DOI:https://doi.org/10.33039/ami.2022.09.002 URL:https://ami.uni-eszterhazy.hu
Two illustrating examples for comparison of uniform and proximal spaces using
relators
Gergely Pataki
Department of Analysis, Institute of Mathematics,
Budapest University of Technology and Economics, Budapest, Hungary pataki@math.bme.hu
Abstract. We introduce (generalized) proximities in the same way as (gen- eralized) uniformities in paper of Weil. We prove the equivalence of our new definitions with classical ones.
Using these analog definitions, we compare the properties of (generalized) proximities and (generalized) uniformities. The main parts of this paper are examples of an (𝑋,ℛ) relator space such that ℛ# is uniformly (and proximally) transitive, but neither ℛ nor ℛΦ is proximally (or uniformly) transitive.
For this, we summarize the essential properties of relators, using their theory from earlier works of Á. Száz.
Keywords:(generalized) uniformities, (generalized) proximities, relators AMS Subject Classification:54E05, 54E15, 54A05, 54G15, 54G20
1. Introduction
At the beginning of the 20th century some mathematicians tried to define abstract topological structures. The most relevant results are Poincaré 1895, Fréchet 1906, Hausdorff 1914, and Kuratowski 1922.
Uniform spaces in terms of relations were introduced by Weil in 1937 [13].
Proximities were first investigated by Riesz in 1909 [9], Effremovič, and Smirnov in 1952 [2] and [10].
After the works of Davis, Pervin, and Nakano [1], [8], and [4] in 1987, Száz [11]
introduced the notion of relator and relator space in the following way.
Definition 1.1. A nonvoid family ℛof relations on a nonvoid set 𝑋 is called a relator on𝑋, and the ordered pair (𝑋,ℛ) is called a relator space.
In the last decades, a few authors investigated the interpretation of well-known topological properties in terms of relators. In 2021, Pataki introduced the gen- eral definition for (generalized) uniformities and (generalized) proximities in [5], moreover quasi-uniformities in [6].
For more details, see, for instance [5–7, 12], but for the readers’ convenience, we summarize the necessary notions and notations.
Remark 1.2. With the usual notations,ℛis a relator on𝑋 means that 𝑋 ̸=∅, ∅ ̸=ℛ ⊂Exp(𝑋2),
where Exp(𝑋) is the power set of𝑋, and𝑋2=𝑋×𝑋.
If𝑅is a relation on𝑋,𝑥∈𝑋, and 𝐴⊂𝑋, then the sets 𝑅(𝑥) ={𝑦∈𝑋 : (𝑥, 𝑦)∈𝑅}, and 𝑅[𝐴] = ⋃︁
𝑥∈𝐴
𝑅(𝑥)
are called the images of 𝑥and𝐴under 𝑅, respectively.
2. Preliminary concepts
Definition 2.1. If𝑅and 𝑆 are relations on𝑋, then the composition of𝑅and 𝑆 can be defined, such that (𝑅∘𝑆)(𝑥) =𝑅[𝑆(𝑥)] for all𝑥∈𝑋.
Moreover, let𝑅−1 = {(𝑦, 𝑥) : (𝑥, 𝑦) ∈𝑅}, 𝑅0 = Δ𝑋 ={(𝑥, 𝑥) : 𝑥∈ 𝑋} and 𝑅𝑛=𝑅∘𝑅𝑛−1, for all𝑛= 1,2, . . .. Finally, we say that𝑅 is
• reflexive if𝑅0⊂𝑅, • symmetric if𝑅−1⊂𝑅, • transitive if 𝑅2⊂𝑅.
Lemma 2.2. If 𝑅is a relation on 𝑋, and𝐴, 𝐵⊂𝑋, then 𝑅[𝐴]⊂𝐵 ⇐⇒ 𝑅−1[𝑋∖𝐵]⊂𝑋∖𝐴.
Definition 2.3. Ifℛis a relator on𝑋, then the relators ℛ*={𝑆⊂𝑋2:∃𝑅∈ ℛ:𝑅⊂𝑆}, and
ℛ#={𝑆⊂𝑋2:∀𝐴⊂𝑋 :∃𝑅∈ ℛ:𝑅[𝐴]⊂𝑆[𝐴]},
are called the uniform and the proximal refinements of ℛ, respectively. For more details, see [7].
Moreover, for all𝑛=−1,0,1,2, . . ., we define ℛ𝑛={𝑅𝑛:𝑅∈ ℛ}.
Remark 2.4. * and # are really refinements as we defined in [7]. That is, they are self-increasing in the sense that
ℛ ⊂ 𝒮* ⇐⇒ ℛ*⊂ 𝒮* and ℛ ⊂ 𝒮# ⇐⇒ ℛ#⊂ 𝒮#,
or equivalently, they are expansive, increasing, and idempotent, in the sense that ℛ ⊂ ℛ*, ℛ ⊂ 𝒮 =⇒ ℛ*⊂ 𝒮*, ℛ**=ℛ*
and
ℛ ⊂ ℛ#, ℛ ⊂ 𝒮 =⇒ ℛ#⊂ 𝒮#, ℛ##=ℛ#, for allℛand𝒮 relators on𝑋.
Moreover, # is*-dominating, *-invariant,*-absorbing, and*-compatible, that is
ℛ*⊂ ℛ#, ℛ#=ℛ#*, ℛ#=ℛ*#, ℛ#*=ℛ*#.
For all𝑛=−1,0,1,2, . . . the mapping ℛ ↦→ ℛ𝑛 of relators on𝑋 is increasing.
Finally,* and # are inversion-compatible, that is, for allℛrelators on𝑋 ℛ*−1=ℛ−1* and ℛ#−1=ℛ−1#.
And we have that for all relators on𝑋 ℛ2*=ℛ*2*.
The following example shows that the analog assertion is not true for #.
Example 2.5. Let𝑋 ={1,2,3,4}, and
ℛ={Δ𝑋∪ {(1,2),(4,2),(2,1),(2,4)},Δ𝑋∪ {(1,3),(4,3),(3,1),(3,4)}}
is an elementwise reflexive and symmetric relator on𝑋. Now, ℛ#2 ̸⊂ ℛ2#, since 𝑅=𝑋2∖ {(1,4),(4,1)} ∈ ℛ#2 however𝑅 /∈ ℛ2#.
Note, thatℛ=
{︃□■□■□□■□■■□■■■□□,□□■■■□■■□■□□■□■□
}︃
, and𝑅=□■■■■■■■■■■■■■■□.
Definition 2.6. Let ℛ be a relator on 𝑋, and □ ∈ {*,#} is a refinement for relators on𝑋. We define the followings.
• ℛis□-reflexive, ifℛ ⊂ ℛ0□;
• ℛis□-symmetric, ifℛ ⊂ ℛ−1□;
• ℛis□-transitive, ifℛ ⊂ ℛ2□;
• ℛis□-fine, ifℛ=ℛ□.
For instance, we say that ℛ is uniformly symmetric or proximally transitive instead of*-symmetric or #-transitive.
Following Weil, we say that the relator ℛ on 𝑋 is a generalized uniformity/
generalized proximity on𝑋, and the ordered pair (𝑋,ℛ) is a generalized uniform space/generalized proximal space if it is
• uniformly/proximally reflexive;
• uniformly/proximally symmetric;
• uniformly/proximally transitive;
• uniformly/proximally fine.
Following the notations of [3], in [6] we introduced quasi-uniformities, and now, we define generalized quasi-uniformities/generalized quasi-proximities.
We say that the relatorℛon 𝑋 is a generalized quasi-uniformity/generalized quasi-proximity on𝑋, and the ordered pair (𝑋,ℛ) is a generalized quasi-uniform space/generalized quasi-proximal space if it is
• uniformly/proximally reflexive;
• uniformly/proximally transitive;
• uniformly/proximally fine.
Definition 2.7. Let𝒜be a family of sets, or equivalently 𝒜 ⊂Exp(𝑋) for some set 𝑋. We call
Φ(𝒜) ={︁⋂︁
ℬ:∅ ̸=ℬ ⊂ 𝒜, and ℬis finite}︁
the filtered family of sets generated by𝒜.
Moreover, we say that𝒜is filtered if Φ(𝒜) =𝒜.
Remark 2.8. Since Φ is a refinement for relators on 𝑋, we write ℛΦ instead of Φ(ℛ), ifℛis a relator on𝑋. Note thatℛis filtered iffℛΦ⊂ ℛ.
Moreover, note that Φ is an inversion-compatible refinement for relators on𝑋.
That is ifℛis a relator on𝑋, thenℛ−1Φ=ℛΦ−1. Finally, ifℛis finite, thenℛΦ*={⋂︀
ℛ}*.
Lemma 2.9. Ifℛis a relator on𝑋, thenℛ*Φ=ℛΦ*,ℛ2Φ⊂ ℛΦ2*, andℛ#2#⊂ ℛΦ2#.
Definition 2.10. If □is a refinement for relators on𝑋, then we say that theℛ relator on𝑋is□-filtered if there exists an𝒮relator on𝑋such that𝒮Φ⊂ 𝒮□=ℛ□. We use the uniformly filtered and proximally filtered notions instead of*-filtered and #-filtered.
Definition 2.11. If the generalized uniformity/generalized proximity (generalized quasi-uniformity/generalized quasi-proximity)ℛon𝑋 is also
• uniformly/proximally filtered,
then we say that ℛis a uniformity/proximity (quasi-uniformity/quasi-proximity) on𝑋, and (𝑋,ℛ) is a uniform space/proximal space (quasi-uniform space/quasi- proximal space).
3. Main example
Example 3.1. For all𝑖∈Nlet𝑋𝑖={3𝑖−2,3𝑖−1,3𝑖}and𝜋𝑖,0,𝜋𝑖,1, . . . ,𝜋𝑖,5 are the all bijections of{1,2,3}to𝑋𝑖 such that𝜋𝑖,0is the only increasing one of them.
Moreover, let
𝑅𝑖,𝑘={(𝜋𝑖,𝑘(1), 𝜋𝑖,𝑘(2)),(𝜋𝑖,𝑘(2), 𝜋𝑖,𝑘(3))} ∪Δ𝑋𝑖
for all𝑖∈Nand𝑘∈ {0, . . . ,5}.
Furthermore, for all 𝑛 ∈ N let𝑆𝑛 = ⋃︀
𝑖∈N𝑅𝑖,𝜈𝑖, where 𝜈𝑖 is the 𝑖th digits of (𝑛−1) in a positional base 6 numeral system, that is𝑛−1 =∑︀
𝑖∈N𝜈𝑖·6𝑖−1. The following figure shows the main part of the graph of𝑆44791, where 44790 =
∑︀5
𝑖=0𝑖·6𝑖 and
𝜋2,1(1) = 4, 𝜋2,1(2) = 6, 𝜋3,2(1) = 8, 𝜋3,2(2) = 7, 𝜋4,3(1) = 11, 𝜋4,3(2) = 12, 𝜋5,4(1) = 15, 𝜋5,4(2) = 13, 𝜋6,5(1) = 18, 𝜋6,5(2) = 17.
𝑋1 𝑋1 𝑅1,0
𝑋2 𝑋2 𝑅2,1
𝑋3
𝑋3 𝑅3,2
𝑋4
𝑋4 𝑅4,3
𝑋5
𝑋5 𝑅5,4
𝑋6
𝑋6 𝑅6,5
𝑋7
𝑋7 𝑅7,0
𝑋8
𝑋8 𝑅8,0
𝑋9
𝑋9 𝑅9,0
𝑋10
𝑋10 𝑅10,0
𝑋11
𝑋11 𝑅11,0
𝑋12
𝑋12 𝑅12,0
𝑋13
𝑋13 𝑅13,0
Finally, let𝑋 =N=⋃︀
𝑖∈N𝑋𝑖andℛ={𝑆𝑛 :𝑛∈N}is an elementwise reflexive relator on 𝑋. Then ℛ# = {Δ𝑋}* is a uniformly transitive relator on 𝑋, but neitherℛnorℛΦis proximally transitive. Namely,
𝑆1= Δ𝑋∪ ⋃︁
𝑖∈𝑋∖3𝑋
{(𝑖, 𝑖+ 1)} ∈ ℛ ⊂ ℛΦ,
but we show that if𝐴={3𝑖−2 :𝑖∈𝑋}, then𝑄2[𝐴]̸⊂𝑆1[𝐴] =𝐴∪(𝐴+ 1) for all 𝑄∈ ℛΦ, that is𝑆1∈ ℛ/ Φ2# and hence𝑆1∈ ℛ/ 2#.
To prove this let ∅ ̸=𝒮 ⊂ ℛ finite, such that 𝑄= ⋂︀
𝒮. Then there exists a greatest𝑚∈𝑋 such that𝑆𝑚∈ 𝒮. If𝑖is large enough then𝜈𝑖= 0 for all elements of𝒮, therefore𝑅𝑖,0⊂⋂︀
𝒮 =𝑄, and then 3𝑖∈𝑅2𝑖,0(3𝑖−2)⊂𝑄2(3𝑖−2)⊂𝑄2[𝐴].
We can change the above example to be symmetric.
Example 3.2. Namely, for all𝑖∈Nlet𝑋𝑖={5𝑖−4,5𝑖−3,5𝑖−2,5𝑖−1,5𝑖} and 𝜋𝑖,0,𝜋𝑖,1, . . . ,𝜋𝑖,119are the all bijections of{1,2,3,4,5}to𝑋𝑖such that𝜋𝑖,0is the only increasing one of them. Moreover, let
𝑅𝑖,𝑘={(𝜋𝑖,𝑘(1), 𝜋𝑖,𝑘(2)),(𝜋𝑖,𝑘(2), 𝜋𝑖,𝑘(1)),(𝜋𝑖,𝑘(2), 𝜋𝑖,𝑘(3)),(𝜋𝑖,𝑘(3), 𝜋𝑖,𝑘(2))}∪Δ𝑋𝑖
for all𝑖∈Nand𝑘∈ {0, . . . ,119}. Furthermore for all 𝑛∈Nlet𝑆𝑛 =⋃︀
𝑖∈N𝑅𝑖,𝜈𝑖, where 𝜈𝑖 is the𝑖th digits of (𝑛−1) in a positional base 120 numeral system, that is 𝑛−1 = ∑︀
𝑖∈N𝜈𝑖·120𝑖−1. Now ℛ={𝑆𝑛 : 𝑛 ∈N} is an elementwise reflexive, elementwise symmetric relator on 𝑋=N=⋃︀
𝑖∈N𝑋𝑖 with the same properties.
4. A finite example
Lemma 4.1. Let ℛbe a finite relator on𝑋. Now,ℛΦ is proximally transitive iff the relation ⋂︀
ℛis transitive.
Proof. Ifℛis finite, then Remark2.4and2.8yield that ℛΦ2*=ℛΦ*2*={︁⋂︁
ℛ}︁*2*
={︁⋂︁
ℛ}︁2*
therefore
ℛΦ2#=ℛΦ2*#={︁⋂︁
ℛ}︁2*#
={︁⋂︁
ℛ}︁2#
={︁⋂︁
ℛ}︁2*
=ℛΦ2*. By using the self-increasingness of* we have that
(︁⋂︁ℛ)︁2
⊂⋂︁
ℛ ⇐⇒ {︁⋂︁
ℛ}︁
⊂{︁⋂︁
ℛ}︁2*
⇐⇒ {︁⋂︁
ℛ}︁*
⊂{︁⋂︁
ℛ}︁2*
⇐⇒
⇐⇒ ℛΦ*⊂ ℛΦ2* ⇐⇒ ℛΦ⊂ ℛΦ2* ⇐⇒ ℛΦ⊂ ℛΦ2#.
Theorem 4.2. If ℛ is a finite relator on 𝑋, then uniform transitivity of ℛ# implies proximal transitivity ofℛΦ.
Proof. Assume to the contrary that ℛΦ is not proximally transitive that is, by using the above Lemma, there exists an (𝑥, 𝑧)∈(⋂︀
ℛ)2∖⋂︀
ℛ. It means that there exist 𝑥, 𝑦, 𝑧∈𝑋 such that
(a) 𝑦∈(⋂︀
ℛ)(𝑥);
(b) 𝑧∈(⋂︀
ℛ)(𝑦);
(c) 𝑧 /∈(⋂︀
ℛ)(𝑥).
For all 𝑆 ∈ ℛ# there exists an 𝑅 ∈ ℛ such that 𝑅(𝑥) ⊂ 𝑆(𝑥). For such an 𝑅∈ ℛ, by using(a), it is easy to see, that𝑦∈(⋂︀
ℛ)(𝑥)⊂𝑅(𝑥).
In a similar way(b)implies that 𝑧∈𝑆(𝑦) for all𝑆 ∈ ℛ#and (c)implies that 𝑧 /∈𝑅(𝑥) for some𝑅∈ ℛ.
In summary we have that
∃𝑅∈ ℛ:∀𝑆∈ ℛ#:𝑆2̸⊂𝑅
that is ℛ ̸⊂ ℛ#2*. This is a contradiction, because uniform transitivity of ℛ# means thatℛ ⊂ ℛ#⊂ ℛ#2*.
The following examples show that uniform transitivity ofℛ# does not imply proximal transitivity ofℛeven if the space𝑋 is finite.
The appendix shows the graphs of all 24𝑅𝜋 elements of the ℛ relator in the following example.
Moreover, we can see the graphs of 𝑅𝜋2 and 𝑆𝑛 elements of 𝒮, which is the smallest relator on𝑋 such that 𝒮*=ℛ#.
Finally, in appendix we can also see𝑆𝑘2∈ 𝒮2 examples for all𝑆𝑛 ∈ 𝒮 such that 𝑆𝑘2⊂𝑆𝑛.
Example 4.3. Let𝑋 ={1,2,3,4} and
𝑅𝜋 ={(𝜋(1), 𝜋(1)),(𝜋(1), 𝜋(2)),(𝜋(2), 𝜋(2)),(𝜋(2), 𝜋(3))}
for all𝜋permutation of𝑋. Moreover, let ℛ={𝑅𝜋:𝜋is a permutation of𝑋}. At first, we investigateℛ#. For this end, let 𝑄∈ ℛ# be arbitrary. By defi- nition, there exists a 𝜋permutation of𝑋, such that𝜋[{1,2,3}] =𝑅𝜋[𝑋]⊂𝑄[𝑋], that is 𝑄[𝑋] has at least 3 elements. Moreover, if 𝐴 ⊂𝑋 has 3 elements, then 𝜋(1)∈𝐴or𝜋(2)∈𝐴when𝜋is a permutation of𝑋 such that𝑅𝜋[𝐴]⊂𝑄[𝐴], that is𝑄[𝐴] has at least 2 elements.
In summary, a𝑄relation on𝑋 is an element ofℛ# iff it satisfies at least one of the following conditions.
• There exists 𝛼 and 𝛽 permutations of 𝑋 such that 𝛽(𝑖) ∈ 𝑄(𝛼(𝑖)) for all 𝑖∈ {1,2,3}.
• There exists 𝛼and 𝛽 permutations of 𝑋 such that{𝛽(𝑖)}⊊𝑄(𝛼(𝑖)) for all 𝑖∈ {1,2} and𝛽(3)∈𝑄(𝛼(1)).
Now, we show thatℛ# is uniformly transitive. For this end, let 𝑄 ∈ ℛ# be arbitrary. This is possible in the following ways.
𝑄⊂Δ𝑋:
(1) There exists an𝛼permutation of𝑋 such that (𝛼(1), 𝛼(1)), (𝛼(2), 𝛼(2))∈𝑄.
In this case
𝑃 ={(𝛼(1), 𝛼(1)),(𝛼(2), 𝛼(2)),(𝛼(3), 𝛼(4))} ∈ ℛ# and𝑃2⊂𝑄.
𝑄⊂𝑋2∖Δ𝑋:
(2) There exists an𝛼permutation of𝑋 such that (𝛼(1), 𝛼(2)), (𝛼(1), 𝛼(3))∈𝑄.
In this case
𝑃={(𝛼(1), 𝛼(2)),(𝛼(1), 𝛼(4)),(𝛼(4), 𝛼(2)),(𝛼(4), 𝛼(3))} ∈ ℛ# and𝑃2⊂𝑄.
Or(3) there exists an𝛼permutation of𝑋such that (𝛼(1), 𝛼(2)), (𝛼(3), 𝛼(4))∈𝑄.
In this case
𝑃 ={(𝛼(1), 𝛼(3)),(𝛼(2), 𝛼(4)),(𝛼(3), 𝛼(2))} ∈ ℛ# and𝑃2⊂𝑄.
Or(4) there exists an 𝛼 permutation of 𝑋 such that (𝛼(1), 𝛼(2)), (𝛼(2), 𝛼(3)), (𝛼(3), 𝛼(1))∈𝑄. In this case
𝑃 ={(𝛼(1), 𝛼(3)),(𝛼(2), 𝛼(1)),(𝛼(3), 𝛼(2))} ∈ ℛ# and𝑃2⊂𝑄.
𝑄̸⊂Δ𝑋 and𝑄̸⊂𝑋2∖Δ𝑋:
(5) There exists an𝛼permutation of𝑋 such that (𝛼(1), 𝛼(1)), (𝛼(1), 𝛼(2))∈𝑄.
In this case
𝑃={(𝛼(1), 𝛼(1)),(𝛼(1), 𝛼(3)),(𝛼(2), 𝛼(4)),(𝛼(3), 𝛼(4))} ∈ ℛ# and𝑃2⊂𝑄.
Or(6) there exists an𝛼permutation of𝑋such that (𝛼(1), 𝛼(1)), (𝛼(2), 𝛼(3))∈𝑄.
In this case
𝑃 ={(𝛼(1), 𝛼(1)),(𝛼(2), 𝛼(4)),(𝛼(4), 𝛼(3))} ∈ ℛ# and𝑃2⊂𝑄.
Finally, we show thatℛis not proximally transitive. For this, note that for any 𝜋permutation of𝑋
𝑅2𝜋[{1,3,4}]̸⊂ {1,2}=𝑅Δ𝑋[{1,3,4}] Note that𝑅Δ𝑋 =□□□□□■□□■■□□■□□□.
Unfortunately, the above relator is neither reflexive nor symmetric. We can make a reflexive one.
Example 4.4. Let𝑋 ={1,2,3,4},𝑌 ={1,2,3,4,5,6,7,8}and 𝑅𝜋 ={(𝜋(1), 𝜋(1)),(𝜋(1), 𝜋(2)),(𝜋(2), 𝜋(2)),(𝜋(2), 𝜋(3))} and
𝑆𝜋 =𝑅𝜋∪ {(𝜋(1) + 4, 𝜋(1)),(𝜋(1) + 4, 𝜋(2)),(𝜋(2) + 4, 𝜋(2)),(𝜋(2) + 4, 𝜋(3))} ∪Δ𝑌
for all𝜋permutation of𝑋.
Moreover, let𝒮 ={𝑆𝜋 : 𝜋is a permutation of𝑋} be an elementwise reflexive relator on 𝑌. Now 𝒮# is uniformly transitive, but𝒮 is not proximally transitive.
Note that
𝑆Δ𝑋 =
□□□□□□□■
□□□□□□■□
□□□□□■□□
□□□□■□□□
□□□■□□□□
□■■□□■□□
■■□□■■□□
■□□□■□□□
=
□□□□□□□□
□□□□□□□□
□□□■□□■□
□■□□■□□□
□□□■□■■□
■■□□■□□□
□□□□□■□□
■■□□■□□□
=
⏞ ⏟ ∅ ⏞ ⏟ ∅
□□□□□□□□
□□□□□□□□
□□□□□□□□
□□□□□□□□
□□□□□■□□
■■□□■□□□
□□□□□■□□
■■□□■□□□
⏟ ⏞
𝑅Δ𝑋 ⏟ ⏞
𝑅Δ𝑋
∪
□□□□□□□□
□□□□□□□□
□□□■□□■□
□■□□■□□□
□□□■□□■□
□■□□■□□□
□□□□□□□□
□□□□□□□□
⏟ ⏞
Δ𝑌
.
We do not know whether if a symmetric finite ℛ relator is not proximally transitive, then can ℛ# be uniformly transitive.
References
[1] A. S. Davis:Indexed Systems of Neighborhoods for General Topological Spaces, The Amer- ican Mathematical Monthly 68.9 (1961), pp. 886–893,issn: 00029890, 19300972,doi:http s://doi.org/10.2307/2311686,url:http://www.jstor.org/stable/2311686(visited on 09/13/2022).
[2] V. A. Efremovič: The geometry of proximity. I. Matematicheskii Sbornik 73.1 (1952), pp. 189–200.
[3] L. Nachbin:Topology and order, Princetown: D. Van Nostrand, 1965.
[4] H. Nakano,K. Nakano:Connector theory, Pacific J. Math. 56.1 (1975), pp. 195–213.
[5] G. Pataki:Investigation of proximal spaces using relators, Axioms 10.3 (2021), pp. 1–10, doi:https://doi.org/10.3390/axioms10030143.
[6] G. Pataki:Investigation of topological spaces using relators, Applied General Topology 23.1 (2022), pp. 45–54,doi:https://doi.org/10.4995/agt.2022.16128.
[7] G. Pataki,Á. Száz:A unified treatment of well-chainedness and connectedness properties, Acta Math. Acad. Paedagog. Nyházi. (N.S.) 19 (2003), pp. 101–166.
[8] W. J. Pervin:Quasi-uniformization of topological spaces, Math. Ann. 147 (1962), pp. 316–
317,doi:https://doi.org/10.1007/BF01440953.
[9] F. Riesz:Stetigkeit und abstrakte Mengenlehre, Rom. 4. Math. Kongr. 2 (1909), pp. 18–24.
[10] Y. M. Smirnov:On proximity spaces, Matematicheskii Sbornik 73.3 (1952), pp. 543–574.
[11] Á. Száz:Basic tools and mild continuities in relator spaces, Acta Math. Hungar. 50.3-4 (1987), pp. 177–201,doi:https://doi.org/10.1007/bf01903935.
[12] Á. Száz:Minimal structures, generalized topologies, and ascending systems should not be studied without generalized uniformities, Filomat 21.1 (2007), pp. 87–97,doi:https://doi .org/10.2298/FIL0701087S.
[13] A. Weil:Sur les espaces a structure uniforme at sur la topologie générale, Paris: Herman and Cie, 1937,doi:https://doi.org/10.1007/978-1-4757-1705-1_27.
Appendix
List of elements ofℛof Example4.3, with using the notation𝜋(1)𝜋(2)𝜋(3)𝜋(4) for the𝜋permutation of𝑋.
𝑅1234=□□□□□■□□■■□□■□□□, 𝑅1243=□■□□□□□□■■□□■□□□, 𝑅1324=□□□□■□■□□□■□■□□□, 𝑅1342=□□■□■□■□□□□□■□□□, 𝑅1423=■□□■□□□□□□□■■□□□, 𝑅1432=■□□■□□□■□□□□■□□□, 𝑅2134=□□□□■□□□□■□□■■□□, 𝑅2143=■□□□□□□□□■□□■■□□, 𝑅2314=□□□□□■■□□■□□□□■□, 𝑅2341=□□■□□■■□□■□□□□□□, 𝑅2413=□■□■□□□□□■□□□□□■, 𝑅2431=□■□■□□□■□■□□□□□□, 𝑅3124=□□□□□□■□■□□□■□■□, 𝑅3142=■□□□□□■□□□□□■□■□, 𝑅3214=□□□□□□■□□■■□□■□□, 𝑅3241=□■□□□□■□□■■□□□□□, 𝑅3412=□□■■□□■□□□□□□□□■, 𝑅3421=□□■■□□■□□□□■□□□□, 𝑅4123=□□□■□□□□■□□□■□□■, 𝑅4132=□□□■■□□□□□□□■□□■, 𝑅4213=□□□■□□□□□■□■□■□□, 𝑅4231=□□□■□■□□□■□■□□□□, 𝑅4312=□□□■□□■■□□□□□□■□, 𝑅4321=□□□■□□■■□□■□□□□□. List of elements ofℛ2of Example4.3, with using the notation𝜋(1)𝜋(2)𝜋(3)𝜋(4) for the𝜋permutation of𝑋.
𝑅21234=□□□□■■□□■■□□■□□□, 𝑅21243=■■□□□□□□■■□□■□□□, 𝑅21324=□□□□■□■□■□■□■□□□, 𝑅21342=■□■□■□■□□□□□■□□□, 𝑅21423=■□□■□□□□■□□■■□□□, 𝑅21432=■□□■■□□■□□□□■□□□, 𝑅22134=□□□□■■□□□■□□■■□□, 𝑅22143=■■□□□□□□□■□□■■□□, 𝑅22314=□□□□□■■□□■□□□■■□, 𝑅22341=□■■□□■■□□■□□□□□□, 𝑅22413=□■□■□□□□□■□□□■□■, 𝑅22431=□■□■□■□■□■□□□□□□, 𝑅23124=□□□□□□■□■□■□■□■□, 𝑅23142=■□■□□□■□□□□□■□■□, 𝑅23214=□□□□□□■□□■■□□■■□, 𝑅23241=□■■□□□■□□■■□□□□□, 𝑅23412=□□■■□□■□□□□□□□■■, 𝑅23421=□□■■□□■□□□■■□□□□, 𝑅24123=□□□■□□□□■□□■■□□■, 𝑅24132=□□□■■□□■□□□□■□□■, 𝑅24213=□□□■□□□□□■□■□■□■, 𝑅24231=□□□■□■□■□■□■□□□□, 𝑅24312=□□□■□□■■□□□□□□■■, 𝑅24321=□□□■□□■■□□■■□□□□.
In Example4.3,𝒮 is the smallest relator on𝑋, such that𝒮*=ℛ#
We draw the graph of elements of 𝒮 with marked the required member of𝒮, which implies the 𝒮 ⊂ 𝒮2* inclusion.
𝑆272 =□□□□■□□□□□□□■□□□⊂𝑆1=□□□□■□□□□■□□■■□□, 𝑆902 =□□□□□□□□□■□□□■□□⊂𝑆2=□□□□■□□□■■□□□■□□,
𝑆292 =■□□□□□□□□□□□■□□□⊂𝑆3=■□□□□□□□□■□□■■□□, 𝑆822 =■□□□□□□□□■□□□□□□⊂𝑆4=■□□□□□□□■■□□□■□□,
𝑆282 =■□□□■□□□□□□□□□□□⊂𝑆5=■□□□■□□□□■□□□■□□, 𝑆952 =□□□□□□□□■□□□■□□□⊂𝑆6=□□□□□■□□■□□□■■□□,
𝑆1912 =□□□□□■□□□□□□□■□□⊂𝑆7=□□□□■■□□■□□□□■□□, 𝑆292 =■□□□□□□□□□□□■□□□⊂𝑆8=■□□□□■□□□□□□■■□□,
𝑆802 =■□□□□■□□□□□□□□□□⊂𝑆9=■□□□□■□□■□□□□■□□, 𝑆282 =■□□□■□□□□□□□□□□□⊂𝑆10=■□□□■■□□□□□□□■□□,
𝑆152 =□□□□□■□□□■□□□□□□⊂𝑆11=□□□□□■□□■■□□■□□□, 𝑆72=□□□□■■□□□■□□■□□□⊂𝑆12=□□□□■■□□□■□□■□□□,
𝑆132 =■□□□□■□□□■□□■□□□⊂𝑆13=■□□□□■□□□■□□■□□□, 𝑆152 =□□□□□■□□□■□□□□□□⊂𝑆14=■□□□□■□□■■□□□□□□,
𝑆152 =□□□□□■□□□■□□□□□□⊂𝑆15=■□□□■■□□□■□□□□□□, 𝑆832 =□■□□□□□□□□□□■□□□⊂𝑆16=□■□□□□□□■□□□■■□□,
𝑆272 =□□□□■□□□□□□□■□□□⊂𝑆17=□■□□■□□□□□□□■■□□, 𝑆842 =□■□□■□□□□□□□□□□□⊂𝑆18=□■□□■□□□■□□□□■□□,
𝑆912 =□■□□□□□□□□□□□■□□⊂𝑆19=■■□□□□□□■□□□□■□□, 𝑆282 =■□□□■□□□□□□□□□□□⊂𝑆20=■■□□■□□□□□□□□■□□,
𝑆252 =□■□□□□□□□■□□□□□□⊂𝑆21=□■□□□□□□■■□□■□□□, 𝑆92=□■□□■□□□□■□□■□□□⊂𝑆22=□■□□■□□□□■□□■□□□,
𝑆252 =□■□□□□□□□■□□□□□□⊂𝑆23=□■□□■□□□■■□□□□□□, 𝑆192 =■■□□□□□□□■□□■□□□⊂𝑆24=■■□□□□□□□■□□■□□□,
𝑆252 =□■□□□□□□□■□□□□□□⊂𝑆25=■■□□■□□□□■□□□□□□, 𝑆102 =□■□□□■□□□□□□□□□□⊂𝑆26=□■□□□■□□■□□□■□□□,
𝑆102 =□■□□□■□□□□□□□□□□⊂𝑆27=□■□□■■□□□□□□■□□□, 𝑆102 =□■□□□■□□□□□□□□□□⊂𝑆28=□■□□■■□□■□□□□□□□,
𝑆102 =□■□□□■□□□□□□□□□□⊂𝑆29=■■□□□■□□□□□□■□□□, 𝑆102 =□■□□□■□□□□□□□□□□⊂𝑆30=■■□□□■□□■□□□□□□□,
𝑆1312 =□□□□□□□□□■□□□□■□⊂𝑆31=□□□□■□□□□■□□□□■□, 𝑆822 =■□□□□□□□□■□□□□□□⊂𝑆32=■□□□□□□□□■□□□□■□,
𝑆372 =□□□□□■□□■□□□□□■□⊂𝑆33=□□□□□■□□■□□□□□■□, 𝑆802 =■□□□□■□□□□□□□□□□⊂𝑆34=■□□□□■□□□□□□□□■□,
𝑆382 =□■□□□□□□□□□□□□■□⊂𝑆35=□■□□□□□□■□□□□□■□, 𝑆382 =□■□□□□□□□□□□□□■□⊂𝑆36=□■□□■□□□□□□□□□■□,
𝑆332 =□□□□■□□□□□■□□■□□⊂𝑆37=□□□□■□□□□□■□□■□□, 𝑆352 =■□□□□□□□□□■□□□□□⊂𝑆38=■□□□□□□□□□■□□■□□,
𝑆1472 =□□□□□□□□□□■□■□□□⊂𝑆39=□□□□□■□□□□■□■□□□, 𝑆352 =■□□□□□□□□□■□□□□□⊂𝑆40=■□□□□■□□□□■□□□□□,
𝑆832 =□■□□□□□□□□□□■□□□⊂𝑆41=□■□□□□□□□□■□■□□□, 𝑆842 =□■□□■□□□□□□□□□□□⊂𝑆42=□■□□■□□□□□■□□□□□,
𝑆272 =□□□□■□□□□□□□■□□□⊂𝑆43=□□□□■□□□□□■□■□■□, 𝑆1632 =□□□□□□□□□□■□□□■□⊂𝑆44=□□□□■□□□■□■□□□■□,
𝑆292 =■□□□□□□□□□□□■□□□⊂𝑆45=■□□□□□□□□□■□■□■□, 𝑆352 =■□□□□□□□□□■□□□□□⊂𝑆46=■□□□□□□□■□■□□□■□,
𝑆282 =■□□□■□□□□□□□□□□□⊂𝑆47=■□□□■□□□□□■□□□■□, 𝑆1632 =□□□□□□□□□□■□□□■□⊂𝑆48=□□□□□■□□□□■□□■■□,
𝑆152 =□□□□□■□□□■□□□□□□⊂𝑆49=□□□□□■□□□■■□□□■□, 𝑆382 =□■□□□□□□□□□□□□■□⊂𝑆50=□■□□□□□□□□■□□■■□,
𝑆252 =□■□□□□□□□■□□□□□□⊂𝑆51=□■□□□□□□□■■□□□■□, 𝑆102 =□■□□□■□□□□□□□□□□⊂𝑆52=□■□□□■□□□□■□□□■□,
𝑆1282 =□□□□□□■□□□□□□■□□⊂𝑆53=□□□□□□■□■□□□□■□□, 𝑆582 =■□□□□□■□□□□□□□□□⊂𝑆54=■□□□□□■□□□□□□■□□,
𝑆312 =□□□□□□■□□■□□■□□□⊂𝑆55=□□□□□□■□□■□□■□□□, 𝑆402 =□□□□□□■□□■□□□□□□⊂𝑆56=■□□□□□■□□■□□□□□□,
𝑆362 =□□□□□□■□□□□□■□□□⊂𝑆57=□■□□□□■□□□□□■□□□, 𝑆542 =□■□□□□■□□□□□□□□□⊂𝑆58=□■□□□□■□■□□□□□□□,
𝑆362 =□□□□□□■□□□□□■□□□⊂𝑆59=□□□□□□■□■□□□■□■□, 𝑆662 =□□□□□□■□□□□□□□■□⊂𝑆60=□□□□■□■□■□□□□□■□,
𝑆292 =■□□□□□□□□□□□■□□□⊂𝑆61=■□□□□□■□□□□□■□■□, 𝑆582 =■□□□□□■□□□□□□□□□⊂𝑆62=■□□□□□■□■□□□□□■□,
𝑆282 =■□□□■□□□□□□□□□□□⊂𝑆63=■□□□■□■□□□□□□□■□, 𝑆402 =□□□□□□■□□■□□□□□□⊂𝑆64=□□□□□□■□□■□□□■■□,
𝑆152 =□□□□□■□□□■□□□□□□⊂𝑆65=□□□□□■■□□■□□□□■□, 𝑆382 =□■□□□□□□□□□□□□■□⊂𝑆66=□■□□□□■□□□□□□■■□,
𝑆252 =□■□□□□□□□■□□□□□□⊂𝑆67=□■□□□□■□□■□□□□■□, 𝑆102 =□■□□□■□□□□□□□□□□⊂𝑆68=□■□□□■■□□□□□□□■□,
𝑆362 =□□□□□□■□□□□□■□□□⊂𝑆69=□□□□□□■□■□■□■□□□, 𝑆272 =□□□□■□□□□□□□■□□□⊂𝑆70=□□□□■□■□□□■□■□□□,
𝑆292 =■□□□□□□□□□□□■□□□⊂𝑆71=■□□□□□■□□□■□■□□□, 𝑆352 =■□□□□□□□□□■□□□□□⊂𝑆72=■□□□□□■□■□■□□□□□,
𝑆282 =■□□□■□□□□□□□□□□□⊂𝑆73=■□□□■□■□□□■□□□□□, 𝑆402 =□□□□□□■□□■□□□□□□⊂𝑆74=□□□□□□■□□■■□□■□□,
𝑆722 =□□□□□□■□□□■□□□□□⊂𝑆75=□□□□□■■□□□■□□■□□, 𝑆542 =□■□□□□■□□□□□□□□□⊂𝑆76=□■□□□□■□□□■□□■□□,
𝑆252 =□■□□□□□□□■□□□□□□⊂𝑆77=□■□□□□■□□■■□□□□□, 𝑆102 =□■□□□■□□□□□□□□□□⊂𝑆78=□■□□□■■□□□■□□□□□,
𝑆342 =□□■□□□□□□□□□□■□□⊂𝑆79=□□■□□□□□■□□□□■□□, 𝑆342 =□□■□□□□□□□□□□■□□⊂𝑆80=□□■□■□□□□□□□□■□□,
𝑆322 =□□■□□□□□□■□□□□□□⊂𝑆81=□□■□□□□□□■□□■□□□, 𝑆322 =□□■□□□□□□■□□□□□□⊂𝑆82=□□■□■□□□□■□□□□□□,
𝑆412 =□□■□□□□□□□□□■□□□⊂𝑆83=□□■□□■□□□□□□■□□□, 𝑆422 =□□■□□□□□■□□□□□□□⊂𝑆84=□□■□□■□□■□□□□□□□,
𝑆412 =□□■□□□□□□□□□■□□□⊂𝑆85=□□■□□□□□■□□□■□■□, 𝑆272 =□□□□■□□□□□□□■□□□⊂𝑆86=□□■□■□□□□□□□■□■□,
𝑆422 =□□■□□□□□■□□□□□□□⊂𝑆87=□□■□■□□□■□□□□□■□, 𝑆422 =□□■□□□□□■□□□□□□□⊂𝑆88=■□■□□□□□■□□□□□■□,
𝑆282 =■□□□■□□□□□□□□□□□⊂𝑆89=■□■□■□□□□□□□□□■□, 𝑆322 =□□■□□□□□□■□□□□□□⊂𝑆90=□□■□□□□□□■□□□■■□,
𝑆342 =□□■□□□□□□□□□□■□□⊂𝑆91=□□■□□■□□□□□□□■■□, 𝑆152 =□□□□□■□□□■□□□□□□⊂𝑆92=□□■□□■□□□■□□□□■□,
𝑆252 =□■□□□□□□□■□□□□□□⊂𝑆93=□■■□□□□□□■□□□□■□, 𝑆102 =□■□□□■□□□□□□□□□□⊂𝑆94=□■■□□■□□□□□□□□■□,
𝑆412 =□□■□□□□□□□□□■□□□⊂𝑆95=□□■□□□□□■□■□■□□□, 𝑆272 =□□□□■□□□□□□□■□□□⊂𝑆96=□□■□■□□□□□■□■□□□,
𝑆422 =□□■□□□□□■□□□□□□□⊂𝑆97=□□■□■□□□■□■□□□□□, 𝑆292 =■□□□□□□□□□□□■□□□⊂𝑆98=■□■□□□□□□□■□■□□□,
𝑆282 =■□□□■□□□□□□□□□□□⊂𝑆99=■□■□■□□□□□■□□□□□, 𝑆322 =□□■□□□□□□■□□□□□□⊂𝑆100=□□■□□□□□□■■□□■□□,
𝑆342 =□□■□□□□□□□□□□■□□⊂𝑆101=□□■□□■□□□□■□□■□□, 𝑆152 =□□□□□■□□□■□□□□□□⊂𝑆102=□□■□□■□□□■■□□□□□,
𝑆342 =□□■□□□□□□□□□□■□□⊂𝑆103=□■■□□□□□□□■□□■□□, 𝑆102 =□■□□□■□□□□□□□□□□⊂𝑆104=□■■□□■□□□□■□□□□□,
𝑆362 =□□□□□□■□□□□□■□□□⊂𝑆105=□□■□□□■□■□□□■□□□, 𝑆272 =□□□□■□□□□□□□■□□□⊂𝑆106=□□■□■□■□□□□□■□□□,
𝑆422 =□□■□□□□□■□□□□□□□⊂𝑆107=□□■□■□■□■□□□□□□□, 𝑆292 =■□□□□□□□□□□□■□□□⊂𝑆108=■□■□□□■□□□□□■□□□,
𝑆422 =□□■□□□□□■□□□□□□□⊂𝑆109=■□■□□□■□■□□□□□□□, 𝑆322 =□□■□□□□□□■□□□□□□⊂𝑆110=□□■□□□■□□■□□□■□□,
𝑆342 =□□■□□□□□□□□□□■□□⊂𝑆111=□□■□□■■□□□□□□■□□, 𝑆152 =□□□□□■□□□■□□□□□□⊂𝑆112=□□■□□■■□□■□□□□□□,
𝑆342 =□□■□□□□□□□□□□■□□⊂𝑆113=□■■□□□■□□□□□□■□□, 𝑆252 =□■□□□□□□□■□□□□□□⊂𝑆114=□■■□□□■□□■□□□□□□,
𝑆1692 =□□□□■□□□□■□□□□□□⊂𝑆115=□□□□■□□□□■□□□□□■, 𝑆822 =■□□□□□□□□■□□□□□□⊂𝑆116=■□□□□□□□□■□□□□□■,
𝑆1332 =□□□□□■□□□□□□□□□■⊂𝑆117=□□□□□■□□■□□□□□□■, 𝑆802 =■□□□□■□□□□□□□□□□⊂𝑆118=■□□□□■□□□□□□□□□■,
𝑆1342 =□■□□□□□□■□□□□□□■⊂𝑆119=□■□□□□□□■□□□□□□■, 𝑆842 =□■□□■□□□□□□□□□□□⊂𝑆120=□■□□■□□□□□□□□□□■,
𝑆1492 =□□□□□□□□□□■□□□□■⊂𝑆121=□□□□■□□□□□■□□□□■, 𝑆352 =■□□□□□□□□□■□□□□□⊂𝑆122=■□□□□□□□□□■□□□□■,
𝑆1332 =□□□□□■□□□□□□□□□■⊂𝑆123=□□□□□■□□□□■□□□□■, 𝑆1492 =□□□□□□□□□□■□□□□■⊂𝑆124=□■□□□□□□□□■□□□□■,
𝑆1442 =□□□□□□■□■□□□□□□□⊂𝑆125=□□□□□□■□■□□□□□□■, 𝑆582 =■□□□□□■□□□□□□□□□⊂𝑆126=■□□□□□■□□□□□□□□■,
𝑆402 =□□□□□□■□□■□□□□□□⊂𝑆127=□□□□□□■□□■□□□□□■, 𝑆542 =□■□□□□■□□□□□□□□□⊂𝑆128=□■□□□□■□□□□□□□□■,
𝑆422 =□□■□□□□□■□□□□□□□⊂𝑆129=□□■□□□□□■□□□□□□■, 𝑆1732 =□□■□■□□□□□□□□□□■⊂𝑆130=□□■□■□□□□□□□□□□■,
𝑆322 =□□■□□□□□□■□□□□□□⊂𝑆131=□□■□□□□□□■□□□□□■, 𝑆1332 =□□□□□■□□□□□□□□□■⊂𝑆132=□□■□□■□□□□□□□□□■,
𝑆1172 =□□□□■□□□□□□■□□□□⊂𝑆133=□□□□■□□□□□□■□■□□, 𝑆1192 =■□□□□□□□□□□■□■□□⊂𝑆134=■□□□□□□□□□□■□■□□,
𝑆1702 =□□□□□■□□□□□□■□□□⊂𝑆135=□□□□□■□□□□□■■□□□, 𝑆802 =■□□□□■□□□□□□□□□□⊂𝑆136=■□□□□■□□□□□■□□□□,
𝑆832 =□■□□□□□□□□□□■□□□⊂𝑆137=□■□□□□□□□□□■■□□□, 𝑆842 =□■□□■□□□□□□□□□□□⊂𝑆138=□■□□■□□□□□□■□□□□,
𝑆1172 =□□□□■□□□□□□■□□□□⊂𝑆139=□□□□■□□□□□□■□□■□, 𝑆1292 =□□□□□□□□□□□■□□■□⊂𝑆140=■□□□□□□□□□□■□□■□,
𝑆1292 =□□□□□□□□□□□■□□■□⊂𝑆141=□□□□□■□□□□□■□□■□, 𝑆382 =□■□□□□□□□□□□□□■□⊂𝑆142=□■□□□□□□□□□■□□■□,
𝑆362 =□□□□□□■□□□□□■□□□⊂𝑆143=□□□□□□■□□□□■■□□□, 𝑆582 =■□□□□□■□□□□□□□□□⊂𝑆144=■□□□□□■□□□□■□□□□,
𝑆1252 =□□□□□□■□□□□■□□□□⊂𝑆145=□□□□□□■□□□□■□■□□, 𝑆542 =□■□□□□■□□□□□□□□□⊂𝑆146=□■□□□□■□□□□■□□□□,
𝑆412 =□□■□□□□□□□□□■□□□⊂𝑆147=□□■□□□□□□□□■■□□□, 𝑆1172 =□□□□■□□□□□□■□□□□⊂𝑆148=□□■□■□□□□□□■□□□□,
𝑆342 =□□■□□□□□□□□□□■□□⊂𝑆149=□□■□□□□□□□□■□■□□, 𝑆1792 =□□■□□■□□□□□■□□□□⊂𝑆150=□□■□□■□□□□□■□□□□,
𝑆272 =□□□□■□□□□□□□■□□□⊂𝑆151=□□□□■□□□□□□■■□□■, 𝑆1172 =□□□□■□□□□□□■□□□□⊂𝑆152=□□□□■□□□■□□■□□□■,
𝑆292 =■□□□□□□□□□□□■□□□⊂𝑆153=■□□□□□□□□□□■■□□■, 𝑆972 =■□□□□□□□■□□□□□□□⊂𝑆154=■□□□□□□□■□□■□□□■,
𝑆282 =■□□□■□□□□□□□□□□□⊂𝑆155=■□□□■□□□□□□■□□□■, 𝑆1332 =□□□□□■□□□□□□□□□■⊂𝑆156=□□□□□■□□□□□■□■□■,
𝑆152 =□□□□□■□□□■□□□□□□⊂𝑆157=□□□□□■□□□■□■□□□■, 𝑆912 =□■□□□□□□□□□□□■□□⊂𝑆158=□■□□□□□□□□□■□■□■,
𝑆252 =□■□□□□□□□■□□□□□□⊂𝑆159=□■□□□□□□□■□■□□□■, 𝑆102 =□■□□□■□□□□□□□□□□⊂𝑆160=□■□□□■□□□□□■□□□■,
𝑆662 =□□□□□□■□□□□□□□■□⊂𝑆161=□□□□□□■□□□□■□□■■, 𝑆722 =□□□□□□■□□□■□□□□□⊂𝑆162=□□□□□□■□□□■■□□□■,
𝑆502 =□□■□□□□□□□□□□□■□⊂𝑆163=□□■□□□□□□□□■□□■■, 𝑆462 =□□■□□□□□□□■□□□□□⊂𝑆164=□□■□□□□□□□■■□□□■,
𝑆1092 =□□■□□□■□□□□□□□□□⊂𝑆165=□□■□□□■□□□□■□□□■, 𝑆1202 =□□□□□□□■□□□□□■□□⊂𝑆166=□□□□□□□■■□□□□■□□,
𝑆1202 =□□□□□□□■□□□□□■□□⊂𝑆167=■□□□□□□■□□□□□■□□, 𝑆792 =□□□□□□□□□■□□■□□□⊂𝑆168=□□□□□□□■□■□□■□□□,
𝑆822 =■□□□□□□□□■□□□□□□⊂𝑆169=■□□□□□□■□■□□□□□□, 𝑆832 =□■□□□□□□□□□□■□□□⊂𝑆170=□■□□□□□■□□□□■□□□,
𝑆1212 =□□□□□□□■■□□□□□□□⊂𝑆171=□■□□□□□■■□□□□□□□, 𝑆1212 =□□□□□□□■■□□□□□□□⊂𝑆172=□□□□□□□■■□□□□□■□,
𝑆1302 =■□□□□□□■□□□□□□■□⊂𝑆173=■□□□□□□■□□□□□□■□, 𝑆1152 =□□□□□□□■□■□□□□□□⊂𝑆174=□□□□□□□■□■□□□□■□,
𝑆382 =□■□□□□□□□□□□□□■□⊂𝑆175=□■□□□□□■□□□□□□■□, 𝑆1352 =□□□□□□□■□□□□■□□□⊂𝑆176=□□□□□□□■□□■□■□□□,
𝑆352 =■□□□□□□□□□■□□□□□⊂𝑆177=■□□□□□□■□□■□□□□□, 𝑆1202 =□□□□□□□■□□□□□■□□⊂𝑆178=□□□□□□□■□□■□□■□□,
𝑆1502 =□■□□□□□■□□■□□□□□⊂𝑆179=□■□□□□□■□□■□□□□□, 𝑆412 =□□■□□□□□□□□□■□□□⊂𝑆180=□□■□□□□■□□□□■□□□,
𝑆422 =□□■□□□□□■□□□□□□□⊂𝑆181=□□■□□□□■■□□□□□□□, 𝑆342 =□□■□□□□□□□□□□■□□⊂𝑆182=□□■□□□□■□□□□□■□□,
𝑆322 =□□■□□□□□□■□□□□□□⊂𝑆183=□□■□□□□■□■□□□□□□, 𝑆952 =□□□□□□□□■□□□■□□□⊂𝑆184=□□□□□□□■■□□□■□□■,
𝑆1212 =□□□□□□□■■□□□□□□□⊂𝑆185=□□□□■□□■■□□□□□□■, 𝑆292 =■□□□□□□□□□□□■□□□⊂𝑆186=■□□□□□□■□□□□■□□■,
