Download important result of the fuzzy tychonoff theorem and

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CHAPTER
(VI)
IMPORTANT RESULT OF THE
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FUZZY TYCHONOFF THEOREM
AND INITIAL AND FINAL
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TOPOLOGIES
CHAPTER-VI
IMPORTANT
RESULTS OF THE FUZZY TYCHONOFF
THEOREM AND INITIAL AND FINAL TOPOLOGIES
Much of topology can be done in a setting where open
sets have “fuzzy boundaries.” To render this precise; the
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paper first describes cl ∞ -monoids; which are used to measure
the degree of membership of points in sets. Then L - or “fuzzy”
sets are defined; and suitable collections of these are called LTopological spaces.
A number of examples and result for such space are given.
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Perhaps most interesting is a version of the Tychonoff theorem which
given necessary and sufficient conditions on L for all
collection with given cardinality of compact L- spaces to have
compact product.
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We know Tychonoff Theorem;-
“Arbitrary products of compact spaces are compact.”
6.1
Introduction
6.2
Definition of Fuzzy Tychonoff Theorem
6.3
Theorems of Fuzzy Tychonoff Theorem
6.4
Initial and Final Topologies
6.5
Theorems of Initial Topologies
6.6
Final Fuzzy Topologies
6.7
Conclusions
A part of this chapter has appeared in International Journal of
Engineering Research and Application (IJERA) ISSN:2248-9622,
Vol.2, Issue 4, July-August 2012, pp.2251-2255.
6.1. Introduction-There seems to be use in certain infinite valued
logics [85] in constructive analysis [55]; and in mathematical
psychology [161] for notions of proximity less restrictive than
those found in ordinary topology
This chapter develops some basic theory for spaces in
which open sets are fuzzy. First we need sufficiently general
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sets of truth values with which to measure degree of
membership. The main thing is no get enough algebraic
structure.
6.2
Definition of Fuzzy Tychonoff Theorem- we define the fuzzy
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Tychonoff theorem as follows-
6.2.1 Definition- A cl ∞ - monoid is a complete lattice L with an
additional associative binary operation ∗ such that the lattice
zero 0 is a zero for ∗. The lattice infinity 1 is a identity for ∗
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and the complete distributive laws
a∗
and
V bi
i ∈I
V bi
i∈I
= V
i∈I
∗a=
( a ∗ bi)
V ( b i ∗ a)
i∈I
Hold for all a i b i ∈ L; and all index sets I.
6.2.2 Definition- An Let A on a set X is a function A:X→ L. X is
called the carrier of A : L is called the truth set of A; and for
x∈X; A(x) is called the degree of membership of x in A.
6.2.3 Definition- An L-topological (or just L-) space is a pair <X;
œ
> such that X is a set ;
‫⊆ﻉ‬
(1)
œ
⇒
œ
(2)
A: B ∈
(3)
OX ; 1X ∈
R
R
R
R
⊆ LX and
∨‫∈ﻉ‬œ
⇒A∗B∈
œ
œ
œ
.
6.2.4 Definition- The purpose of this paper is to prove a Tychonoff
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theorem in the so-called “intuitionistic fuzzy topological spaces.”
After giving the fundamental definitions, such as the definitions of
intuitionistic fuzzy set, intuitionistic fuzzy topology, intuitionistic
fuzzy topological space fuzzy continuity, fuzzy compactness, and
fuzzy dicompactness, we several preservation properties and some
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ciharacterizations concerning fuzzy compactness. Lastly we give a
Tychonoff-like theorems.
Let X be a nonempty fixed set. An IFS A is an object having
the form
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A = {<X,µ A (x),ɣ A (X)>: X∈X},
R
R
R
R
(2.1)
Where the functions µ A : X→ I and ɣ A :X→ I denote the degree of
R
R
R
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membership namely, µ A (X)) and the degree of nonmember ship
R
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(namely, ɣ A (X)) of each element X
R
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and 0 ≤ µ A (X) + ɣ A (X) ≤ 1 for each X
R
R
R
R
∈X
to the set A, respectively,
∈ X.
6.2.5 Definition- Let {A i : i∈J} be an arbitrary family of IFSs in X.
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Then
(a)
∩A i = {<X, ∧µ A i(X), ∨ ɣ i (X)> : X∈X};
(b)
∪A i = {<X, ∨µ A i(X), ∧ ɣ i (X)> : X∈X}.
R
R
R
R
R
R
R
R
R
R
R
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The ordinary complement of A=
<X,µA,ɣ > is defined by
AC =<X,1-ɣ , 1-µ A
A
>.
6.3 Theorem of Fuzzy Tychonoff Theorem- For solving fuzzy
Tychonoff theorem we firstly define the following lemma.
6.3.1 Lemma- For a:b elements of a cl ∞ - monoid L;
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a ∗ b ≤ a ∧ b.
a ≤ 1 , so 1 ∗ b = (a ∨1 ) ∗ b
Proof-
= a ∗ b ∨1 ∗ b
That is
b = (a∗b) ∨b;
and therefore
similarly
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a ∗ b ≤ b.
a ∗ b ≤ a.
6.3.2 Theorem- Every product of α -compact L- spaces is compact iff
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1 is α-isolated in L.”
Proof – with the help of theorem
if φ is a sub-base for <x:
œ > and every cover of X by sets in φ
has a finite sub-cover; then < X.
œ > is compact.
Now we have to proof fuzzy-Tychonoff theorem :We have < X i
œ i > compact for i ∈ I; |I| ≤ α and
œ > = π i∈I . <X i : œ i >.
< x:
Where
œ has the sub-base.
φ = {p1 i (A i )| A i ∈
P
PR
R
R
R
R
R
œ i; i ∈ I }
So by the above theorem it suffices to show that no  ⊆ φ with
F UP (finite union property) is a cover.
Let  ⊆ φ with F U P be given: and for i∈I let  i = { A∈
1
œi | Pi-
(A) ∈  }. Then each  i has FUP;
For if ∨n j=1 A j = 1xi;
Where P i -1 (A j ) ∈ 
Then
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V j=1 P-1 i (A j ) = 1x
Since P i -1 (1xi) = 1x and P-1 i preserves
Therefore  i is a non-cover for each i ∈ I and there exits
x i ∈ X i such that
(V i ) (x i ) = a i <1
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Now let
x = < x i > ∈ X and
 I ’ = { P-1 i (A)| A ∈
œi } ∩ 
Then
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 ⊆ φ implies  = ∪ I  I ’ and
( V  i ) (xi) = a i implies
( V i ) (x) = a i
Since
( V  I ’ ) (x) = V { (P i -1 (A)) (x) | A ∈ œ
and P i -1 (A) ∈  }
= V { A (P i (x))| P i -1 (A) ∈ 
and A∈œ i } = (V  i )a (xi).
Therefore
(V ) (x) = V ((V’ I ) (x))
i∈I
P
= V ai < |
i
PR
R
i
since I is α- isolated in I; each
a i < 1 and |I| ≤ α .
6.3.3 Theorem-
if
1 is not α –isolated in L; then there is a
collection <xi:
œ i > for i ∈ I and |I| = α of compact L-spaces
such that the product <x:œ > is non-compact.
such that V i a i = 1 .
Let
X i = ℕ for i∈I and
Let
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Proof- non α- isolation given I with |I|= α and ai <1 for i ∈ I
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œ i = { 0, (ai) [n] . 1 }*; where
As ; for a ∈ L and S ⊆ Xi; denote the function equal to a on S
and O or X i -S; where [n] = { 0,1 ……………. N-1} and where φ ∗
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indicates the L- topology generated by φ as a sub-basis.
Then < ℕ,
œ i > is a compact. As every A ∈ œ i except 1 is
contained in (a i ) N ; and (a i ) N <1; that
 ⊆
œ i is a cover iff 1 ∈  therefore every cover has {1} as a
sub-cover.
But <x :
œ > = π i ∈I <X i : œ i > is non compact. Let A in denote
the open L-set P i -1 ((ai) [n]) ∈
œ : for i ∈ I
at is given for x ∈ X by the formula
A in (x) = a i if x i <n;
and
A in (x) = O otherwise
{ A in } i ∈ I; n ∈ ℕ }
is an open cover with no finite sub-cover. First
V n A in = (a i ) x . since
V n (ai) [N] = (ai) ℕ in <xi:œ i ;>
and P i -1 preserves suprema so that
V in A in = Vi (V n A in ) = Vi (a i ) x = l
Second no finite sub-collection
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{A in | < i;n>∈ J } ; |J| < ω can
Cover; since V<i,n> ∈ J A in (x) = O
For any x with
X i ≥ V {n| <i; n> ∈ J }
For any i
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and infinitely many such x always exist.
6.4 Initial and Final Topologies- In this sections we introduce the
notations of initial and final topologies. For this firstly we
defined initial and final topologies then their theorems. We
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show that from a categorical point of view they are the right
concept of generalize the topological ones. We also show that
the Tychonoff theorem is safeguarded.
We show that with our notion of fuzzy compactness the
Tychonoff product theorem is safeguarded. We also show that
this is not the case for weak fuzzy compactness.
6.4.1 Initial Fuzzy Topologiesif we consider a set E; a fuzzy topological space (f:y) and a
function
ƒ : E → F then we can define
ƒ-1 (Y) = { ƒ-1 (‫ )ג‬: ‫ ∈ ג‬Y }
P
P
P
P
It is easily seen that ƒ-1 (Y) is the smallest fuzzy topology
P
P
making ƒ fuzzy continuous.
More generally: consider a family of fuzzy topological space
(F j : Y j ) j∈J and for each j∈ J a function
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R
R
R
ƒj : E → fj
R
R
R
R
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then it is easily seen that the union
∪ ƒ-1j(Y j )
P
P
R
R
j∈ J
is a sub-base for a fuzzy topology on E making every ƒ i fuzzy
continues . we denote it by
∪ ƒ-1 j (Y j )
PR
RP
P
R
R
R
or sup ƒ-1 j (Y j )
j∈J
P
PR
R
R
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P
R
j∈ J
6.4.2 Definition- The initial fuzzy topology on E for the family of
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fuzzy topological space (Fj: Yj) j ∈ J and the family of
functions
ƒ j : E → (F j : y j )
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R
R
R
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is the smallest fuzzy topology on E making each function ƒ j
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fuzzy continuous.
Within the notations of the definition let (G:δ) be a fuzzy
topological space and let ƒ be a function from G to E then ƒ
will be fuzzy continuous iff all the compositions ƒ j oƒ are fuzzy
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continues. Moreover the initial fuzzy topology on E is the
finest one possessing this property; it is equally clear that the
transitivity property for initial topologies holds for initial fuzzy
topologies.
6.4.3 Definition- If (E j , δ j ) j ∈ J is a family of fuzzy topological
space; then the fuzzy product topology on π j∈J E j is defined as
the initial fuzzy topology on π j∈J E j for the family of spaces (E j .
Where
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δ j ) j ∈ J and function ƒi: π j∈J E j → Ei
∀i∈J
ƒ i = P ri ,
the protection on the ith coordinate.
6.4.4 Definition- if (E j , δ j ) j∈J is a family of fuzzy topological space
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then the fuzzy product topology, on π j∈J E j is defined as the
initial fuzzy topology on π j∈J E j for the family of spaces
(E j ,δ j ) j∈J and functions ƒi: π j∈J π j∈J E j → Ei where ∀ i ∈J fi =
the projection
on ith
coordinate we denote this fuzzy
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p ri
topology π j∈J δ j .
6.5. Theorems of Initial Topologies- The Theorems related to initial
fuzzy topologies is as follows-
6.5.1 Lemma- if ƒ: E → F, δ then
i (ƒ-1 (δ )) = ƒ-1 (i(δ )) = i (ƒ-1(δ)).
Proof - ƒ:E; ƒ-1 (i (δ)) → F.
i (δ) is continuous: thus ƒ: E; W (ƒ-1(i(δ )))→ F;
δ is fuzzy continuous.
This implies W (ƒ-1 (i(δ ))) ⊃ ƒ-1 (δ) ⊃ ƒ-1 (δ) ⇒ ƒ-1 (i(δ)) ⊃ i (ƒ-1 (δ
))) ⊃ i (ƒ-1 (δ)).
Further: ƒ: E ; ƒ-1 ( δ ) → F
δ is fuzzy continuous thus
ƒ E : i (ƒ-1 (δ)) → F.
i (δ) is continuous and this implies
i (ƒ-1(δ)) ⊃ ƒ-1 (i(δ)).
6.5.2 Theorem- If E is a set (F j : T j ) j∈J is a family of topological space
and
ƒ j : E → E j is a family of functions then we have
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Sup ƒ-1 j (w(T j ))= w (supƒ j -1 (T j ))
j∈J
j∈J
Proof- we can prove this theorem By the following lemma:-
First Lemma 1:-
if ƒ: E → F : δ then
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i (ƒ-1 (δ )) = ƒ-1(i(δ)) = i (ƒ-1(δ)).
Lemma 2:-
if ƒ: E → F: W (T) then
W (ƒ-1(T)) = ƒ-1 (W(T)).
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Consider now a set E; a family of fuzzy topological spaces (F j :
δ j ) j∈J and a family of functions ƒ j : E → F j
Then on the one hand: we can consider the initial fuzzy
topology as defined and take the associated topology
i.e.
i (sup) ƒ-1 (δ j ))
j∈J
on the other hand we can consider the family of topological
spaces (F j ; i (δ j )) j∈J and take the initial topology i.e.
Sup ƒ-1 j (i(δ j )).
j∈J
Showing that these two topologies are equal is trivial.
6.5.3 Theorem- if E is a set (F j . δ j ) j∈J is a family of fuzzy topological
spaces: and
ƒ j : E → F j is a family of function: then we have
i (sup ƒ-1 j (δ j )) = sup ƒ j -1(i(δ j )).
j∈J
j∈J
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Proof- we can prove this theorem by the help of lemma 1if ƒ: E → F: δ then
i (ƒ-1(δ)) = ƒ-1(i(δ)) = i (ƒ-1(δ)).
It follows that
∀ j∈J
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ƒ j -1 (i(δ j )) = i ƒ-i j (δ j ).
and the result then follows analogously by the lemma 3-
Lemma 3:- if E is a set and (T j ) j∈J is a family of topologies on E then
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Sup ω(T j ) = ω(sup T j ).
j∈J
j∈J
6.6 Final Fuzzy TopologiesConsider a set F a fuzzy topological space (E:δ): and a
function
ƒ: E → F
Then we can define
ƒ (δ) = <{∨ :ƒ-1 (∨ ) ∈ δ }>
more generally consider a family of fuzzy topological space (E j :
δ j ) j∈J and for each j∈J a function
ƒ j : E j → F.
It is easily seen that the intersection
∩ ƒ j (δ j )
j∈J
is the finest fuzzy topology on F making all the functions f j
fuzzy continuous.
6.6.1 Definition- the final fuzzy topology on F: for the family of fuzzy
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topological spaces (E j :δ j ) j∈J and the family of functions
ƒ j : (E j :δ j ) → F
is the finest fuzzy topology on F making each functions ƒ j : fuzzy
continuous.
And ∀ j ∈ J :
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Let δ j = ω (T j )
Then on the other hand we have
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∩ ƒ j (ω (T j )).
j∈J
and on the other hand we have
W(∩
j∈J
ƒ j (T j ))
again we can show the two fuzzy topological to be the same.
6.6.2 Lemma 1- if ƒ:E δ → F
Then we have
i (ƒ(δ)) = ƒ(i(δ)) ⊃ i (ƒ(δ)).
Proof-
to show the inclusions
i (ƒ(δ )) ⊃ ƒ (i(δ)) ⊃ i (ƒ(δ))
we proceed as we proceeded in lemma:if ƒ: E → F: δ then
i (ƒ-1(δ)) = ƒ-1 (i(δ)) = i (ƒ-1 (δ )).
The remaining inclusion is shown using the latter one upon δ.
i.e
i (ƒ(δ)) ⊂ ƒ(i(δ)) = ƒ(ioω (i(δ)))=ƒ (i(δ)).
6.6.3 Lemma 2- if ƒ:E: ω (T) → F then we have
ƒ (ω(T)) = ω (ƒ (T)).
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Proof- from lemma as proved above. at follows with
δ = ω (T) then
ƒ (ω (T)) = W (ƒ (T))
However: ƒ (ω (T) is topologically generated indeed it is the
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finest fuzzy topology making
ƒ: E : ω (T) → F:
ƒ (W(T)) fuzzy continuous
But since ω (T) is topologically generated
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ƒ: E: ω (T) → F:
ƒ (ω(T)) still is fuzzy continuous and thus
ƒ (ω(T) = ƒ (w(T)).
6.5. Conclusion- In this section we discussed about the fuzzy
Tychonoff theorems and initial and final fuzzy topology.fuzzy
Tychonoff theorems is describes as –
“ Arbitrary products of compact spaces are compact”. We show
that with our notion of fuzzy compactness the Tychonoff product
theorem is safeguarded.
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