Improved Nearness Research

In the realm of Bounded Topology we now consider supernearness spaces as a common generalization of various kinds of topological structures. Among them the so-called Lodato spaces are of significant interest. In one direction they are standing in one-to-one correspondence to some kind of topological extensions. This last statement also holds for contiguity spaces in the sense of Ivanova and Ivanov, respectively and moreover for bunch-determined nearness spaces as Bentley has shown in the past. Further, Doîtchînov proved that the compactly determined Hausdorff extensions of a given topological space are closely connected with a class of supertopologies which he called b-supertopologies. Now, the new class of supernearness spaces—called paranearness spaces—generalize all of them, and moreover its subclass of clan spaces is in one-to-one correspondence to a certain kind of symmetric strict topological extension. This is leading us to one theorem which generalize all former mentioned.

As usual PX denotes the power set of a set X , and we use X PX ⊂  to denote a collection of bounded subsets of X , also known as B -sets, i.e.X  has the following properties: (b 1 ) Definition 1.1 For a set X , we call a triple ( )  and an operator ( ) ( ) a prehypernear space iff the following axioms are satisfied, i.e.
(shn) X B ∈  and ( ) 2 Note, that shn-maps between prehypernear spaces are always hn-maps.We denote by PHN • respectively PHN the corresponding categories.(ii) For a b -filter space ( ) , , ) we consider the triple ( ) , where for each is defined by setting: (iii) For a set-convergence space ( ) , , X X q  ( [3] we consider the triple ( ) , where for each is defined by setting: (iv) For a generalized convergence space ( ) , X q [4], we consider the triple ( ) ( ) for x X ∈ with ( ) { } : q N ∅ = ∅ ; alternately we look at the following triple ( ) , , q X PX N , where ) ( ) → are continuous maps such that f is bounded, and the following diagram commutes: they can be composed according to the rule: where "  " denotes the composition of maps.
Remark 1.6 Observe, that axiom ( ) tx in this definition is automatically satisfied if : e X Y → is a topological embedding.Moreover we admit an ordinary B -set X  on X which need not be necessary coincide with the power PX .In addition we mention that such an extension is called ⊂ forms a base for the closed subsets of Y [7]; (2) symmetric iff x X ∈

Fundamental Classes of Prehypernear Spaces
With respect to above examples, first let us focus our attention to some important classes of prehypernear spaces.
Hence, the above mentioned connections are functoriell, and thus it remains to prove that the following two statements are valid, i.e. (i) is symmetric and saturated by hypothesis.Consequently, ( ) where the following inclusion is valid: . In the symmetric case these two operators coincide, moreover we have ( ) , and finally ( ) , M Y cl .Hence, the above men- tioned connections are functoriell, and thus it remains to prove that the following two statements are valid, i.e.: (i) , and x A ∈ follows.
To (ii): Now, let be without restriction ( ) ( ) Now, we will show that , hence concluding the proof.Remark 2.11 Now, in the following another important class of prehypernear spaces will be examined, being fruitful in considering convergence problems and having those properties, which are characterizing topological universes.

Grill-Spaces
is called a prehypergrill space iff N satisfies (gri), i.e. (gri) We denote by G-PHN the category, whose objects are the prehypergrill spaces with hn-maps between them and by G-PHN • the category, whose objects are the prehypergrill spaces with shn-maps between them.
Proof.According to Theorem 2.2 we already know that ( ) , , X PX N ξ is a symmetric saturated prehypernear space, hence additionally it is a prehypergrill space by hypothesis.Conversely, M η is grill-determined by sup- position.
 Theorem 3. 4 The category SETCONV ([3]) of set-convergence spaces and related maps is isomorphic to a full subcategory of G-PHN • .
Proof.According to Example 1.3.(iii) we claim that the triple ( ) is a set-defined prehypergrill space.Conversely, we consider for such proposed space ( ) defined by setting: for each X B ∈  .Hence, the above mentioned connections are functoriell with respect to shn-maps.Thus, it remains to prove that the following two statements are valid, i.e. (i) , hence there exists ( ) ( )  Corollary 3. 5 The category GCONV of generalized convergence spaces and related maps is isomorphic to the category DISG-PHN • , whose objects are the discrete prehypergrill soaces and whose morphisms are the sected hn-maps.
Remark 3.6 Now, in this connextion it is interesting to note that there exists and alternate description of generalized convergence spaces in the realm of prehypergrill spaces.Analogously, how to describing set convergence on arbitrary B-sets we offer now a corresponding one for the point convergence as follows: Let be given a point-convergence space ( ) , X q , where ( ) is satisfying some natural conditions.Then we consider the following pointed prehyergrill space ( ) Conversely let be given a pointed saturated prehypergrill space ( ) then we naturally define a pointconvergence space ( ) . As a consequence we obtain the result that point convergence can be essentially expressed by means of its corresponding pointed saturated prehypergrill spaces and sected hn-maps.
Hence, the last mentioned category also is isomorphic to DISG-PHN • .Remark 3.7 Another interesting fact is the following one.As Wyler has shown in [3] supertopological spaces in the sense of Doîtchînov can be regarded as special set-convergence spaces.Hence it is also possible for describing them in the realm of prehypergrill spaces.Concretely let be given a supertopological space (see [10]) or more generally a neighborhood space ( ) in the sense of [6], in the following referred as to presupertopological space.Then we consider the triple ( ) , where is a conic pseudohypergrill space.Hereby, a prehypergrill space ( ) is called pseudohypergrill space iff N satisfies (is) (see also Definition 2.4).By CG-PSHN respectively CG-PSHN • we denote the corresponding categories.At last we point out that conic pseudohyper-near spaces are even set-defined.
Theorem 3.8 The category PRESTOP of presupertopological spaces and continuous maps is isomorphic to the category CG-PSHN • .
Proof.According to Remark 3.7 we consider conversely for a conic pseudohypergrill space ( ) , , , where for each is defined by setting: is a presupertopological space.Hence, the above mentioned connections are functoriell with respect to shn-maps.Thus, it remains to prove that the following two statements are valid, i.e. (i) implies the existence of ( )  Remark 3.9 b -proximities (see [6]) are of significant importance when considering topological extensions.Here we will give two interesting examples in that direction as follows: (1) For a symmetric topological space ( ) , Y t (given by a closure operator t) let X  be a B -set with X Y ⊂ , then we define a b -proximity ⊂ .Now, it is easy to verify that t δ is t -compatible, which means the equality t cl t δ = holds by re- stricting t on X , where t cl δ denotes the closure-operator induced by t δ .

{ }
: : and define a near- Then t δ defines a b-proximity with the same properties as mentioned above.Now, we recall the definition of a b-proximity respectively b-proximity space as follows: Definition 3.10 A b -proximity space consists of a triple ( ) satisfying the following conditions: (bp 1 ) A δ ∅ and Bδ∅ (i.e.∅ is not in relation to A , and analogously this is also holding for B ); (bp 2 ) ( ) Remark 3.11Here we point out that b-proximities are in one-to-one correspondence with presupertopologies.In the symmetric case, if δ additionally satisfies (sbp), i.e.
(sbp) 1 2 , δ and moreover X  equals PX , then symmetric b-proximities coincide with the Čech-proximities mentioned by Deák ([11]).Definition 3.12 For b -proximity spaces ( ) , where is a conic pseudohypergrill space.Conversely let be given such a space ( ) , then we consider the triple ( ) , , , where is a b-proximity space.The above mentioned connections are functoriell, and thus it remains to prove that the following two statements are valid, i.e. (i) To (i): " ⊂ ": B A δ implies { } ( ) To (ii): " ≤ ": ( ) . We will show that  Résumé 3.14 Respecting to former advisements we note that we have established only some topological concept in which some important classical ones can be now expressed and studied in a very natural way.Moreover, the fundamental categories how as GRILL, b-PROX, PRESTOP, GCONV and SETCONV can be regarded as special subcategories of G-PHN.(see also the Theorem 3.3, 3.4, 3.8 and 3.13 respectively).

Bonding in Prehypernear Spaces
A slight modification of the definition for being a prehypergrill space leads us to the following notation.
, where , hence being a pretopology on its underlying set.On the other hand ,

The category SNEAR of seminearness spaces and related maps is isomorphic to a full subcategory of SHN.
Proof.According to Theorem 2.2 we firstly show that N ξ is bonded.Without restriction bet be ( ) showing that N ξ is satisfying (b).On the other hand let be 1 by hypothesis.Since M is bounded we have ( ) . By symmetry of M we obtain the statement { } ( ) If N is symmetric then the two operators coincide, and moreover we claim the following equalities for each is called a psb-hull operator, and the triple ( ) is called a psb-hull space iff h satisfies the following conditions: (bh 1 ) ( ) For psb-hull spaces ( ) is a pseudohull space.Hence, the above mentioned connections are functoriell.Thus it remains to prove that the following two statements are valid, i.e. (i) according to (hn 1 ).Consequently, the above mentioned inclusion is valid, showing that ( )  Corollary 4.9 In the saturated case CL-PSHN and PSHU are isomorphic categories.Proof.We refer to Theorem 2.6, Definition 2.10 and Theorem 4.8 respectively.Definition 4.10 A prehypernear space ( ) 11 We note that each pointed prehypernear space is connected, moreover this also is holding for any symmetric semihypernear space.Consequently, the underlying psb-hull operator N − additionally satisfying (ad), i.e.
(ad) ( ) ( ) ( ) . Now, let us call such an operator h b-hull operator, and we denote by b-HULL the corresponding full subcategory of Psb-HULL with related objects.In the saturated case we claim that b-HULL and CL-SHN are isomorphic categories.Hereby CL-SHN denotes the full subcategory of SHN, whose objects are the closed semihypernear spaces.

Hypernear Spaces
As already observed, hypertopologies appear in connexion with certain interior operators studied by Kent and Min ([12]).Hereby a function : PX PX − → is called a hypertopology on X , and the pair ( ) , X − is called a hypertopological space iff "−" satisfies the following conditions: For hypertopological spaces ( ) . By HYTOP we denote the corresponding subcategory of C  -CLO.
Evidenly, the category TOP of topological spaces and continuous maps can be now regarded as a special case of HYTOP.On the other hand certain nearnesses play an important role in the realm of unifications and extensions, respectively.This is holding for distinguished nearness spaces and b-proximity spaces in fact.Moreover, certain supertopologies are involved, too.Now, in the following we will give a common description of them all by introducing the so called concept of a hypernear space.
. We denote by HN the corresponding full subcategory of PSHN.Note, that in this case N cl is a hypertopol- ogy on X .
Theorem 5.2 CL-HN denotes the full subcategory of CL-PSHN, whose objects are the closed hypernear spaces, then CL-HN and HYTOP are isomorphic.
Proof.The reader is referred to Theorem 2.6 and Definition 2.10, respectively. Remark 5.3 As pointed out in Remark 3.6, point convergence can be described by certain pointed prehypernear spaces.To obtain a result more closer related to hypertopologies we will give the following definition.

Lemma 5.6 For a hypernear space ( )
, , the following statements are equivalent: Proof.The only remaining implication "(ii) ⇒ (i)" will be shown now: ρ ∈ results according to (hn).Remark 5.7 Now, if we consider a bounded hypertopology, this is a psb-hull operator h on a B-set X  , which additionally satisfies (bh 4 ), i.e.
(bh 4 ) B X ⊂ and ( ) , then the corresponding category is isomorphic to the full subcategory SR-HN of HN, whose objects are the surrounded hypernear spaces.In this connexion we consider the restriction of N cl on the B-set X  .Conversely, for a bounded hypertopological space ( ) we define the corresponding sourrounded hypernear space ( ) , , , otherwise.In the saturated case then we can recover all hypertopological spaces.So, in general it is now possible to study those closure operators not only on PX , but also on arbitrary B-sets even in the realm of the broader concept of hypernear spaces.
Remark 5.8 In this connexion another concept of closure operators seems to be of interest, and it is playing an important rule when considering classical nearness structures.In the following we will give some notes in this direction.Definition 5.9 We call a prehypernear space ( ) 10 We note that each surrounded prehypernear space is neartopological.On the other hand let be given a symmetric bounded hypertopological space ( ) , , , where in addition h is satisfying (sym), i.e. (sym) , x z Y ∈ and { } ( ) then we define the corresponding neartopological hypernear space ( ) by setting: round.Analogously, we can consider roundbounded symmetric hypertopological spaces, i.e. spaces ( ) Then the corresponding category is isomorphic to the full subcategory RNT-HN of HN, whose objects are the round neartopological hypernear spaces.As above defined we only verify the following two statements: by definition.Hence there exists ( ) ( ) implies the existence of ( ) : results.In the saturated case then we can recover all symmetric hypertopological spaces.

Supernear and Paranear Spaces
Now, based on former advisements we are going to consider two special classes of hypernear spaces, which are being fundamental in the theory of topological extensions.Definition 6.1 We call a bonded hypernear space a supernear space and denote by SN the corresponding full subcategory of HN.

Corollary 6.2 The category TOP of topological spaces and continuous maps is isomorphic to a full subcategory of SN.
Proof.According to Example 1.3.(v), Theorem 2.6, Theorem 4.4 and Definition 5.1 we only have to verify that N − is satisfying (hn).Now, let be For F ρ ∈ we have ( )  (a) For a SY-TEXT-object ( ) , , Then F : SY-TEXT → CLA-PN is a functor.
Proof.We already know that the image of F lies in CLA-PN.Now, let ( ) ( ) ( )  be a TEXT-morphism; it has to be shown that f preserves B-near collections for each

Strict Topological Extensions
Remark 8.1 In the previous section we have found a functor from SY-TEXT to CLA-PN.Now, we are going to introduce a related one in the opposite direction.

Lemma 8.2 Let ( )
, , X X N  be a paranear space.We set { } : : , and for each C C A X ⊂ we put: Proof.We first note that ( ) . Consequently there exists : Then the following statements are valid: (1) C f is a continuous map from ( ) and Y e f  coincide, where : C X e X X → denotes the function which assigns the { } x -clan N x to each x X ∈ .Proof.First, let  be a B -clan in N .We will show that ( ) , which satisfies (cla 2 ) in Definition 7.1.In order to establish (cla 1 ) we observe that ( )

B N B ∈ ∈
 by hypothesis.We will now verify that (Note, that f is a hn-map by assumption.)For any ( )  , and all together we con- clude that

4 1 e
For a C  ech-closure space ( ) , X − ([5]) let X  be B -set.Then we consider the triple ( ) In preparing the next two important examples we give the following definitions.Definitions 1.5 TEXT denote the category, whose objects are triples topological spaces (given by closure operators) with B -set X − denotes the inverse image under e ; (tx 2 ) means that the image of X under e is dense in Y .Morphisms in TEXT have the form ( ) (

γ.
By b-PROX we denote the corresponding category.Theorem 3.13 The category b-PROX and CG-PSHN are isomorphic.Proof.For a b-proximity space ( ) 2 M A ∈  leads us to a contradiction. Remark 4.6 A pseudohypernear space ( ) underlying psb-hull operators by setting for each X B ∈  :

3 a
The category STOP of supertopological spaces and continuous maps is isomorphic to a subcategory of SN.Proof.The reader is referred to Remark 3.7, Theorem 3.8 and Remark 4.2 respectively. Remark 6.4 b-proximities (see Definition 3.10) are playing an important rule when considering topological extensions (see Remark 3.9).In this connexion we are now giving two special cases of them.First of all we call a b-proximity space () preLEADER space iff δ in addition satisfies (bp 5 ), i.e. tx 1 ).Alltogether, the equality now results.Secondly, it is easy to verify that ( ) for being a semihypernear space.N e is symmetric, since

5
space.It remains to prove N e satisfies the axiom (cla).For  is the desired B -clan in e We denote by SY-TEXT the full subcategory of TEXT, whose objects are the symmetric topological extensions and by CLA-PN the full subcategory of PN, whose objects are the paraclan spaces.Theorem 7.6 Let : F SY-TEXT → CLA-PN be defined by: results, since (f, g) is a TEXT-morphism by assumption.Now, consider some A ρ ∈  is valid.To (2): Let x be an element of X .We will prove the validity of

Theorem 8 . 4 Theorem 8 . 5
D D f e x fe x ∈  , since by hypothesis f is a hn-map, we obtain the desired equality.We obtain a functor G : CLA-PN to SY-TEXT by setting: With respect to Corollary 6.2 it is straight forward to verify that N cl is a topological closure operator on X .We also have the topological closure operator establish (tx 1 ), let A be a subset of X and suppose establishes that the composition of hn-maps is preserved by G.At last we will show that the image of G also is contained in STR-TEXT, whose objects are the strict topological extensions.Consider Let F : SY-TEXT → CLA-PN and G : CLA-PN → SY-TEXT be the above defined func- tors.For  each object ( ) t F G →  is natural equivalence from G F  to the identity functor PN CLA− 1 , i.e.

2
The category PNEAR of prenearness spaces and related maps is isomorphic to the category SY-PHN S of saturated symmetric prehypernear spaces and hn-maps.

CL-PHSN the full subcategory of PSHN, whose objects are closed pseudohypernear spaces. Proof of Theorem 2.6.
Now, before showing the above mentioned theorem we give the following definition.

2
Each prehypergrill space is bonded.Now, we call a bonded pseudohypernear space a semihypernear space and denote by SHN the full subcategory of PSHN.
4heorem 4.4The category PrTOP of pretopological spaces and continuous maps is isomorphic to a full subcategory of SHN.Proof.According to Theorem 2.6 respectively Definition 2.10 it is evident that M cl additionally satisfies We denote by Psb-HULL the corresponding category.
N B ρ ∈ .We denote by PSHU the full subcategory of PSHN, whose objects are the pseudohull spaces.Theorem 4.8 The categories Psb-HULL and PSHU are isomorphic.Proof.According to Remark 4.6 we already know that ( ) and F ∈  results, which leads us to a contradiction.
 Theorem 8.3 For paranear spaces