_{1}

^{*}

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, Do?tch?nov 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

(b_{1})

(b_{2})

(b_{3})

Then, for

(b)

Definition 1.1 For a set

(hn_{1})

(hn_{2})

(hn_{3})

(hn_{4})

If

(hn)

(shn)

Remark 1.2 Note, that shn-maps between prehypernear spaces are always hn-maps. We denote by PHN^{·} respectively PHN the corresponding categories.

Examples 1.3 (i) For a prenearness space

(ii) For a

is defined by setting:

(iii) For a set-convergence space

(iv) For a generalized convergence space

for

and

(v) For a

with

(vi) For a

for each

(vii) For a neighborhood space

Remark 1.4 In preparing the next two important examples we give the following definitions.

Definitions 1.5 TEXT denote the category, whose objects are triples

(tx_{1})

(tx_{2})

Morphisms in TEXT have the form

If

where “

Remark 1.6 Observe, that axiom

(1) strict iff

(2) symmetric iff

Examples 1.7 (i) For a topological extension

if

(ii) For a symmetric topological extension

if

With respect to above examples, first let us focus our attention to some important classes of prehypernear spaces.

Definitions 2.1 A prehypernear space

(i) saturated iff

(ii) discrete iff

(iii) symmetric iff

(iv) pointed iff

(v) conic iff

(vi) set-defined iff

Theorem 2.2 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.

Proof. According to Example 1.3. (i) we claim that

Hence, the above mentioned connections are functoriell, and thus it remains to prove that the following two statements are valid, i.e.

(i)

(ii)

To (i): “

“

and

Since

To (ii): “

For

“

Remark 2.3 In this context we point out that each prehypernear space

(1)

(2)

where the following inclusion is valid:

Definition 2.4 A prehypernear space

Remark 2.5 In this context we refer to Examples 1.3. (i), (iv), (v), (vi), (vii), respectively Examples 1.7. (i), (ii).

Theorem 2.6 The category Č-CLO of Čech-closure spaces and continuous maps is isomorphic to a full subcategory of PSHN.

Remark 2.7 Now, before showing the above mentioned theorem we give the following definition.

Definition 2.8 A prehypernear space

(sec)

Remark 2.9 In this connexion we point out that each pointed prehypernear space (see Remark 3.6) is always sected.

Moreover, sected prehypernear spaces are already pseudohypernear spaces.

Definition 2.10 A sected conic saturated prehypernear space is called closed, and we denote by CL-PHSN the full subcategory of PSHN, whose objects are closed pseudohypernear spaces.

Proof of Theorem 2.6.

According to Example 1.3. (v) we claim that

(i)

(ii)

To (i): Now let be

Secondly,

To (ii): Now, let be without restriction

Conversely,

Now, we will show that

Choose

is valid which implies

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.

Definitions 3.1 A prehypernear space

(gri)

where

(gri_{1})

(gri_{2})

We denote by G-PHN the category, whose objects are the prehypergrill spaces with hn-maps between them and by G-PHN

Remark 3.2 We refer to Examples 1.3. (ii), (iii), (iv), (vi), (vii) respectively and to Examples 1.7. (i), (ii).

Theorem 3.3 The category GRILL of grill-determined prenearness spaces and nearness preserving maps is isomorphic to a full subcategory of G-PHN.

Proof. According to Theorem 2.2 we already know that

Theorem 3.4 The category SETCONV ([

Proof. According to Example 1.3. (iii) we claim that the triple

(i)

(ii)

To (i) “

“

To (ii): “

“

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

if

Conversely let be given a pointed saturated prehypergrill space

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 [^{·} we denote the corresponding categories. At last we point out that conic pseudohypernear 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

for each

(i)

(ii)

To (i): “

“

is valid.

To (ii): “

“

Remark 3.9

(1) For a symmetric topological space

(2) Let being the same hypothesis as in (1). We set

ness relation

Definition 3.10 A

(bp_{1})

(bp_{2})

(bp_{3})

(bp_{4})

Remark 3.11 Here we point out that b-proximities are in one-to-one correspondence with presupertopologies. In the symmetric case, if

(sbp)

Definition 3.12 For

(p)

Theorem 3.13 The category b-PROX and CG-PSHN are isomorphic.

Proof. For a b-proximity space

for each

versely let be given such a space

is defined by setting

(i)

(ii)

To (i): “

“

To (ii): “

“

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).

A slight modification of the definition for being a prehypergrill space leads us to the following notation.

Definition 4.1 A prehypernear space

(b)

Remark 4.2 Each prehypergrill space is bonded.

Proof. evident.

Definition 4.3 Now, we call a bonded pseudohypernear space a semihypernear space and denote by SHN the full subcategory of PSHN.

Theorem 4.4 The 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

bonded, because

hence there exist

But

Theorem 4.5 The category SNEAR of seminearness spaces and related maps is isomorphic to a full sub- category of SHN.

Proof. According to Theorem 2.2 we firstly show that

and

tain

Remark 4.6 A pseudohypernear space

whereby the inclusion

(bh_{1})

(bh_{2})

(bh_{3})

For psb-hull spaces

Definition 4.7 Now, we call a conic pseudohypernear space

(h)

Theorem 4.8 The categories Psb-HULL and PSHU are isomorphic.

Proof. According to Remark 4.6 we already know that

psb-hull space

Then

(i)

(ii)

To (i): “

“

To (ii): “

“

_{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

(cnc)

Remark 4.11 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

(ad)

As already observed, hypertopologies appear in connexion with certain interior operators studied by Kent and Min ([

(hyt_{1})

(hyt_{2})

(hyt_{3})

(hyt_{4})

For hypertopological spaces

iff

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.

Definition 5.1 A pseudohypernear space

(hn)

We denote by HN the corresponding full subcategory of PSHN. Note, that in this case

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.

Definition 5.4 A prehypernear space

(sr)

Remark 5.5 Here we claim that each pointed prehypernear space is surrounded, hence sected, too. (See also Definition 2.8).

Lemma 5.6 For a hypernear space

(i)

(ii)

Proof. The only remaining implication “(ii)

hence

Remark 5.7 Now, if we consider a bounded hypertopology, this is a psb-hull operator _{4}), i.e.

(bh_{4})

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

(nt)

Remark 5.10 We note that each surrounded prehypernear space is neartopological. On the other hand let be given a symmetric bounded hypertopological space

(sym)

then we define the corresponding neartopological hypernear space

(d)

This can be seen as follows: Without restriction let be

by hypothesis.

hence

results, since

(ron)

A detailed description of this fact will be given in some forthcoming papers. Then evidently saturated spaces are round. Analogously, we can consider roundbounded symmetric hypertopological spaces, i.e. spaces

(rd)

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:

(i)

(ii)

To (i): Let be

To (ii): Without restriction let be

that

In the saturated case then we can recover all symmetric hypertopological 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 sub- category 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

For

ator we get

Corollary 6.3 The category STOP of supertopological spaces and continuous maps is isomorphic to a sub- category 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 _{5}), i.e.

(bp_{5})

By pLESP we denote the corresponding full subcategory of b-PROX.

In the saturated case (if

Corollary 6.5 The category pLESP is isomorphic to a full subcategory of SN.

Proof. According to Example 1.3. (vi), Remark 3.11 and Theorem 3.13 respectively it remains to verify that _{5}) respectively.

To (hn):

results, showing that _{5}) we get

To (bp_{5}): Conversely, let be

By hypothesis

implies_{1}) we obtain

Remark 6.6 At this point we note that certain supernear spaces are in one-to-one correspondence to strict topological extensions which we study in a forthcoming paper. Here, we will examine the case if a symmetric topological extension is presumed (see Example 1.7. (ii)). In this connexion bunch-determined nearness and certain preLODATO spaces are playing an important role. Now, we will give the definition of a preLODATO space:

Definition 6.7 A preLEADER space

(bp_{6})

(bp_{6})

(bp_{6})

By pLOSP we denote the corresponding full subcategory of pLESP.

Remark 6.8 In the saturated case LODATO proximity spaces then can be recovered as special objects. More- over, we note that each b-supertopological space then can be regarded as special preLODATO space. A slight specialization lead us to the so-called LODATO space by adding the axiom (bp_{9}), i.e.

(bp_{9})

Once again, in the saturated case the two definitions coincide, and LODATO proximity spaces then can be recovered as special objects.

But in general the two definitions differ, and the reader is referred to Remark 3.9 in connexion with Remark 5.10. In a forthcoming paper we will show that the corresponding category LOSP of LODATO spaces can be re- garded as a full subcategory of SN, whose objects are symmetric. On the other hand nearness also leads us to a certain symmetric supernear space, hence we give the following definition.

Definition 6.9. A symmetric supernear space is called a paranear space and we denote by PN the corresponding full subcategory of SN.

Theorem 6.10. The category NEAR of nearness spaces and related maps is isomorphic to a full subcategory of PN.

Proof. According to Example 1.3. (ii) and Theorem 4.5 respectively it remains to verify that

To (hn): Without restriction let be

But

nearness axiom: we get

Since M is dense (see Remark 5.10) we get _{1}). But then

Corollary 6.11. For a saturated paranear space

(i)

(ii)

Proof. evident according to Remark 5.10.

“Relationship between important categories”

Taking into account Example 1.7.(ii), Remark 3.9, 6.6 and 6.8 respectively we will now consider the problem for finding a one-to-one correspondence between certain topological extensions and their related paranear spaces. In this connexion we point out that certain grill-spaces come into play.

Definition 7.1 Let be given a supernear space

(cla_{1})

(cla_{2})

Remark 7.2 For a supernear space

Definitions 7.3 A supernear space respectively paranear space

(cla)

Remark 7.4 In giving some examples we note that each surrounded supernear space is a superclan space, and each neartopological paranear space is a paraclan space. This is analogical valid for the spaces considered in 1.7.

Proof of Example 1.7. (ii)

First, we prove the equality of the corresponding closure operators. So, let be _{1})

metric. Consequently, _{1}). Alltogether, the equality now results. Secondly, it is easy to verify that ^{e} is symmetric, since

since

because

consequently ^{e} satisfies the axiom (cla). For

that

Convention 7.5 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

(a) For a SY-TEXT-object

(b) for a TEXT-morphism

Then

Proof. We already know that the image of

tion let

Our goal is to verify the existence of

we have

by assumption. Now, consider some

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

where

Proof. We first note that

Then,

an element

other hand,

of

plies

Theorem 8.3 For paranear spaces

Then the following statements are valid:

(1)

(2) The composites

Proof. First, let _{1}) we observe that

(Note, that

Since

clude that

To (1): Let

hence

To (2): Let

Thus

spect to

is a hn-map, we obtain the desired equality.

Theorem 8.4 We obtain a functor

(a)

(b)

Proof. With respect to Corollary 6.2 it is straight forward to verify that _{1}), let

_{2}), let be

satisfies (cla_{2}). But this is a contradiction, and thus

symmetric let x be an element of

hypothesis we have

and N is symmetric we get _{1}).

But

hypothesis

To this end let

“

Since

“

Finally, this 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

Theorem 8.5 Let

Then

is a hn-map in both directions for each object

Proof. The commutativity of the diagram is obvious, because of

is a hn-map in both directions. To fix the notation let

It suffices to show that for each

get

Now

(1)

(2)

To (1): By definition of

To (2): Let A be an element of

Remark 8.6 Making the theorem more transparent we claim that a paranear space is a paraclan space if it can be embedded in a topological space

Corollary 8.7 If

(sep)