1. Dr. H.S.Nayal, Manoj Kabadwal / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2251-2255
ON FUZZY TYCHONOFF THEOREM AND INITIAL AND
FINAL TOPOLOGIES
Dr. H.S.Nayal, Manoj Kabadwal
Department of Mathematics,Govt.Postgradute College,Ranikhet (Uttrakhand)
Abstract:
Much of topology can be done in a setting Hold for all ai bi  L; and all index sets I.
where open sets have fuzzy boundaries. To render 1.2 Definition- An Let A on a set X is a function
this precise; the paper first describes cl∞ - monoids; A:X→ L. X is called the carrier of A : L is
which are used to measure the degree of membership called the truth set of A; and for xX; A(x)
of points in sets. Then L- or fuzzy sets are defined; is called the degree of membership of x in
and suitable collections of these are called L- A.
Topological spaces. 1.3 Definition- An L-topological (or just L-)
A number of examples and result for such space space is a pair <X; œ > such that X is a set ; œ 
are given. Perhaps most interesting is a version of the LX and
Tychonoff theorem which given necessary and
sufficient conditions on L for all collection with (1) ‫ع‬ œ  ‫ ع‬œ
given cardinality of compact L- spaces to have (2) A: B  œ  A  B  œ
compact product. (3) OX ; 1X  œ .
We know Tychonoff Theorem as Arbitrary products
of compact spaces are compact. Introduction-There 1.4 Theorem of Fuzzy Tychonoff Theorem- For
seems to be use in certain infinite valued logics [2] in solving fuzzy Tychonoff theorem we firstly
constructive analysis [1]; and in mathematical define the following lemma.
psychology [6] for notions of proximity less 1.5 Lemma- For a:b elements of a cl∞ - monoid
restrictive than those found in ordinary topology L;
This paper develops some basic theory for spaces in
a  b ≤ a  b.
which open sets are fuzzy. First we need sufficiently
Proof- a ≤ 1 , so 1  b = (a 1 )  b
general sets of truth values with which to measure
degree of membership. The main thing is no get = a  b 1  b
enough algebraic structure. That is b = (ab) b;
1.1 Definition- A cl∞ - monoid is a complete lattice L and therefore a  b ≤ b.
with an additional associative binary operation  similarly a  b ≤ a.
1.6 Theorem- Every product of α -compact L-
such that the lattice zero 0 is a zero for . The spaces is compact iff 1 is α-isolated in L.”
lattice infinity 1 is a identity for  and the
complete distributive laws 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 < Xi œi > compact for i  I; |I| ≤ α
and
And < x: œ > = πiI. <Xi: œi >.
Where œ has the sub-base.
φ = {p1i (Ai)| Ai  œ i; i  I }
So by the above theorem it suffices to show
that no   φ with FUP (finite union
property) is a cover.
Let   φ with F U P be given: and for iI
let  = { A œi | Pi-1 (A)   }. Then each
i
 has FUP;
i
2251 | P a g e

2. Dr. H.S.Nayal, Manoj Kabadwal / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2251-2255
For if nj=1 Aj = 1xi; First Vn Ain = (ai)x. since
Where Pi-1 (Aj)   Vn(ai)[N] = (ai)ℕ in <xi:œi ;>
Then n and Pi-1 preserves suprema so that
Vj=1 P-1i (Aj) = 1x VinAin= Vi (Vn Ain) = Vi (ai)x = l
Since Pi (1xi) = 1x and P-1i preserves
-1
Second no finite sub-collection
Therefore  is a non-cover for
i {Ain | < i;n> J } ; |J| <  can
each i  I and there exits xi  Xi such that Cover; since V<i,n>  J Ain (x) = O
(V) (xi) = ai <1
i For any x with
Now let x = < xi >  X and Xi ≥ V {n| <i; n>  J } For any i
’ = { P-1i (A)| A  œi } ∩ 
I and infinitely many such x always exist.
Then  φ implies = I ’ and I 1.8 Initial and Final Topologies- In this paper we
( V  ) (xi) = ai implies
i introduce the notations of initial and final
( V) (x) = ai
i topologies. For this firstly we defined initial
Since ( V ’ ) (x) = V { (Pi-1 (A)) (x) | A
I and final topologies then their theorems. We
œi show that from a categorical point of view
and Pi-1 (A)  } they are the right concept of generalize the
topological ones.
We show that with our notion of
= V { A (Pi(x))| Pi-1 (A)   fuzzy compactness the Tychonoff product
and Aœ i } = (V )a (xi).
i theorem is safeguarded. We also show that
’ this is not the case for weak fuzzy
Therefore (V  (x) = V ((VI) (x))
)
iI compactness.
= V ai < 1 1.9 Initial Fuzzy Topologies- If we consider a set E;
i a fuzzy topological space (f:y) and a function
since I is α- isolated in I; each ƒ : E → F then we can
a i < 1 and |I| ≤ α . define
ƒ-1 (Y) = { ƒ-1 (‫ ג : )ג‬ Y }
1.7 Theorem- If 1 is not α –isolated in L; then there It is easily seen that ƒ-1 (Y) is the smallest
is a collection <xi: œ i> for i  I and |I| = α fuzzy topology making ƒ fuzzy continuous.
of compact L-spaces such that the product More generally: consider a family of fuzzy
<x:œ > is non-compact. topological space
Proof- non α- isolation given I with |I|= α (Fj : Yj) jJ and for each j J a function
and ai <1 for i  I such that Vi ai = 1. ƒj : E → fj
then it is easily seen that the union
Let Xi = ℕ for iI and
Let œ i = { 0, (ai)[n]. 1 }*; where
As ; for a  L and S  Xi; denote the  ƒ-1j(Yj)
function equal to a on S and O or Xi-S; j J
where [n] = { 0,1 ……………. N-1} and
where φ  indicates the L- topology
generated by φ as a sub-basis. is a sub-base for a fuzzy topology on E
making every ƒi fuzzy continues . we denote
Then < ℕ, œi> is a compact. As every A 
it by
œi except 1 is contained in (ai)N; and (ai)N
<1; that
  œi is a cover iff 1   therefore every
cover has {1} as a sub-cover.  ƒ-1j (Yj) or sup
But <x : œ > = π iI <Xi : œi > is non -1
ƒ j (Yj)
compact. Let Ain denote the open L-set Pi-1
((ai) [n])  œ : for i  I j J jJ
at is given for x  X by the formula
Ain (x) = a i if xi<n; 1.10 Definition- The initial fuzzy topology on E for
and Ain (x) = O otherwise the family of fuzzy topological space (Fj:
{ Ain} i  I; n  ℕ } Yj) j  J and the family of functions
is an open cover with no finite sub-cover.
ƒj : E → (Fj : yj)
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3. Dr. H.S.Nayal, Manoj Kabadwal / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2251-2255
is the smallest fuzzy topology on E making ƒ E : i (ƒ-1 ()) → F.
each function ƒj fuzzy continuous.
i () is continuous and this implies
Within the notations of the
definition let (G:) be a fuzzy topological i (ƒ-1())  ƒ-1 (i()).
space and let ƒ be a function from G to E
then ƒ will be fuzzy continuous iff all the 1.15 Theorem- If E is a set (Fj: Tj)jJ is a family of
compositions ƒjoƒ are fuzzy continues. topological space and
Moreover the initial fuzzy topology on E is
the finest one possessing this property; it is ƒj : E → Ej is a family of
equally clear that the transitivity property for functions then we have
initial topologies holds for initial fuzzy
topologies. Sup ƒ-1j (w(Tj))= w (supƒj-
1
(Tj))
1.11 Definition- If (Ej, j) j  J is a family of fuzzy
topological space; then the fuzzy product topology on jJ jJ
πjJ Ej is defined as the initial fuzzy topology on πjJ
Proof- we can prove this theorem By the following
Ej for the family of spaces (Ej. j) j  J and function lemma:-
ƒi: πjJ Ej → Ei
Lemma 1:-if ƒ: E → F :  then
Where i  J
i (ƒ-1 ( )) = ƒ-1(i()) = i (ƒ-1()).
ƒ i = Pri,
Lemma 2:-if ƒ: E → F: W (T) then
the protection on the ith coordinate.
W (ƒ-1(T)) = ƒ-1 (W(T)).
1.12 Definition- if (Ej, j)jJ is a family of fuzzy
topological space then the fuzzy product Consider now a set E; a family of fuzzy
topology, on πjJ Ej is defined as the initial topological spaces (Fj: j)jJ and a family of
fuzzy topology on πjJ Ej for the family of functions ƒj : E → Fj
spaces (Ej,j)jJ and functions ƒi: πjJ πjJ Ej
→ Ei where  i J fi = pri the projection Then on the one hand: we can consider the
on ith coordinate we denote this fuzzy initial fuzzy topology as defined and take
topology πjJ j. the associated topology
1.13 Theorems of Initial Topologies- The i.e. i (sup) ƒ-1 (j))
Theorems related to initial fuzzy topologies
is as follows- jJ
1.14 Lemma- if ƒ: E → F,  then on the other hand we can consider the family
of topological spaces (Fj; i (j))jJ and take
i (ƒ-1 ( )) = ƒ-1 (i( )) = i (ƒ-1()). the initial topology i.e.
Proof - ƒ:E; ƒ-1 (i ()) → F. Sup ƒ-1j (i(j)).
i () is continuous: thus ƒ: E; W (ƒ- jJ
1
(i( )))→ F;
Showing that these two topologies are equal
 is fuzzy continuous. is trivial.
This implies W (ƒ-1 (i( )))  ƒ-1 ()  ƒ-1 () 1.16 Theorem- if E is a set (Fj. j)jJ is a family of
 ƒ-1 (i())  i (ƒ-1 ( )))  i (ƒ-1 ()). fuzzy topological spaces: and
Further: ƒ: E ; ƒ-1 (  ) → F ƒj : E → Fj is a family of function:
then we have
 is fuzzy continuous thus
2253 | P a g e

4. Dr. H.S.Nayal, Manoj Kabadwal / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2251-2255
i (sup ƒ- ƒj: (Ej:j) → F
1
j (j)) = sup ƒj-1(i(j)).
is the finest fuzzy topology on F making
jJ each functions ƒj: fuzzy continuous.
jJ
And j  J :
Proof- we can prove this theorem by the help of
lemma 1 Let j =  (Tj)
if ƒ: E → F:  then Then on the other hand we have
i (ƒ-1()) = ƒ-1(i()) = i (ƒ- ∩ ƒj ( (Tj)).
1
()).
jJ
It follows that jJ
and on the other hand we have
ƒj-1 (i(j)) = i ƒ-ij (j).
W ( ∩ ƒj (Tj))
and the result then follows
analogously by the lemma 2 jJ
again we can show the two fuzzy
Lemma 3:- if E is a set and (Tj)jJ is a family of
topological to be the same.
topologies on E then
1.19 Lemma 1- if ƒ:E  → F
Sup (Tj) = (sup Tj).
Then we have i (ƒ()) = ƒ(i())  i (ƒ()).
jJ jJ
Proof- to show the inclusions
1.17 Final Fuzzy Topologies- Consider a set F a
i (ƒ( ))  ƒ (i())  i (ƒ())
fuzzy topological space (E:): and a function
we proceed as we proceeded in lemma:-
ƒ: E → F
if ƒ: E → F:  then
Then we can define
i (ƒ-1()) = ƒ-1 (i()) = i (ƒ-1 ( )).
ƒ () = <{ :ƒ-1 ( )  
}> The remaining inclusion is shown using the
latter one upon .
more generally consider a family of fuzzy
topological space (Ej: j)jJ and for each jJ i.e i (ƒ())  ƒ(i()) = ƒ(io
a function
(i()))=ƒ (i()).
ƒj : Ej → F.
1.20 Lemma 2- if ƒ:E:  (T) → F then we have
It is easily seen that the intersection
ƒ ((T)) =  (ƒ (T)).
∩ ƒj (j)
Proof- from lemma as proved above. at follows with
jJ
 =  (T) then
is the finest fuzzy topology on F making all
the functions fj fuzzy continuous. ƒ ( (T)) = W (ƒ (T))
1.18 Definition- the final fuzzy topology on F: for the However: ƒ ( (T) is topologically
family of fuzzy topological spaces (Ej:j)jJ generated indeed it is the finest fuzzy
and the family of functions topology making
2254 | P a g e

5. Dr. H.S.Nayal, Manoj Kabadwal / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2251-2255
ƒ: E :  (T) → F:
ƒ (W(T)) fuzzy continuous
But since  (T) is topologically
generated
ƒ: E:  (T) → F:
ƒ ((T)) still is fuzzy continuous and thus
ƒ ((T) = ƒ (w(T)).
1.21 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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[1] E. Bishop, “Foundations of Constructive
Analysis,” McGraw-Hill, New York, 1967.
[2] J.A. Goguen; Catogories of fuzzy sets; Ph.
D. Thesis; university of cal. At Berkeley;
Dept. of Math. 1968. (available through
University microfilm).
[3] J.A. Goguen; L-fuzzy sets; J.math Anal.
Appl. 18 (1967); 145 -174.
[4] J.L. Kelley “ General Topology” Van-
Nostrand: Princetor: new Jersey : 1955.
[5] L.A. Zadeh; fuzzy sets inform and control 8
(1965); 338-358
[6] S. Mac Lane, “Categories for the Working
Mathematician,” Spriner-Verlag New York,
1971
[7] P.M. Cohn, “universal Algebra,” Harper and
Row, New York, 1965.
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