1. Basic Concepts
As usual
denotes the power set of a set
, and we use
to denote a collection of bounded subsets of
, also known as
-sets, i.e.
has the following properties:
(b1)
;
(b2)
imply
;
(b3)
implies
.
Then, for
-sets
a function
is called bounded iff
satisfies
, i.e.
(b)
.
Definition 1.1 For a set
, we call a triple
consisting of
,
-set
and an operator
a prehypernear space iff the following axioms are satisfied, i.e.
(hn1)
and
imply
, where
iff
;
(hn2)
implies
;
(hn3)
implies
;
(hn4)
implies
.
If
for some
, then we call
a
-near collection in
. For prehypernear spaces
,
a bounded function
is called a hypernear map, shortly hn-map iff it satisfies (hn), i.e.
(hn)
and
imply
; a sected hn-map, shortly shn-map iff it satisfies (shn), i.e.
(shn)
and
imply
with
and
.
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
([1] ) let
be
-set. Then we consider the triple
where
and
, otherwise.
(ii) For a
-filter space
([2] ) we consider the triple
, where for each ![]()
is defined by setting:
;
(iii) For a set-convergence space
([3] we consider the triple
, where for each
is defined by setting:
;
(iv) For a generalized convergence space
[4] , we consider the triple
, where
and
for
with
; alternately we look at the following triple
, where
,
and
;
(v) For a
ech-closure space
([5] ) let
be
-set. Then we consider the triple ![]()
with
for each
;
(vi) For a
-proximity space
([6] ) we consider the triple
, where
![]()
for each
with
;
(vii) For a neighborhood space
([6] ) we consider the triple
, where for each
.
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
—called topological extensions—where
are topological spaces (given by closure operators) with
-set
and
is a function satisfying the following conditions:
(tx1)
implies
, where
denotes the inverse image under
;
(tx2)
, which means that the image of
under
is dense in
.
Morphisms in TEXT have the form
, where
are continuous maps such that
is bounded, and the following diagram commutes:
![]()
If
and
are TEXT-morphisms, then they can be composed according to the rule:
![]()
where “
” denotes the composition of maps.
Remark 1.6 Observe, that axiom
in this definition is automatically satisfied if
is a topological embedding. Moreover we admit an ordinary
-set
on
which need not be necessary coincide with the power
. In addition we mention that such an extension is called
(1) strict iff
forms a base for the closed subsets of
[7] ;
(2) symmetric iff
and
imply
[8] .
Examples 1.7 (i) For a topological extension
we consider the triple
, where
![]()
if
and
;
(ii) For a symmetric topological extension
we consider the triple
, where
![]()
if
and
.
2. Fundamental Classes of Prehypernear Spaces
With respect to above examples, first let us focus our attention to some important classes of prehypernear spaces.
Definitions 2.1 A prehypernear space
is called
(i) saturated iff
;
(ii) discrete iff
;
(iii) symmetric iff
and
imply
and
;
(iv) pointed iff
implies
;
(v) conic iff
implies
;
(vi) set-defined iff
implies
.
Theorem 2.2 The category PNEAR of prenearness spaces and related maps is isomorphic to the category SY-PHNS of saturated symmetric prehypernear spaces and hn-maps.
Proof. According to Example 1.3. (i) we claim that
is a symmetric saturated prehypernear space. Conversely, we consider for such proposed space
the following prenearness space
defined by setting:
.
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
imply
, hence
, and
is valid which shows
.
“
”:
and without restriction
. Choose
, hence
by hypothesis,
and
follows.
Since
we claim
.
To (ii): “
”: Without restriction let be
.
For
we have to verify
. So, let be
, hence
since
is symmetric and saturated by hypothesis. Consequently,
is valid.
“
” Conversely, let be
, hence
. Choose
(according to
re- spectively
). Thus
holds, and
follows by hypothesis. But
, hence
is valid. ![]()
Remark 2.3 In this context we point out that each prehypernear space
induces in general the following
ech-closure operators by setting:
(1)
;
(2)
,
where the following inclusion is valid:
implies
. In the symmetric case these two operators coincide, moreover we have
iff
, and finally
defines a symmetric
ech-closure space.
Definition 2.4 A prehypernear space
is called a pseudohypernear space iff
is isoton, i.e.
satisfies (is)
imply
. We denote by PSHN the corresponding full subcate- gory of PHN.
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
is called sected iff
satisfies (
), i.e.
(sec)
and
imply
.
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
is a closed pseudohypernear space. Conversely, we consider for such proposed space
the
ech-closure space
. 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): Now let be
, we have to verify
. Firstly,
implies
, hence
, and
results.
Secondly,
implies
, hence
, and
follows.
To (ii): Now, let be without restriction
.
and
imply
according to
, hence
by hypothesis, and
results.
Conversely,
implies
.
Now, we will show that
.
implies
by hypothesis.
Choose
with
, hence
, since
satisfies (is). But then
![]()
is valid which implies
, 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.
3. Grill-Spaces
Definitions 3.1 A prehypernear space
is called a prehypergrill space iff N satisfies (gri), i.e.
(gri)
and
imply there exists
,
where
, and
is called grill (Choquet [9] ) iff it satisfies
(gri1)
;
(gri2)
iff
or
.
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.
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
is a symmetric saturated prehypernear space, hence additionally it is a prehypergrill space by hypothesis. Conversely,
is grill-determined by supposition. ![]()
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
the following set-convergence space
defined by setting:
iff
for each
. 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)
;
(ii)
.
To (i) “
”
implies
, evidently.
“
”:
implies
, hence there exists
and
. Since
we conclude with
.
To (ii): “
”:
implies the existence of
with
. Consequently,
and
results, hence
is valid.
“
”:
implies that
and
hold. Hence
, and
re- sults. ![]()
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
, where
is satisfying some natural conditions. Then we consider the following pointed prehyergrill space
by setting
and
![]()
if
.
Conversely let be given a pointed saturated prehypergrill space
then we naturally define a point- convergence space
by setting
iff
. 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
for each
. Hence the triple
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 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
the space
, where for each
is defined by setting:
![]()
for each
. Then
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)
;
(ii)
.
To (i): “
”: For
let be
.
implies the existence of
with
, hence
follows. Consequently
is valid, showing that
.
“
”: Since
is grill we get
, hence
![]()
is valid.
To (ii): “
”
implies the existence of
with
by hypothesis, hence
.
“
”
implies
, hence
according to
. ![]()
Remark 3.9
-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
(given by a closure operator t) let
be a
-set with
, then we define a
-proximity
by setting:
iff
for each
and
. Now, it is easy to verify that
is
-compatible, which means the equality
holds by restricting
on
, where
denotes the closure-operator induced by
.
(2) Let being the same hypothesis as in (1). We set
and define a near-
ness relation
by setting:
iff
. Then
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
-proximity space consists of a triple
, where
is set, ![]()
-set and
satisfying the following conditions:
(bp1)
and
(i.e.
is not in relation to
, and analogously this is also holding for
);
(bp2)
iff
or
;
(bp3)
implies
;
(bp4)
and
imply
.
Remark 3.11 Here we point out that b-proximities are in one-to-one correspondence with presupertopologies. In the symmetric case, if
additionally satisfies (sbp), i.e.
(sbp)
and
imply
and moreover
equals
, then symmetric b-proximities coincide with the Čech-proximities mentioned by Deák ([11] ).
Definition 3.12 For
-proximity spaces
,
a bounded function
is called p-map iff
satisfies (p), i.e.
(p)
and
imply
. 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
we consider the triple
, where
![]()
for each
with
. Then
is a conic pseudohypergrill space. Con-
versely let be given such a space
, then we consider the triple
, where ![]()
is defined by setting
iff
for each
and
. Hence,
is a b-pro- ximity space. 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): “
”:
implies
, hence
.
“
”:
implies
, hence
follows.
To (ii): “
”:
and
imply
, hence
, and
results.
“
”
implies
. We will show that
.
implies
by hypothesis, hence
, and
results which concludes the proof. ![]()
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).
4. Bonding in Prehypernear Spaces
A slight modification of the definition for being a prehypergrill space leads us to the following notation.
Definition 4.1 A prehypernear space
is called bonded iff
satisfies (b), i.e.
(b)
and
imply
or
, where
.
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
additionally satisfies
, hence being a pretopology on its underlying set. On the other hand
is
bonded, because
implies
. We suppose that
,
hence there exist
,
such that
. Consequently,
.
But
leads us to a contradiction. ![]()
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
is bonded. Without restriction bet be ![]()
and
, hence
. Since
we ob-
tain
. Thus
or
results, showing that
is satisfying (b). On the other hand let be
and without restriction
. We suppose
, hence there exist
,
with
and
. Consequently
follows. Then we get
by hypothesis. Since M is bounded we have
, or
. By symmetry of
we obtain the statement
or
. Consequently,
or
leads us to a contradiction. ![]()
Remark 4.6 A pseudohypernear space
induces two underlying psb-hull operators by setting for each
:
![]()
whereby the inclusion
is valid for each
. If
is symmetric then the two operators coincide, and moreover we claim the following equalities for each
, i.e.
. Hereby, a function
is called a psb-hull operator, and the triple
is called a psb-hull space iff
satisfies the following conditions:
(bh1)
;
(bh2)
implies
;
(bh3)
imply
.
For psb-hull spaces
,
let
be a bounded function, then
is called b- continuous iff
implies
. We denote by Psb-HULL the corresponding category.
Definition 4.7 Now, we call a conic pseudohypernear space
a pseudohull space iff
satisfies (h), i.e.
(h)
and
imply
. 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
is a psb-hull space. Conversely, for a
psb-hull space
we consider the triple
by setting for each
.
Then
is a pseudohull space. Hence, the above mentioned connections are functoriell. Thus it re- mains to prove that the following two statements are valid, i.e.
(i)
;
(ii)
.
To (i): “
”: For
let be
, hence
. Consequently,
follows, showing that
is valid.
“
”: evident.
To (ii): “
”:
and
imply
by hypothesis, hence
is valid.
“
”:
; we will show that
is valid.
implies
by hypothesis. Choose
with
, hence
according to (hn1). 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
is called connected if
satisfies (cnc), i.e.
(cnc)
and
imply
.
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
additionally satisfying (ad), i.e.
(ad)
. Now, let us call such an operator
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.
5. Hypernear Spaces
As already observed, hypertopologies appear in connexion with certain interior operators studied by Kent and Min ([12] ). Hereby a function
is called a hypertopology on
, and the pair
is called a hypertopological space iff “−” satisfies the following conditions:
(hyt1)
;
(hyt2)
implies
;
(hyt3)
imply
;
(hyt4)
implies
.
For hypertopological spaces
,
let
be a function, then
is called continuous
iff
implies
. By HYTOP we denote the corresponding subcategory of
-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.
Definition 5.1 A pseudohypernear space
is called a hypernear space iff
satisfies (hn), i.e.
(hn)
and
imply
.
We denote by HN the corresponding full subcategory of PSHN. Note, that in this case
is a hypertopology on
.
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
is called surrounded, iff
satisfies (sr), i.e.
(sr)
and
imply there exists
.
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
the following statements are equivalent:
(i)
is pointed;
(ii)
is surrounded.
Proof. The only remaining implication “(ii)
(i)” will be shown now:
and
imply the existence of
with
. Consequently,
,
hence
follows, and
results according to (hn).
Remark 5.7 Now, if we consider a bounded hypertopology, this is a psb-hull operator
on a B-set
, which additionally satisfies (bh4), i.e.
(bh4)
and
imply
, 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
on the B-set
. Conversely, for a bounded hypertopological space
we define the corresponding sourrounded hypernear space
by setting
; and
, otherwise. In the saturated case then we can recover all hy- pertopological spaces. So, in general it is now possible to study those closure operators not only on
, 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
neartopological iff
is satisfying (nt), i.e.
(nt)
and
imply
.
Remark 5.10 We note that each surrounded prehypernear space is neartopological. On the other hand let be given a symmetric bounded hypertopological space
, where in addition
is satisfying (sym), i.e.
(sym)
and
imply
,
then we define the corresponding neartopological hypernear space
by setting:
and
, otherwise. By definition
is automatically symmetric (see Definition 2.1. (iii)). At this point we mention the fact that symmetric hypernear spaces are always dense, which means
is satisfying (d), i.e.
(d)
and
imply
.
This can be seen as follows: Without restriction let be
,
implies
![]()
by hypothesis.
,
hence
follows, and
is valid according to (hn). But then
![]()
results, since
is symmetric. Consequently,
can be deduced according to (hn
). Now, we point out that in some cases
is round which means
additionally satisfies (ron), i.e.
(ron)
implies
. (see also Remark 3.9.(2)).
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
, where
is satisfying (rb), i.e.
(rd)
implies
.
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
and
, then
by definition. Hence there exists
with
. Since
is symmetric we get
, and
results.
To (ii): Without restriction let be
.
implies the existence of
such
that
. Consequently,
, and
results.
In the saturated case then we can recover all symmetric hypertopological spaces.
6. 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 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
is satisfying (hn). Now, let be
,
with
, hence
.
For
we have
. But
is valid. Since “−” is a topological closure oper-
ator we get
, and consequently
results. ![]()
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
a preLEADER space iff
in addition satisfies (bp5), i.e.
(bp5)
and
with
imply
, where
.
By pLESP we denote the corresponding full subcategory of b-PROX.
In the saturated case (if
) LEADER proximity spaces then can be recovered as special objects.
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
satisfies (hn) and
(bp5) respectively.
To (hn):
,
and
imply
. We have to verify that
. So let be
, hence
, and consequently
is valid. The inclusion
holds, because
implies
, hence
, and ![]()
results, showing that
is valid. According to (bp5) we get
, and the proposed inclusion holds.
To (bp5): Conversely, let be
and
with
, we have to verify
.
By hypothesis
is valid.
, since
. Because ![]()
implies
,
leads us to the statement
. According to (hn1) we obtain
, and
results by axiom (hn). Consequently
is valid. ![]()
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
is called a preLODATO space iff
in addition satisfies the following axioms, i.e.
(bp6)
and
imply
or
;
(bp6)
and
imply
;
(bp6)
and
imply
.
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 (bp9), i.e.
(bp9)
implies
.
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
satisfies (hn) and
the nearness axiom.
To (hn): Without restriction let be
,
and
, hence
.
But
, because for
and
we have
, hence
is valid. Consequently
results which shows
. Since
satisfies the
nearness axiom: we get
, and
results. Conversely, let be
for
. We have to verify
. So let be
, our goal is to show
. By hypothesis we get
. But
, since
implies
, hence
follows, and
is valid, which shows
. Consequently,
with
is valid according to (is) of Definition 2.4.
Since M is dense (see Remark 5.10) we get
according to (hn1). But then
follows by (hn), which concludes the proof. ![]()
Corollary 6.11. For a saturated paranear space
the following statements are equivalent:
(i)
is topological nearness;
(ii)
is neartopological.
Proof. evident according to Remark 5.10.
“Relationship between important categories”
![]()
7. Topological Extensions and Their Corresponding Paranear Spaces
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
. For
,
is called a B-clan in
iff it satisfies
(cla1)
;
(cla2)
and
imply
.
Remark 7.2 For a supernear space
and each
with
is a B-clan in
.
Definitions 7.3 A supernear space respectively paranear space
is called superclan space respectively paraclan space iff
satisfies (cla), i.e.
(cla)
and
imply the existence of a B-clan
in
with
.
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
and
, then by (tx1)
with
, hence
follows, which shows
. Conversely, let
, hence
follows, which implies the existence of
with
. But now,
holds, because the presumed extension is sym-
metric. Consequently,
follows, which shows that
according to (tx1). Alltogether, the equality now results. Secondly, it is easy to verify that
fulfills the axioms for being a semihypernear space. Ne is symmetric, since
for
implies the existence of
with
, hence
follows. Now,
implies
,
since
by supposition, hence
symmetric.
is a supernear space,
because
and
imply the existence of
with
.
implies
, hence
,
consequently
follows, which shows that
is a paranearness space. It remains to prove Ne satisfies the axiom (cla). For
let be
, hence
for some
. We set
, consequently
is the desired
-clan in
proving
that
is a paraclan space.
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
SY-TEXT
CLA-PN be defined by:
(a) For a SY-TEXT-object
we put
;
(b) for a TEXT-morphism
we put
.
Then
: SY-TEXT
CLA-PN is a functor.
Proof. We already know that the image of
lies in CLA-PN. Now, let
be a TEXT-morphism; it has to be shown that
preserves B-near collections for each
. Without restric-
tion let
and
, hence we can choose
with
.
Our goal is to verify the existence of
such that
. By hypothesis
we have
, consequently
results, since (f, g) is a TEXT-morphism
by assumption. Now, consider some
, because
, we have
, which results in
. ![]()
8. 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
be a paranear space. We set
is a B-clan in
for some
, and for each
we put:
![]()
where
. (By convention
if
). Then
is a topological closure operator.
Proof. We first note that
, since
for each
. Let
be a subset of
and consider
. Then
implies
, hence
. Now, let be
.
Then,
, which implies
. For arbitrary subsets
we consider
an element
such that
. Then we have
and
. Choose
with
and
with
. Because
we get
. On the
other hand,
implies
. At last, let
be an element
of
and suppose
. Choose
with
. By assumption we have
, hence
. Consequently there exists
with
But this im-
plies
, and
results, which leads us to a contradiction. ![]()
Theorem 8.3 For paranear spaces
,
let
be a hn-map. Define a func- tion
by setting for each
:
![]()
Then the following statements are valid:
(1)
is a continuous map from
to
;
(2) The composites
and
coincide, where
denotes the function which assigns the
-clan
to each
.
Proof. First, let
be a
-clan in
. We will show that
is a
-clan in
. It is easy to verify that
, which satisfies (cla
) in Definition 7.1. In order to establish (cla1) we observe that
by hypothesis. We will now verify that
![]()
(Note, that
is a hn-map by assumption.) For any
we have
, hence
.
Since
and
we get
, and all together we con-
clude that
defines a
-clan in
. Consequently,
results.
To (1): Let
,
and suppose
. Then
,
hence
for some
, which means
. Since
, we get
for some
. Consequently
results, which leads us to a contradiction, because
is valid.
To (2): Let
be an element of
. We will prove the validity of
. To this end, let
. Then,
, hence
, and consequently
.
Thus
, proving the inclusion
. Since
is maximal with re-
spect to
and moreover
, since by hypothesis ![]()
is a hn-map, we obtain the desired equality. ![]()
Theorem 8.4 We obtain a functor
: CLA-PN toSY-TEXT by setting:
(a)
for any paraclan space
with
and
;
(b)
for any hn-map
.
Proof. With respect to Corollary 6.2 it is straight forward to verify that
is a topological closure operator on
. We also have the topological closure operator
on
. Therefore we obtain topological spaces with
-set
, and
is a continuous map, which can be seen as follows: Let
for
, we have to verify that
.
implies
and
for some
by supposition. Consequently,
follows which shows
. To establish (tx1), let
be a subset of
and suppose
. Then we get
, hence
which means that
. Conversely, let
be an element of
. Then by definition we have
, and consequently
. This implies
, which means
. To establish (tx2), let be
and suppose
. By definition we get
, so that there exists a set
with
. But
follows. Since
for some
, we get
, hence
, because ![]()
satisfies (cla2). But this is a contradiction, and thus
is valid. In showing
is
symmetric let x be an element of
such that
. We have to prove
. By
hypothesis we have
and moreover
for some
. Since ![]()
and N is symmetric we get
with
, hence
follows according to (hn1).
But
is maximal with respect to
, which means that
coincides with
. By
hypothesis
is a hn-map, so that
is continuous and bounded with respect to the given
-sets and corresponding closure operators. It remains to prove that the following diagram commutes:
![]()
To this end let
be an element of
. We must show
.
“
”
implies
, which means
, hence
.
Since
is continuous we have
and
follows.
“
”
implies
, hence
follows and consequently
. Thus,
, which means
.
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
and let
be closed in
with
. Then
, hence
. We can find some
such that
. Now, for each
we have
, which implies
, and therefore we conclude
. On the other hand since
, we have
, hence
, which put an end of this. ![]()
Theorem 8.5 Let
: SY-TEXT
CLA-PN and
: CLA-PN
SY-TEXT be the above defined func- tors. For each object
of CLA-PN let
denote the identity map
![]()
Then
is natural equivalence from
to the identity functor
, i.e.
![]()
is a hn-map in both directions for each object
, and the following diagram commutes for each hn-map
:
![]()
Proof. The commutativity of the diagram is obvious, because of
. It remains to prove that
![]()
is a hn-map in both directions. To fix the notation let
be such that
.
It suffices to show that for each
we have
. To this end assume
, then there exists
such that
, hence
. We
get
and
for some
. Since N is symmetric, we obtain
with
, hence
. But
implies
, hence
with
.
Now
results, which shows
. Conversely, let
. Since
is a paraclan space we can choose a
-clan in
such that
. In order to show
we need to verify
(1)
;
(2)
implies
.
To (1): By definition of
it suffices to establish
. So let
be an element of
, hence
follows which implies
. But
is B-clan in N, consequently we get
.
To (2): Let A be an element of
and D be an element of
, hence
. Since
by hypothesis, we get
and analogously as above we infer
, which concludes the proof. ![]()
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
such that the B-near collections are characterized by the fact that the closures of its members meet in
. Therefore this theorem generalize in one direction the Bentley- characterization of bunch-determined nearness spaces, in another the description of Doitchinov’s b-superto- pologies by compactly determined topological extensions and moreover the analogous existing correspondence respected to LODATO spaces involving the famous theorem of LODATO.
Corollary 8.7 If
is separated that means
satisfies (sep), i.e.
(sep)
and
imply
, then
is injective. Conversely, for a
extension
, where
is a topological embedding and
a
-space, then
is separated.