quasi-uniformity, whereU−1 := {(y, x) : (x, y) ∈ U}. If S is a non-empty family of subsets of X, we mayconsider the following three convergences on PX. Let (At) be a net of subsets ofX and A ⊂ X. We say that the net (At):

S−-converges to A, and we write AtS−−→ A provided for each S ∈ S and

U ∈ U there exists t0 such that A ∩ S ⊂ U−1(At) for every t ≥ t0;

S+-converges to A, and we write AtS+

−→ A, provided for each S ∈ S andU ∈ U there exists t0 such that At ∩ S ⊂ U(A) for every t ≥ t0;

S-converges to A provided it S−-converges to A and it S+-converges to A,where U(A) := {y ∈ X : (x, y) ∈ U for some x ∈ A}, U−1(A) := {y ∈ X :(x, y) ∈ U−1 for some x ∈ A}.

Note that if X is a metric space and S = {X} then the S-converges on the spaceof closed bounded subsets of X is simply the H-convergence, i.e. the convergencein the Hausdorff metric. The just presented ideas are going back to A. Lechicki,S. Levi and A. Spakowski [13] where in their work Bornological converges one alsocan find additional examples and corresponding references, see also [24].

Now, to tackle our problem we have to alter the definition of a µ-Cauchy filter[17] so that, in the symmetric case, they coincide.

2. Basic notions

Definition 2.1. Let (X,BX , µ) be a b-uniform filter space, C ∈ FIL(X)\{PX}is called pre-Cauchy filter (shortly pre-Chy filter) in (BX , µ), provided C satisfiesthe following condition,

(pChy) ∃ U ∈ µ s.t. ∀ R ∈ U ∃ B ∈ BX\{∅} s.t. R(B) ∈ C.Here R(B) := {z ∈ X : ∃ x ∈ B s.t. (x, z) ∈ R}.

Remark 2.2. Here we should note that in the discrete case for a symmetricb-uniform net space (X,BX , µ) and a filter C ∈ FIL(X)\{PX} the followingstatements are equivalent:

(i) C is pre-Chy filter in (BX , µ);(ii) C is µ-Chy filter [17].

Here, C ∈ FIL(X)\{PX} is called µ-Chy filter iff C × C ∈ µ is valid. Thus, ifreturning to the discrete case C defines a Cauchy-filter as usual on the associateduniform limit space, [22]. Moreover, we note that roughly spoken the b-uniformlycontinuous image of a pre-Chy filter is a pre-Chy filter again.

Definition 2.3. Let (X,BX , µ) be a b-uniform filter space. Then, F ∈ FIL(X)

is called setconvergent in (BX , µ) iff there exists B ∈ BX\{∅} s.t.•B ×F ∈ µ.

Remark 2.4. By the definition used in Introduction, this can be expressed bythe fact that F pµ B is valid for some B ∈ BX\{∅}.


Now it is clear that every setconvergent filter F ∈ FIL(X)\{PX} in (BX , µ) isa pre-Chy filter. In this context we also note that for an reordered set-convergencespace (X,BX , q) and a filter F ∈ FIL(X) the following statements are equivalent:

(i) F q B for some B ∈ BX\{∅};(ii) F q {x} for some x ∈ X.

Definition 2.5. A b-uniform filter space (X,BX , µ) is called ultracomplete,provided that every pre-Chy filter in (BX , µ) is setconvergent.

Example 2.6. Every non-empty bornological bounded b-uniform filter space(X,BX , µ) is ultracomplete.

Proof. Let C ∈ FIL(X)\{PX} be a pre-Chy filter in (BX , µ), hence we canfind U ∈ µ with the corresponding property. But by the hypothesis (BX , µ) isbounded, thus B × B ∈ U for some B ∈ BX . Then, we can choose D ∈ BX\{∅}s.t. (B×B)(D) ∈ C by applying the pre-Cauchy property, hence D∪B ∈ BX\{∅},since BX is bornology. Consequently, (D

•∪ B)× (D

•∪ B) ∈ µ follows. It remains

to prove (D•∪ B) ⊂ C, because then C qµ D ∪B is valid.

A ∈ (D•∪ B) implies A ⊃ D ∪ B. We will show that the inclusion D ∪ B ⊂

(B × B)(D) holds. z ∈ (B × B)(D) implies the existence of x ∈ D such that(x, z) ∈ B ×B, and z ∈ D ∪B follows, proving the claim. �

Example 2.7. Every non-empty b-uniform convergence space (X,BX , µ) isultracomplete.

Proof. Let C ∈ FIL(X)\{PX} be a pre-Chy filter in (BX , µ), hence we canfind an U ∈ µ with the corresponding property. Since (X,BX , µ) is b-uniformconvergence space, there exists B ∈ BX and a filter F ∈ FIL(X) such that

U ⊃•B ×F ∈ µ are valid. B is not empty and F is subfilter of C and thus

F 6= PX. Because F ∈ F implies B×F ∈ U , hence we can find D ∈ BX\{∅} with(B × F )(D) ∈ C. Since (B × F )(D) ⊂ F holds, F ∈ C follows, and consequently•B ×C ∈ µ results, which shows C qµ B. �

Now, we introduce two further important notions which are closed to the formerpresented concept.

Definition 2.8. A b-uniform filter space (X,BX , µ) is called

(i) ultracompact provided that each ultrafilter F ∈ FIL(X)\{PX} is setcon-vergent in (BX , µ);

(ii) ultrabounded provided that each ultrafilter F ∈ FIL(X)\{PX} is a pre-Chy filter in (BX , µ).

Remark 2.9. Evidently, each non-empty finite b-uniform filter space is ul-tracompact. Moreover, every ultracompact b-uniform filter space is ultrabounded,and in addition we get that every ultrabounded and ultracomplete b-uniform filterspace is ultracompact again.

Furthermore we note that, supposing a symmetric b-uniform net space(X,BX , µ), then F ∈ FIL(X) is setconvergent in (BX , µ) iff F qτµ B is valid

for some B ∈ BX\{∅}.


Here F qτµ B iff F∩•B∈ τµ [15]. Thus we get that for a symmetric b-uniform

net space the terms compactness and ultracompactness are essentially the same.Then, finally, if considering the discrete case we obtain the fundamental resultthat a symmetric b-uniform net space is ultracompact iff it is ultrabounded andultracomplete. Now taking all these facts into account we resume that the newterms introduced generalize the older ones of compactness, precompactness andcompleteness, respectively in a rather natural way.

3. More about set-convergence

Returning to the concept of reordered set-convergence we should explain how itplays an important role for further studies of point-convergences, not only re-stricted to the discrete case.

Now, if one considers point-convergence on arbitrary B-sets, such as the set EXof all finite subsets or the set τX of all totally bounded subsets or the set CX ofall compact subsets of a set X we extend the basics to the following one, i.e.

Definition 3.1. We call a reordered set-convergence (BX , q) pointset- conver-gence (on X), and the triple (X,BX , q) pointset - convergence space, provided thatthe following condition is satisfied, i.e.

(pset) F ∈ FIL(X), B ∈ BX\{∅} and F q {x} ∀ x ∈ B imply F q B.

Remark 3.2. Here we point out, that each discrete set-convergence space isa pointset-convergence space. Convergence in the usual sense like limit spaces orpretopological spaces, respectively are being involved [6]. Consequently, all pos-sible point-convergences on any B-sets can be now subsumed under the conceptof pointset-convergence spaces. If we denote by P-SETCONV the correspond-ing full subcategory of RO-SETCONV, then we claim that P-SETCONV isbireflective in RO-SETCONV.

In fact for a reordered set-convergence space (X,BX , q) we put for F ∈ FIL(X)

and B ∈ BX\{∅} F•q B iff F q {x} ∀ x ∈ B, and F

•q ∅ iff F = PX. Then,

(X,BX ,•q) is a pointset-convergence space such that the demand for the bireflection

is satisfied.Another point of view is considering the set-convergence (BX , qµ) being induced

by a given generated b-uniform filter space (X,BX , µ). As already seen [17, 22]generated b-uniform filter spaces are in one-to-one correspondence with principalpreuniform convergence spaces or preuniform spaces or diagonal filters in the senseof Weil [1] by assuming the discrete case. So we are coming quite naturally to thefollowing definition:

Definition 3.3. A set-convergence (BX , q) is called set-surrounding and thetriple (X,BX , q) set-surrounding space, provided that the following property isvalid,

(ss) B ∈ BX\{∅} implies⋂{F : F q B} q B.

Remark 3.4. Let a neighborhood space (X,BX ,Θ) be given [23], then thetriple (X,BX , qΘ) defines a set-surrounding space, where

F qΘ B iff F ⊃ Θ(B) ∀ B ∈ BX .


Lemma 3.5. If (X,BX , µ) is a generated b-uniform filter space, then the un-derlying set convergence (BX , qµ) is a set-surrounding on X and determines theset-surrounding space (X,BX , qµ).

Proof. Let B ∈ BX\{∅} and consider all F ∈ FIL(X) s.t. F qµ B is valid, hence•B ×F ∈ µ holds for those F ∈ FIL(X). It remains to show that the statement•B × ∩ {F ∈ FIL(X) : F qµ B} ∈ µ can be deduced. Therefore, it suffices to

show that the inclusion ∩µ ⊂•B × ∩ {F ∈ FIL(X) : F qµ B} holds, since by the

hypothesis µ is generated. So let R ∈ µ, hence R ⊃ B×FF for some FF ∈ F withF qµ B. Consequently, R ⊃ ∪{B×FF : F qµ B} ⊃ B×∪{FF : F qµ B}, because(x, z) ∈ B × ∪{fF : F qµ B} implies x ∈ B, and there exists F ′ ∈ FIL(X)with F ′ qµ B and z ∈ F ′F . Consequently, (x, z) ∈ B × FF ′ implies (x, z) ∈∪{B×FF : F qµ B}. But B×∪{FF : F qµ B} ∈

•B ×∩{F ∈ FIL(X) : F qµ B},

and the claim follows. �

Remark 3.6. Here we point out, that in the discrete case (X,BX , qµ) evendefines a pretopological space. Then, a related expression aims at so-called closureoperators. Indeed, let a b-uniform filter space (X,BX , µ) be given, then we candefine the following two closure spaces (X, clµ) and (X, clµ) by setting:

clµ(∅) := ∅ and

clµ(A) := {x ∈ X : ∃ F ∈ FIL(X)(•x ×F ∈ µ and A ∈ secF)} ∀(∅ 6= A) ∈ PX,

where secF := {D ⊂ X : ∀ F ∈ F F ∩D 6= ∅}.Respectively, we put:

clµ(∅) := ∅ andclµ(A) := {x ∈ X : ∃ U ∈ µ s.t. {x} ×A ∈ sec U}, ∀ (∅ 6= A) ∈ PX

Here, in general we note that clµ(A) ⊂ clµ(A) is true for every A ∈ PX. In thecase of (X,BX , µ) being b-uniform convergence space then the equality of boththe closures results. The latter closure defined will be used in 5.1 in obtaining thatX is dense in X∗.

4. b-uniform ring spaces

Before coming to the core of this article, we introduce the following importantnotion.

Definition 4.1. For a b-uniform filter space (X,BX , µ) a pair (AX , η) with∅ 6= AX ⊂ PX and ∅ 6= η ∈ FIL(X × X) is called a base for (BX , µ), providedthat the following equations hold,

(bas1) BX = {B ⊂ X : ∃ A ∈ AX s.t. B ⊂ A} and(bas2) µ = {V ∈ FIL(X ×X) : ∃ U ∈ η s.t. U ⊂ V} (compare with [1]).

Remark 4.2. Here we should note that a pair (AX , η) is a base for a b-uniformfilter structure on X iff it satisfies the following conditions:

(bsuf1) ∅ ∈ AX ;(bsuf2) x ∈ X implies {x} ∈ AX ;


(bsuf3) ∀ B ∈ BX\{∅} ∃ U ∈ η s.t. U ⊂•B ×

•B.

Lemma 4.3. For b-uniform filter spaces (X,BX , µX), (Y,BY , µY ) let f : X −→Y be a map. Let us denote by (AX , ηX) respectively (AY , ηY ) bases for the corre-sponding spaces. Then, the following statements are equivalent:

(i) f : (X,BX , µX) −→ (Y,BY , µY ) is b-uniformly continuous;(ii) (1) A ∈ AX implies ∃ AY ∈ AY s.t. f [A] ⊂ AY ;

(2) U ∈ µX implies ∃ UY ∈ ηY s.t. UY ⊂ (f × f)(U).

Proof. By straightforward executing. �

As pointed out by many authors in the past, quasi-uniform spaces, quasiuniformconvergence spaces, Cauchy spaces or point-convergence spaces, respectively arealso of interest if one considers suitable extensions of the given constructs [4, 5, 7,8, 11,14].

Here we note again that all the above mentioned spaces can be simply de-scribed by the associated b-uniform filter spaces. Moreover each of them fulfillsan additional common property, which we will be now described as follows:

Definition 4.4. A b-uniform filter structure (BX , µ) is called b-uniform ringstructure (on X) and the space (X,BX , µ) b-uniform ring space, provided it sat-isfies the following condition, i.e.

(rg) U ∈ µ implies U ◦ U ∈ µ, where in general for filters U ,V ∈ FIL(X ×X),U ◦ V := {R ⊂ X ×X : ∃ U ∈ U ∃ V ∈ V s.t. R ⊃ U ◦ V }.

Examples 4.5. (i) Let (X,U) be a quasi-uniform space. Then, the as-sociated space (X,DX , µU ) defines a b-uniform ring space, where µU :={V ∈ FIL(X×X) : V ⊃ U}. Note, that the pair (DX , {U}) defines a basefor (DX , µU );

(ii) Let (X, JX) be a quasiuniform convergence space. Then, the associatedspace (X,DX , JX) defines a b-uniform ring space;

(iii) for a b-filter space (X,BX ,Γ) [11, 15, 17] the space (X,BX , µΓ) defines ab-uniform ring space, where the pair (BX , {F ×F : F ∈ Γ}) forms a basefor (BX , µΓ);

(iv) Every b-uniform net space is a b-uniform ring space [17];(v) Every bounded b-uniform filter space (X,BX , µ) is a b-uniform ring space,

where (BX , {B ×B : B ∈ BX}) forms a base for (BX , µ);(vi) For a reordered set-convergence space (X,BX , q) the space (X,BX , µq)

defines a b-uniform ring space, where the pair (BX , {•B ×F : F q B,B ∈

BX}) forms a base for (BX , µq);(vii) For a neighbourhood space (X,BX ,Θ), [23] the space (X,BX , µΘ) defines

a b-uniform ring space, where (BX , {•B ×Θ(B) : B ∈ BX}) forms a base

for (BX , µΘ).

Proposition 4.6. A pair (AX , η) is a base for a b-uniform ring structure iff itsatisfies the conditions in 4.2. and in addition the following one, i.e.

(rbas) U ∈ η implies the existence of V ∈ η s.t. V ⊂ U ◦ U .


Remark 4.7. Here we again assume that all mentioned bases in 4.5 even definethe condition (rbas). So let us call a pair (AX , η) satisfying these conditions a ringbase. Now in applying this new definition we characterize pre-Chy filters in a b-uniform ring space (X,BX , µ) by a given ring base (AX , η) as follows:

C ∈ FIL(X) is a pre-Chy filter in (BX , µ) iff ∃ U ∈ η ∀ R ∈ U ∃ A ∈ AX s.t.R(A) ∈ C.

In 4.5(iv) we pointed out that b-uniform net spaces are in fact b-uniform ringspaces. Now, for generated b-uniform filter spaces the following statements areequivalent:

(i) (X,BX , µ) is a b-uniform net space;(ii) (X,BX , µ) is a b-uniform ring space.

Proof. Evident. �

Remark 4.8. In this context we also mention that for a generated b-uniformfilter space (X,BX , µ), (X,BX , µ) is a b-uniform net space iff (X,BX ,∩µ) definesa quasi-uniform space, provided that BX is discrete with ∩µ := ∩{U : U ∈ µ}.

Note that (X,U) is a quasi-uniform space iff the principal preuniform conver-gence space (X, [U ]) defines a quasiuniform convergence space [1]. At the endof this section we add the fact that each b-uniform filter space (X,BX , µ) in-

duces a b-uniform ring space (X,BX ,◦µ) by defining a ring base (BX ,

◦b) with

◦b:= {U ∈ µ : U ⊂ U ◦ U}. The identity map 1X : (X,BX ,

◦µ) −→ (X,BX , µ)

is b-uniform continuous by applying 4.3, and moreover if (Y,BY , µY ) is b-uniformring space and f : (Y,BY , µY ) −→ (X,BX , µ) an injective b-uniformly continuous

map, then f : (Y,BY , µY ) −→ (X,BX ,•µ) is b-uniformly continuous, too.

Thus, each b-uniform filter space (X,BX , µ) has a restricted co-universal mapwith respect to the inclusion functor F : b-URING −→ b-UFIL, where b-URING denotes the full subcategory of b-UFIL, whose objects are the b-uniformring spaces.

Theorem 4.9. b-URING forms a topological construct [22].

Proof. For any set X, the class {(Y,BY , µ) ∈ |b-URING| : X = Y } of allb-URING objects with underlying set X is a set, because of (BY , µ) ∈ P (PX)×P (FIL(X ×X)).

The only b-uniform ring structure on a set X with CardX = 1 is the pair

({∅, {x}, {•x × •x, P ({x} × {x})}), where x denotes the element of X. If X is

empty, then ({∅}, {{∅}}) represents the only b-uniform ring structure on X.For a set X, let I be a class, (Xi,BXi , µi)i∈I a family of b-uniform ring spaces

and (fi : X −→ Xi)i∈I a family of maps. Then, (BXI , µIX) is the initial b-URINGstructure on X, where

BXI := {B ⊂ X : ∀ i ∈ I fi[B] ∈ BXi} and

µXI := {U ∈ FIL(X ×X) : ∀i ∈ I (fi × fi)(U) ∈ µi}.

Then, the remaining is clear. �


Remark 4.10. The initial b-uniform ring structure on a set X with respect to(X, fi, (Xi,BXi , µi), I)) is the coarsest b-uniform ring structure on X such that fiis b-uniformly continuous for each i ∈ I.

Specifically, let (X,BX , µ) be a b-uniform ring space and A ⊂ X. Then,(BA, µA) is b-uniform ring structure on A, where

BA := {B ∩A : B ∈ BX} and

µA := {UA : U ∈ µ} with UA := {R ∩ (A×A) : R ∈ U},

such that (A,BA, µA) represents the b-uniform ring subspace of (X,BX , µ) in b-URING.

5. The ultracompletion of a b-uniform ring space

Now, in answering the main question of this paper, we will construct an ultra-completion for an arbitrary non-empty b-uniform ring space and then apply thisresult to some former treated special constructs. In addition we study certainseparation properties especially those the space may possess to carry over to theultracompletion. Here, we extend an idea for quasi-uniform spaces due to Carlsonand Hicks [3] as indicated in the following:

Construction 5.1. Let (X,BX , µ) be a non-empty b-uniform ring space. Then,we put X∗ := X ∪ {∞} with ∞ /∈ X. For U ∈ µ we are setting:

U∗ := {R∗ ⊂ X∗ ×X∗ : ∃ R ∈ U R∗ ⊃ R ∪ {(∞, x) : x ∈ X∗}} and

BX∗ := BX ∪ {{∞}}. Then, (BX∗ , b∗) forms a base for a b-uniform ring structure(BX∗ , µ∗) on X∗, where b∗ := {U∗ : U ∈ µ}, compare with 4.7.

Proof. Evidently BX∗ defines a B-set onX∗. Also note that U∗ ∈ FIL(X∗×X∗)holds. As next we infer that for U ∈ µ{i} (R ∪ {∞, x) : x ∈ X∗})({∞}) = X∗ for each R ∈ U and{ii} (R∪{∞, x) : x ∈ X∗})(B) = R(B) for every R ∈ U and each B ∈ BX\{∅}

are valid.

(X∗,BX∗ , µ∗) is ultracomplete, because the following holds:Let C∗ be a pre-Chy filter in (BX∗ , µ∗). We will show that the statement

C∗ qµ∗ {∞} can be deduced, meaning that•∞ ×C∗ ∈ µ∗ is true. Therefore, it

suffices to verify that the inclusion•∞ ×C∗ ⊃ U∗ holds for some U∗ ∈ b∗.

By the hypothesis and according to 4.7 we can find V∗ ∈ b∗ with the corre-sponding property. Since b∗ is a ring base we can find U∗ ∈ b∗ with U∗ ⊂ V∗ ◦ V∗.Our goal is to verify that U∗ ⊂ •∞ ×C∗ holds.R∗ ∈ U∗ implies R∗ ⊃ V ∗ ◦ V ∗ for some V ∗ ∈ V∗, where V ∗ ⊃ V ∪ {(∞, x) :

x ∈ X∗} for some V ∈ V. By applying the corresponding property for C∗ we canfind D ∈ BX∗ with V ∗(D) ∈ C∗. We claim that {∞} × V ∗(D) ⊂ V ∗ ◦ V ∗ is true.But (y, z) ∈ {∞} × V ∗(D) implies the existence of x′ ∈ D such that (x′, z) ∈ V ∗with y =∞.

Since V ∪ {(∞, x) : x ∈ X∗} ⊂ R∗ we obtain (y, x′) ∈ V ∗. Thus (y, z) ∈V ∗ ◦ V ∗ ⊂ R∗ follows, concluding the proof.


Next we infer that the inclusion map i : (X,BX , µ) −→ (X∗,BX∗ , µ∗) is b-uniformly continuous, and (X,BX , µ) is b-uniform ring subspace of (X∗,BX∗ , µ∗),compare with 4.10. But this can be done in a straightforward manner. Finally, wehave to verify that X is dense in X∗, which means that the equation clµ

∗(X) = X∗

can be deduced, compare with 3.6. So let z ∈ X∗ and without restriction z =∞.Choose U ∈ µ, hence U∗ ∈ b∗ follows. It remains to verify that {z}×X ∈ sec U∗

holds. For R∗ ∈ U∗ we can find R ∈ U s.t. R∗ ⊃ {(∞, x) : x ∈ X∗} ∪ R. Bychoosing x ∈ X we obtain (z, x) ∈ {z} × (X ∩ R∗), and the claim immediatelyfollows. �

Definition 5.2. Let (X,BX , µ) be a non-empty b-uniform ring space and(X∗,BX∗ , µ∗) the ultracomplete b-uniform ring space as constructed in 5.1, thenthe pair (i, (X∗,BX∗ , µ∗)) is called the ultracompletion of (X,BX , µ) (sometimesonly the space (X∗,BX∗ , µ∗) will be called as above stated).

Separation properties come into play if one is considering convergence in a moresuitable sense. This is also of importance if universal properties are examined. Inthe next we are giving some fundamental definitions in this direction.

Definition 5.3. A set-convergence (BX , q) is called

(i) T0 set- convergence, and the triple (X,BX , q) T0 set-convergence space iffit satisfies the following condition, i.e.

(T0) B1, B2 ∈ BX\{∅},•B1 q B2 and

•B2 q B1 imply B1 = B2;

(ii) T1 set-convergence, and the triple (X,BX , q) T1 set-convergence space iffit satisfies the following condition, i.e.

(T1) B1, B2 ∈ BX\{∅} and•B1 q B2 imply B1 = B2;

(iii) T2 set-convergence, and the triple (X,BX , q) T2 set-convergence space iffit satisfies the following condition, i.e.T2) F ∈ FIL(X), B1, B2 ∈ BX\{∅} with F q B1 and F q B2 imply

B1 = B2.

Consequently, we call a b-uniform filter space (X,BX , µ) T0 space (T1 space,T2 space, respectively) iff (X,BX , qµ) is a T0 set- convergence space (T1 set-convergence space, T2 set-convergence space, respectively). Then, related to thelatter constructs we also speak of a T0 ring space (T1 ring space, T2 ring space, re-spectively), and these should be also done when considering b-uniform net spaces.

Remark 5.4. Now, it can be easily seen that T2 implies T1 and T1 implies T0.On the other hand we point out that in the discrete case these definitions coincidewith those occurring in the theory of point-convergence spaces in the sense ofPreuss [22].

Lemma 5.5. For a reordered set-convergence space (X,BX , q) each successivepair of conditions are equivalent:

{i} (BX , q) is T0 set-convergence;

{ii} x, z ∈ X, •x q {z} and•z q {x} imply x = z;

{iii} (BX , q) is T1 set-convergence;

{iv} x, z ∈ X and•x q {z} imply x = z;


{v} (BX , q) is T2 set-convergence;{vi} F ∈ FIL(X), x, z ∈ X and F q {x}, F q {z} imply x = z.

Proof. Evident. �

Proposition 5.6. For a T0 ring space (X,BX , µ) the completion (X∗,BX∗ , µ∗)is an T0 ring space.

Proof. Straightforward. �

Remark 5.7. At this point we note that (X∗,B∗, µ∗) is never T1 space, andthus it is also not T2 space. Now, the following question naturally arises. Doesa T2 ring space or T1 ring space have an T2 ultracompletion or T1 ultracompletion,respectively? But this problem is not the aim of our present paper.

Proposition 5.8. Let (X,BX , µ) be a non-empty generated b-uniform ringspace and (X∗,B∗, µ∗) its ultracompletion. Then, the ultracompletion is generated,too.

Proof. Note that the following inclusion (∩µ)∗ ⊂ ∩µ∗ holds. �

Remark 5.9. Taking 4.8 into account we point out that in the discrete casethe associated quasi-uniform space of the ultracompletion is up to isomorphismthe one- point completion of a given quasi-uniform space in the sense of Carlsonand Hicks [3], and thus it is even strongly complete in their terminology. In thiscontext we mention that for a discrete b-uniform filter space (X,BX , µ) and forevery F ∈ FIL(X) the following statements are equivalent:

(i) F is pre-Chy filter in (BX , µ);(ii) ∃ U ∈ µ ∀ R ∈ U ∃ x ∈ X with R({x}) ∈ F .

By taking 5.8 into account, F ∈ FIL(X) is pre-Chy filter in a generated b-uniformfilter structure (BX , µ) iff ∀ R ∈ ∩µ, ∃ B ∈ BX\{∅} s.t. R(B) ∈ F .Thus if combining both statements in the discrete case we obtain the usual classicalproperty of a filter for being Cauchy.

Theorem 5.10. For a generated final b-uniform filter space (X,BX , µ) thefollowing statements are, equivalent:

(i) (X,BX , µ) is ultrabounded;(ii) ∀ R ∈ ∩µ ∃ B ⊂ X finite R(B) = X.

Proof. to (ii)⇒ (i): Let F ∈ FIL(X)\{PX} be an ultrafilter and R ∈ ∩µ.Then, we can find points x1, . . . , xn ∈ X such that X = R(x1) ∪ ... ∪ R(xn) byapplying the hypothesis.

But F is ultrafilter, and thus R(xi) ∈ F for some xi. Consequently, F ispre-Chy filter in (BX , µ).

to (i)⇒ (ii): If (i) is not true, then we can find a relation R ∈ ∩µ such thatR(B) 6= X for all B ⊂ X finite.

The set {X\R(B) : B ⊂ X finite} forms a base for a filter F ∈ FIL(X)\{PX}.Let C be an ultrafilter containing F . Then, by the hypothesis C is pre-Chy filterin (BX , µ). Hence we can find B1 ∈ BX\{∅} with R(B1) ∈ C. But this is a con-tradiction because by the hypothesis B1 is finite, and thus the claim follows. �


Theorem 5.11. For a non-empty generated final ultrabounded b-uniform ringspace (X,BX , µ) its ultracompletion (X∗,BX∗ , µ∗) is final ultrabounded, too.

Proof. By the construction and hypothesis (X∗,BX∗ , µ∗) is final. In additionwe have that ∩µ ∈ µ is valid, hence (∩µ)∗ ∈ b∗ follows. So let R∗ ∈ (∩µ)∗, thenR∗ ⊃ R ∪ {(∞, x) : x ∈ X∗} for some R ∈ ∩µ. By applying 5.10 we can findB ⊂ X finite with R(B) = X. We put B∗ := B ∪ {∞}, hence B∗ ⊂ X∗ is finite.So it remains to verify that the equation R∗(B∗) = X∗ holds.

Let z ∈ X∗, in the case of z ∈ X, we can choose x ∈ B with (x, z) ∈ R.Consequently, x ∈ B∗ and (x, z) ∈ R∗ follow, showing that z ∈ R∗(B∗) is true. Inthe other case, z =∞ implies z ∈ B∗, and (z, z) ∈ R∗ follows, showing the claim,too. �

Theorem 5.12. For a non-empty generated final ultrabounded b-uniform ringspace (X,BX , µ) the b-uniform ring space (X∗,BX∗ , µ∗) is ultracompact.

Proof. By applying 2.9 and the former obtained results. �

Remark 5.13. The outcome just obtained may be of importance, if one in-tends to consider generalized proximities defined in terms of final ultraboundedb-uniform ring structures. But this line of vision may be left to the reader.

Applied Resume 5.14. At the end of this section we still mention the factsthat each non-empty b-uniform net space as well as every non-empty merotopicallyb-uniform filter space has an ultracompletion, too [17]. Consequently, as a corol-lary we obtain the result that each quasiuniform limit space has an ultracompletionif supposing the discrete case. Secondly, we can state, that a non-empty merotopi-cally b-uniform filter space possesses at least two different completions, namelythe one mentioned above and, in addition, the Cauchy-completion, dealt with in[17].

6. Some important links to generalized approach spaces

The central idea in approach spaces in the sense of Lowen is that of a distance d,which is a function onX×2X to [0,∞]. Here of fundamental interest is the fact thata distance can be defined not only in a metric space, but also in a topological space,a uniform space and so on. This setting is in fact well motivated by Lowens originalaxioms for an approach space in terms of its point-set function d : X × 2X −→[0,∞] listed in [18,21]. Now, we get some interesting links to our former presentedconcepts of b-uniform filter spaces, set-convergence spaces, b-topological spaces,see 6.4 or bornological spaces, respectively, see also [10].

First we will give the definition of a so-called approach-bornology on a set X.

Definition 6.1. For a set X a pair (BX , d) consisting of non-empty sub-set BX ⊂ PX and a distance function d : X × BX −→ [0,∞] is called anapproach-bornology, shortly apbornology and the triple (X,BX , d) approach-bor-nological space (shortly apbornological space or apb space, respectively), providedthat the following conditions are satisfied,

(apb1) B1 ⊂ B ∈ BX imply B1 ∈ BX ;(apb2) x ∈ X implies {x} ∈ BX ;


(apb3) B1, B2 ∈ BX imply B1 ∪B2 ∈ BX ;(apb4) x ∈ X implies d(x, ∅) =∞;(apb5) x ∈ X implies d(x, {x}) = 0;(apb6) x ∈ X and B1, B2 ∈ BX imply d(x,B1 ∪B2) = min{d(x,B1), d(x,B2)}.If (X,BX , dX), (Y,BY , dY ) are apb spaces then a function f : X −→ Y is calledb-contracted map provided it satisfies following conditions,

(bct1) f is bounded;(bct2) for each x ∈ X and for each B ∈ BX the unequality dY (f(x), f [B]) ≤

dX(x,B) holds.

By APB we are denoting the category of apb spaces and b-contracted maps.

Remark 6.2. Here, we point out that BX defines a bornology on X in thesense of Hogbe–Nlend [9]. In addition we note that the operator cld : BX −→ PXdefined by setting cld(B) := {x ∈ X : d(x,B) = 0} forms a b-closure operator onBX provided that cld(B) ∈ BX is valid for every B ∈ BX [14].

In this context we can specify the term b-contracted by adding that f is alsorebounded, if necessary, which means for each D ∈ BY , f−1[D] ∈ BX is valid. Wealso point out that in the case of BX = PX, the corresponding space is calledpre-approach space by F. Mynard and E. Pearl in their book Beyond Topology[20]. Here one can find additional propositions concerning this construct.

Furthermore, motivated by Lowen’s idea to define a distance in a topologicalspace we will generalize this concept to a b-closure operator(b-closure) as indicatedin the following:

Definition 6.3. In accordance with Leseberg [14], we call a pair (BX , h), whereBX is bornology and h : BX −→ PX a function, called b-closure operator, a b-closure structure (on BX) and the triple (X,BX , h) b-closure space, provided thatthe following conditions are satisfied:

(bclo1) B ∈ BX implies h(B) ∈ BX ;(bclo2) h(∅) = ∅;(bclo3) x ∈ X implies x ∈ h({x});(bclo4) B1 ⊂ B ∈ BX imply h(B1) ⊂ h(B);(bclo5) B1, B2 ∈ BX imply h(B1 ∪B2) ⊂ h(B1) ∪ h(B2).

If (X,BX , hX), (Y,BY , hY ) are b-closure spaces, then a function f : X −→ Y iscalled b-continuous map provided f satisfies the following conditions:

(bc1) f is bounded;(bc2) f is rebounded;(bc3) B ∈ BX implies f [hX(B)] ⊂ hY (f [B]).

By b-CLO we are denoting the category of b-closure spaces and b-continuousmaps.

Remark 6.4. As an important supplement we should note that a b-closurestructure (BX , h) is called b-topology and the triple (X,BX , h) b-topological spaceprovided that, in addition, the following condition holds:

(btop) B ∈ BX implies h(h(B)) ⊂ h(B).

Then, h is called b-topological operator. By b-TOP we denote the full subcategoryof b-CLO whose objects are the b-topological spaces [14].


Now, we will give an intrinsic example for the former introduced concept asfollows:

Example 6.5. Let a bornological space (X,BX) be given and x ∈ X be a pointof X. Then, we define a b-topological operator tx : BX −→ PX with fix-point xby setting:

tx(∅) := ∅ andtx(B) := {x} ∪B for B ∈ BX\{∅}.

Remark 6.6. With the now introduced concept it is possible to consider topolo-gies not only on the power set PX of X, but also on subsets such as the set of allfinite sets, compact sets or totally bounded sets, respectively. In the case of BXbeing saturated, meaning that X ∈ BX holds, b-topological spaces and topologicalspaces are essentially the same (up to isomorphism). On the other hand classicalclosure spaces are extensively examined in the book of Cech, so that the conceptof b-closure operators also makes sense.

Proposition 6.7. In a b-closure space (X,BX , h) we set δh(x,B) := 0 iffx ∈ h(B) and δh(x,B) := ∞, otherwise. Then, the pair (BX , δh) satisfies theconditions for an apbornology, and it is compatible with the b-closure operator h.

Proof. Evidently, clδh defines a b-closure operator, see also 6.2, and the follow-ing sequence of equivalences is valid, i.e. x ∈ clδh(B) ⇔ δh(x,B) = 0 ⇔ x ∈h(B). �

Remark 6.8. As seen above to make sure that in general cld : BX −→ PXforms a b-closure operator it is necessary to provide the condition in 6.2. Now thequestion arises, whether the associated b-closure (BX , cld) is compatible with thegiven apbornology (BX , d). To this end, we are giving the following condition:

Definition 6.9. An apbornology (BX , d) is called covered, and the triple(X,BX , d) covered apbornological space (shortly covered apb space) provided thatthe following conditions are satisfied:

(cov1) B ∈ BX implies cld(B) ∈ BX ;(cov2) B ∈ BX and x /∈ cld(B) imply d(x,B) =∞.

By COV-APB we denote the subcategory of APB, whose objects are covered.

Proposition 6.10. For a covered apbornological space(X,BX , d) the followingequation holds, i.e. δcld = d.

Proof. Let x ∈ X and B ∈ BX . In the case of x ∈ cld(B), we have d(x,B) = 0and δcld(x,B) = 0, which shows the equality. If x /∈ clδ(B), then d(x,B) =∞ byapplying (cov2). But on the other hand δcld(x,B) =∞ also follows, thus implyingthe claim. So it remains to verify that δh is satisfying (cov2), because by 6.3 wehave clδh(B) = h(B) ∈ BX , thus implying (cov1). x /∈ clδh(B) implies x /∈ h(B),hence δh(x,B) =∞, so that δh fulfills (cov2). �

Proposition 6.11. For b-closure spaces (X,BX , hX), (Y,BY , hY ) let f : X −→Y be a function, then the following statements are equivalent:

(i) f : (X,BX , hX) −→ (Y,BY , hY ) is b-continuous;(ii) f : (X,BX , δhX ) −→ (Y,BY , δhY ) is b-contracted.


Proof. Consider remark 6.2.to (i)⇒ (ii): So let x ∈ X and B ∈ BX . In the case of x ∈ hX(B), δhX (x,B) =

0, and by the hypothesis f(x) ∈ hY (f [B]) follows, showing that δhY (f(x), f [B]) =0 is true, and the claim in (ii) results. In the case of x /∈ hX(B), δhX (x,B) = ∞is valid. Since δhY (f(x), f [B]) ∈ [0,∞] holds we obtain δhY (f(x), f [B]) ≤ ∞ =δhX (x,B), and the claim follows.

to (ii)⇒ (i): Let B ∈ BX and y ∈ f [hX(B)]. Our goal is that δhY (y, f [B]) = 0can be deduced.

We have y = f(x) for some x ∈ hX(B), and consequently δhX (x,B) = 0 follows.But by the hypothesis f is b-contracted, i.e. δhY (f(x), f [B]) ≤ δhX (x,B) holds,and the claim results. �

Theorem 6.12. The categories COV-APB and b-CLO are isomorphic.

Proof. Evident, by applying 6.4, 6.5, 6.6, 6.7, 6.8, 6.9 and 6.10, respectively. �

To obtain a corresponding embedding of b-TOP into APB we have to specifythe concept of a covered apbornology as follows:

Definition 6.13. A covered apbornology (BX , d) is called topoform, and thetriple (X,BX , d) topoform apb space, provided that the following condition is sat-isfied,

(top) x ∈ X and B ∈ BX imply d(x,B) ≤ d(x, cld(B)).

By TOP-APB we denote the full subcategory of CLO-APB, whose objects aretopoform.

Remark 6.14. It is easy to see that if (BX , d) forms a topoform apbornology,then cld : BX −→ PX defines a b-topological operator on BX .

Theorem 6.15. The categories TOP-APB and b-TOP are isomorphic.

Proof. By applying the results obtained earlier and taking into account thefollowing last reflection, i.e. for a b-topological operator t : BX −→ PX, x ∈X and B ∈ BX we consider the two distances, i.e. δt(x,B) and δt(x, clδt(B)),respectively. In the case of x ∈ t(B), δt(x,B) = 0. On the other hand weget x ∈ t(t(B)), hence δt(x, t(B)) = 0 implying x ∈ clδt(t(B)) = clδt(B)), thusδt(x, clδt(B)) = 0, and the claim follows. If secondly x /∈ t(B),then δt(x,B) =∞ is valid. On the other hand we have x /∈ t(t(B)), since t satisfies (btop).Consequently,∞ = δt(x, t(B)) = δt(x, clδt(B)) follows, showing the claim too. �

Remark 6.16. Our last result now makes sure that the category APB andthe category of generalized near spaces, denoted by SD [17] have a non-emptyintersection, which contains b-TOP.

Next we will give a slight modification for the conditions of being an apbornol-ogy.

Definition 6.17. For a bornology BX let d : X ×BX −→ [0,∞] be a functionso that (BX , d) satisfies the following conditions, i.e.

(pre-apb1) x ∈ X implies d(x, ∅) =∞;(pre-apb2) x ∈ B ∈ BX imply d(x,B) = 0.


Then, (BX , d) is called a pre-apbornology (on X) and the triple (X,BX , d) pre-apbornological space (shortly pre-apb space). By PRE-APB we denote the cat-egory, whose objects are the pre-apb spaces and whose morphisms between themsatisfy the conditions in 6.1.

Remark 6.18. Evidently, every apbornological space is a pre-apb space.

Now, let us return to the concept of pointset- convergence spaces, compare alsowith [19].

Proposition 6.19. For a pointset- convergence space (X,BX , q) (see also 3.2),we put:

dq(x,B) = 0 iff•x q B and

dq(x,B) =∞ otherwise.

Then, (BX , dq) forms a pre-apbornology.

Proof. to (pre-apb1) dq(x, ∅) = ∞ holds, since•x is not in relation with ∅.

Otherwise•x= PX is a contradiction.

to (pre-apb2) For x ∈ B ∈ BX•B q B and

•B⊂

•x are valid, implying

•x q B,

hence dq(x,B) = 0 follows.Conversely, for a pre-apbornology (BX , δ) we are setting:

F pδ ∅ iff F = PX andF pδ B iff ∀ x ∈ B ∃ F ∈ F ∩ BX δ(x, F ) = 0 for B ∈ BX\{∅}.

Then, obviously (BX , pδ) defines a pointset-convergence. �

In this context we mention that convergence on approach spaces is extensivelystudied by Lowen or Brock [2,19], respectively under name of convergence approachspaces. So it was shown that the axiom (F) defined for limit spaces by Cook andFischer can be extended to an axiom (F) for convergence approach spaces so thatthe full subcategory of CAP (convergence approach spaces), whose objects satisfy(F) is isomorphic to the category AP of approach spaces.

Now, the question arises, whether (BX , dq) is compatible with the pointset-convergence (BX , q), meaning that pδq = q can be deduced. But this seems onlyto be possible by adding suitable properties for the given pointset-convergence asdone in the following:

Definition 6.20. A pointset-convergence (BX , q) is called rotary and the triple(X,BX , q) rotary pointset-convergence space, provided that the following condi-tions are satisfied, i.e. For F ∈ FIL(X) and each x ∈ X let being valid:

(rot1) F ∈ F ∩ BX and•x q F imply F q {x};

(rot2) F q {x} implies ∃ F ∈ F ∩ BX s.t.•x q F .

By RP-SETCONV we denote the full subcategory of P-SETCONV, whoseobjects are rotary.

Proposition 6.21. Let a rotary pointset- convergence space (X,BX , q) be given,then the pre-apbornology (BX , dq) as defined in 6.19 is compatible.


Proof. So let without restriction B ∈ BX\{∅} and at first F pdq B. Then, for

each x ∈ B we can find F ∈ F ∩BX such that dq(x, F ) = 0 is valid. Hence•x q F

follows, and F q {x} results by applying (rot1). Consequently, we obtain F q B,since by the hypothesis (BX , q) is a pointset-convergence.

Conversely, for F q B let x ∈ B. Since (BX , q) is reordered F q {x} follows

implying the existence of F ∈ F ∩ BX with•x q F by applying (rot2), hence

dq(x, F ) = 0, and F pdq B results. �

To ensure that the pointset-convergence (BX , pδ) is rotary the pre-apbornology(BX , δ) has to be reflexive in the following sense:

Definition 6.22. A pre-apbornology (BX , d) is called reflexive and the triple(X,BX , d) an reflexive pre- apb space provided that the following condition holds:

(ref) z ∈ X,F ∈ BX and d(z, F ) = 0 imply x = z for every x ∈ F .

By RPRE-APB we denote the full subcategory of PRE-APB, whose objectsare reflexive.

Remark 6.23. For a pre-apbornology (BX , d) let d be injective, then (BX , d)is reflexive. On the other hand for a reflexive apbornology (BX , d) the boundedsets of BX are closed meaning that cld(B) = B holds for every B ∈ BX .

Proposition 6.24. For a reflexive pre-apbornology (BX , δ) the pointset-con-vergence (BX , pδ) is a rotary T1 set-convergence.

Proof. to (rot1): For F ∈ FIL(X), x ∈ X and F ∈ F ∩BX let•x pδ F . Choose

z ∈ F , hence by the hypothesis we can find F1 ∈•x ∩BX with δ(x, F1) = 0. But

x ∈ F1 implies z = x, because (BX , d) is reflexive. Consequently, x ∈ F follows,and according to (pre-apb2) δ(x, F ) = 0 results implying F pδ {x}.

to (rot2): F pδ {x} implies the existence of F ∈ F ∩ BX with δ(x, F ) = 0. Let

z ∈ F , hence x = z by the hypothesis, and x ∈ F follows, showing that F ∈•x istrue. But (BX , pδ) is a pointset-convergence, and the claim results.

to (T1): For x, z ∈ X let•x pδ {z}, hence we can find F ∈•x ∩BX with δ(z, F ) = 0

implying x ∈ F , and by the hypothesis x = z follows (compare also with 5.5). �

Proposition 6.25. For a T1 pointset-convergence (BX , q), (BX , dq) is reflexive.

Proof. z ∈ X,F ∈ BX and dq(x, F ) = 0 imply•x q F by the definition. Let

x ∈ F , then we get•x q {z} according to (RO), and x = z follows, since (BX , q) is

T1 set-convergence. �

Definition 6.26. A reflexive pre-apbornology (BX , δ) is called convergent, andthe triple (X,BX , δ) convergent pre-apb space, provided that the following condi-tion is satisfied:

(conv) x ∈ X,B ∈ BX and•x is not in relation under pδ withB imply δ(x,B) =∞.

By CPRE-APB we denote the full subcategory of RPRE-APB, whose objectsare convergent.


Proposition 6.27. For a convergent pre-apbornology (BX , δ) the equation

dpδ = δ

holds.

Proof. Let x ∈ X and B ∈ BX . In the first case of•x pδ B, we have dpδ (x,B) =

0. On the other hand choose z ∈ B, hence δ(z,B) = 0. Moreover by the definition

of pδ we can find F ∈•x ∩BX with δ(z, F ) = 0. But x ∈ F implies x = z, since(BX , δ) is reflexive, and δ(x,B) = 0 follows which shows the claim. Secondly, if•x is not in relation under pδ with B then dpδ(x,B) = ∞ = δ(x,B) are true byapplying the hypothesis. �

Now if we denote by T1RP-SETCONV the full subcategory of

RP-SETCONV

whose objects are T1 set- convergence spaces we obtain the following result:

Theorem 6.28. The categories T1RP-SETCONV and CPRE-APB areisomorphic.

Proof. Evident, by taking into account the last verified equations. So it onlyremains to prove the following equivalence of statements, i.e., for convergent pre-apb spaces (X,BX , dX), (Y,BY , dY ) let f : X −→ Y be a map. Then, followingstatements are equivalent,

(i) f : (X,BX , dX) −→ (Y,BY , dY ) is b-contracted;(ii) f : (X,BX , pdX ) −→ (Y,BY , pdY ) is b-continuous.

to (i)⇒ (ii): Let F pdX B and y ∈ f [B], hence y = f(x) for some x ∈ B.By the hypothesis there exists F ∈ F ∩ BX with dX(x, F ) = 0. Since f is b-contracted dY (f(x), f [F ]) ≤ dX(x, F ) = 0 and f [F ] ∈ f(F)∩BY follow, implyingdY (y, f [F ]) = 0, and the claim results.

to (ii)⇒ (i): For x ∈ X and B ∈ BX we consider dX(x,B) as well as

dY (f(x), f [B]).

But dX(x,B) = dpdX (x,B) and dY (f(x), f [B]) = dpdY (f(x), f [B]).

In the case of•x pdX B, we have dX(x,B) = dpdX (x,B) = 0 and

•f(x) pdY f [B].

But dY (f(x), f [B]) = dpdY (f(x), f [B]) = 0 implies dY (f(x), f [B]) ≤ dX(x,B).

Secondly if•x is not in relation under pdx withB we get dX(x,B) = dpdX (x,B) =∞

with dY (f(x), f [B]) ≤ ∞, and the claim follows. �

Our next definition provides a closed connection between bornological spacesand apbornological spaces in the sence that BORN can be regarded as a full em-bedded subcategory of APB. Hence in view of 1 and 4.5, respectively b-URINGand APB intersects.

Definition 6.29. An apbornology (B, δ) is called elementary, and the triple(X,BX , δ) an elementary apbornological space provided that the following condi-tion is satisfied,

(e) x /∈ B implies δ(x,B) =∞.


By E-APB we denote the full subcategory of APB, whose objects are elementary.

Theorem 6.30. The categories E-APB and BORN are isomorphic.

Proof. For a bornological space (X,BX) we define a distance function dXb :X × BX −→ [0,∞] by setting for each pair (x,B) ∈ X × BX :

dXb (x,B) := 0 iff x ∈ B;dXb (x,B) :=∞, otherwise.

Then, the natural corresponding assignments are functorial and determine theannounced isomorphism between the two categories in question. �

Remark 6.31. By applying 2.7 we can state that every non-empty elementaryapbornological space is ultracomplete, up to isomorphism.

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Dieter Leseberg, Ernst Reuter Gesellschaft Berlin, Germanye-mail : [email protected]

Zohreh Vaziry, Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj,

Irane-mail : z−m−[email protected]

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