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International Journal of Applied Information Systems (IJAIS) – ISSN : 2249-0868
Foundation of Computer Science FCS, New York, USA
Volume 11 – No. 6, November 2016 – www.ijais.org
Separation of Fuzzy Topological Space
Mohammad Arshaduzzaman, PhD
M.Sc (Mathematics)
Associate Professor
Department of Mathematics
Al Baha University, Al Baha
Kingdom of Saudi Arabia (KSA)
ABSTRACT
2. THEOREM (I)
In this paper I discussed about some separation properties of
fuzzy topological space. The definition of fuzzy compactness
does not hold good on the definition of fuzzy Hausdorff space,
we introduce a new definition of proper compactness and prove
some interesting results related with it.
Every fuzzy T2- space is a fuzzy T1- space.
Keywords
Fuzzy set, Fuzzy topological space, Fuzzy Hausdorff space,
compactness, T2- space.
Let X be any set. A fuzzy set A in X is characterized by
 A : X  0,1 A fuzzy singleton p in X is
p
a fuzzy set with the membership function
p (X)  y if x  x 0
defined by
= 0 otherwise
where
y (0,1). x 0
Hence
pAiff p x 0   A x 0 
(i)
fuzzy set in X, if
(ii)
with
p Gpqn

qn Gpqn for all n, as (X, T) is fuzzy T
2.

If
P   Gpqn
then
p
is
a
p x q  0 and p x p  1
closed
set
with
p Vpqn
qP' p'
Consequently,
p'
is
1
then there exist open sets and Gp and

pGp and qGq,pGp
p x p  q x q 
there exist an

open Gp with
nN
p is closed. So, (X, T) is fuzzy T -space.
an open set i.e.
xp  x q
xp = xq and
pqn
can find a sequence of open sets
So, P' is an open set with

Gq,
G 
Case (ii) : Let p be a crisp point and converges to zero, then we
p x 0    A x 0 
p, q X ,
If
(p) x p  Gq x p 
and
Definition 1 :
A fuzzy topological space is said to be Housdorff of fuzzy T2 if
the following conditions are satisfied :
For any
qp'
Case (i) : Let p be a fuzzy point in X, and let
be
arbitrary. Then, there exist an open set Gq containing q such that
Hence Gq {p}. So, {p} being the union of open sets, is an open
set. This implies that {p} is closed.
1. INTRODUCTION
membership function
Proof : Let (X, T) be a fuzzy T2- space.
pGp but qGp
We now introduce the concept of properly compact space in a
fuzzy
topological space and prove an analogous result of general
topology which holds for a compact Hausdroff space.By
Palaniappan[3] & Zadeh[4]
Definition 4 :
.
By Srivastava & Lal [8].
Definition 2 :
A fuzzy set A is said to be open for each , there exists an open
pG  A
set G with
. Out definition of open set is
equivalent to that given by Change [1].
Definition 3 :
A fuzzy topological space is said to be a fuzzy T1- space if the
singletons are closed. By Hutton & Reilly [2].
G :iI
1
A family
open sets in fuzzy topological space X is
called a proper open cover of a fuzzy set A in X if for every
x X
there exists a member
this family such that
The family
Gix
of this family such that of
Gix (x)   A (x)
Gi :iI is called a proper open sub cover of
Gi :iI if it is itself a proper open cover of A.
By Lowen[5].
1
International Journal of Applied Information Systems (IJAIS) – ISSN : 2249-0868
Foundation of Computer Science FCS, New York, USA
Volume 11 – No. 6, November 2016 – www.ijais.org
Definition 5 :
A fuzzy set A in a fuzzy topological space X is said to be
properly compact if every proper open cover of A is reducible to
a finite proper open sub cover.
Then Fp is closed and
We also have
p x p  Fp x p for all x q X
p x p  Fp x p 
(iv)
By Lowen [6] & Rodabaugh[7].
Then, for all satisfying the condition (i), the family
3. THEOREM 2
F 'p
Every properly compact set in a fuzzy T2- space is closed.
Proof : Let A be a properly compact set in a fuzzy T2-space X.
We choose a point with
X
is
 A x p  Gp x p 
Thus,
G
for
each
pq : x q  X
exists
G
family
with (iv)
pq
an
open
Gp
with
there
exists
a
family
of open sets
for all
of open sets
x q X
Then,
p
with establishes that A is closed.
4. CONCLUSION
(iii)
X
such
: x X
From the above theorems and definitions it is clear that A fuzzy
set in a fuzzy topological space is said to be properly compact if
every proper open cover of a fuzzy set is reducible to a finite
proper open sub cover and every properly compact set in a fuzzy
T2-space is closed. Thus the definition of fuzzy compactness
does not hold good on the definition of fuzzy Hausdorff space.
5. REFERENCES
there exists a
 A x q  Gpq x q 
sup
 A x q  x q Gpq x p 
[1] Chang, C.L. : Fuzzy topological spaces, J.Math. Anal.
Appl. 24, 182-190, 1960.
[2] Hutton, B & Reilly, I. : Seperation axioms in fuzzy
topological space, Fuzzy Sets & systems 3, 99-104, 1980.
[3] Palaniappan, N. : Fuzzy topology, Narosa Publ. House,
New Delhi, 2002.
[4] Zadeh, L.A. : Fuzzy sets, Inform, Control, 8, 338-353.
[5] Lowen R : Fuzzy topological spaces and fuzzy
compactness,J. Math. Anal. Appl.,56 (1976), pp.621–633.
A   Gpq
This implies that
X, satisfies the conditions (v) & (vi).
we
have
A   Fp
(ii)
 A x q  Gpq x q 
with
So,
there
p x p  Gp  x p 
and
p
 A x q  Inf Fp x q for all x q X.
(i)
T2,
F  where
p
p x p   A x p 
since
is open in
Consequently
(v)
x q X
G
Thus, the family
[6] Lowen R: Compact Hausdorff fuzzy topological spaces
are topological, Topology and Appl., 12 (1981), pp. 65–
74.
G
[7] Rodabaugh S :The Hausdorff separation axiom for fuzzy
topological spaces,Topol. Appl.,11 (1980),pp. 319–334.
pq : x q  X
is a proper cover of A. Since
A is properly compact, there exists a finite subfamily, say,
,Gpq2 ,....., Gpqmof Gpq : xqXofGpq : x q X
pq1
m 
A   Gpqk .Then A   Gpqk  Fp (sup pose)
K 1
with
[8] Srivastava R, Lal S N, and Srivastava A K, “Fuzzy
Hausdorff topological spaces,” Journal of Mathematical
Analysis and Applications, vol. 81, no. 2, pp. 497–506,
1981.
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