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Fuzzy pre-continuous and fuzzy pre*-continuous function of fuzzy pre-compact space in fuzzy topological space |
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Munir Abdul Khalik Al-Khafaji*, Marwah Flayyih Hasan |
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Department of Mathematics, College of Education, Al-Mustansiriya University, Baghdad, Iraq *Corresponding author E-mail: mnraziz@yahoo.com |
Copyright © 2015 Munir Abdul Khalik Al-Khafaji, Marwah Flayyih Hasan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The aim of this paper is to introduce and study the notion of a fuzzy pre-continuous function, fuzzy pre*- continuous function, fuzzy pre-compact space and some properties, remarks related to them.
1. Introduction
The concept of fuzzy set was introduced by Zadeh (1965) in classical paper. Chang (1968) introduced the notion of a fuzzy topology. Also in (1992) Chakraborty M.K and T.M.G Ahsanullah Introduction fuzzy topology on fuzzy sets in (1993) Chauldhuri M.K and P.Das Introduced some results on fuzzy topology on fuzzy set.
In (1997) Ganesan Balasubamanian introduction on fuzzy β-compact and fuzzy β- extermally disconnected space Also in (2004) I.M.Hanafy Introduced fuzzy β- compactness and fuzzy β-closed space. Also, in (2007) M.K.uma, EROJA and G.BALASUBAL MANINA Introduced on fuzzy β-compact * spaces and fuzzy filters in (2009) Atallah Th.AL-Ani Introduced Z-compact space. Also in (2011) J.Karnel introduced fuzzy, b-compact and fuzzy b-closed spaces.
2. Fuzzy pre-compact space
In this section we study some definitions, remarks, Propositions and theorems about fuzzy Pre- compact space in fuzzy topological spaces.
Definition 2.1: [1] [2] A collection 
of a
sub set of 
that
is ⊆ p () is
said to be fuzzy topology on if
satisfies the following condition:
1)
∊
2)
If 
,
∊, Then 
∊
3)
If i ∊ , Then
i i∊𝝀
The pair (
, ) is
said to be fuzzy topological, every member of is
called fuzzy open (
-open)
set in and
the complement is called fuzzy ( –closed)
set .
Definition 2.2: [8] Let (,) be a
fuzzy topological spaces a family W of fuzzy sets is pre - open cover of a
fuzzy set 
if
and only if ⊆ ∪ {
:
∊
} and
each member of W is a fuzzy pre- open set. A sub cover of W is a sub family
which is also cover.
Definition 2.3: [8] A fuzzy topological spaces (,) is
fuzzy pre- compact if and only if every fuzzy pre- open cover of has a
finite sub cover.
Definition 2.4: [4] Let is
fuzzy sub set of a fuzzy topological space ( ,
), is
said to be fuzzy pre-open relative to 
if for
every fuzzy pre- open cover {
: 𝝀∊
}
such that is
fuzzy pre-open sets in ( ,)
having finite sub cover.
Definition 2.5: [10] A fuzzy topological space (,) is
said to be fuzzy (pre-
) i.e.
(fuzzy pre- Housdorff) if for each pair of distinct point
, 
of (,),
there exists disjoint fuzzy pre-open sets 
and 
such
that 
and 
Remark 2.6: [8] Every fuzzy open cover is a fuzzy pre-open cover.
Proposition 2.7: [4] Every fuzzy pre-compact is fuzzy compact space.
Proof:
Let ( ,) is
fuzzy pre-compact space
And { : 𝝀 ∊𝚲} is open cover to
∴ { : 𝝀 ∊𝚲} is pre-open cover
to
∵ is
fuzzy pre-compact space.
Such that 
(x)
=max {
(x): 𝝀 ∊𝚲}, i=1, 2, 3… n
∴ is
fuzzy compact space ■
Remark 2.8: [4] Fuzzy compact space need not to be fuzzy pre-compact space Proposition (2.9) [7]
If every fuzzy pre-open subset of a fuzzy topological space ( ,) is a
fuzzy pre-compact, then every subset of is a
fuzzy pre-compact.
Proof:
Obvious ■
Proposition (2.10) [4]
A fuzzy pre-closed subset of fuzzy pre-compact space is fuzzy pre-compact
Proof:
Suppose that ( ,) be a
fuzzy pre-compact space
And be a
fuzzy pre-closed subset of ( ,)
And {
(x): 𝝀 ∊𝚲} is pre-open cover
to
Since 
is a
fuzzy pre-open cover.
Such that max {{(x): 𝝀 ∊𝚲}, 
(x)}
is fuzzy pre-open cover of .
Since ( ,) is fuzzy
pre-compact space.
Hence is
fuzzy pre-compact space ■
Corollary (2.11) [4]
A fuzzy closed subset of a fuzzy pre-compact is fuzzy compact.
Proof:
Suppose that 
(x) =
max {
(x): 𝝀 ∊ 𝚲} be fuzzy pre-open cover to
Suppose that 
(x) = max
{(x) (x)}
∴ 
is
fuzzy open cover to ( ,)
∵ is a
fuzzy pre-compact.
Such that 
(x)
≤max {max (
(x),
(x)}
∴ 
(x)
≤max {
(x)}
Then 
is
fuzzy pre- compact ■
Remark 2.12: [6] A fuzzy pre- closed subset of a fuzzy compact space is needed not to be fuzzy compact.
Theorem 2.13: [8] Let (
,
)
be a fuzzy topological space if and
are
two Fuzzy pre- compact subsets of , then
⋃ is
also fuzzy pre- compact.
Proof :
Suppose that { : 𝝀 ∊𝚲} be a fuzzy pre-
open cover of ⋃
Then max {
(x), 
(x)} ≤
max { (x): 𝝀 ∊𝚲}
Since(x) ≤
max {(x), (x)}
And (x) ≤
max {(x), (x)}
Also {: 𝝀 ∊ 𝚲} is fuzzy pre- open cover of and a
fuzzy pre- open cover of 
Since, and are
two pre- compact sets, then there exists a finite sub cover
(,
, 
,……. 
) and
(x) ≤
max {
(x)}
,i=1,2,3,…..n
Hence (x) ≤
max {(x)}
And max {(x), (x)} ≤
max {
(x)}, k=1,
2, 3…n+m
Thus, ⋃ 
is
fuzzy pre compact ■
Remark 2.14: [8] If and are a
fuzzy pre-compact subsets of a fuzzy topological space ( ,) then ⋂ is
need not to be fuzzy pre-compact space.
Theorem 2.15: [4] Every fuzzy pre- closed off ( ,) is
fuzzy pre-compact if and only if ( ,) is
fuzzy pre- compact.
Proof:
Let { : 𝝀 ∊ 𝚲 } is fuzzy
pre-open cover in
Then 
(x) =
max {
(x): 𝝀 ∊ }
Suppose that 
(x) =
max {
: 𝝀 ∊ }
Then 
is
fuzzy closed.
And 
≤ max
{ (x): 𝝀 ∊𝚲 – { 
}}
Then there exist 
fuzzy
subset finite on 𝚲
– { }
Such that ≤ max
{ (x): 𝝀 ∊ 
}
Then (x) =
max {
,
}
And 
≤ max
{ (x): 𝝀 ∊𝚲 – { }}
Then is
fuzzy pre-compact.
Conversely ←
Let is
fuzzy pre-compact
Suppose that 
is
fuzzy closed set in 
Then is
fuzzy pre-compact ■
Theorem 2.16: [3] A fuzzy topological space ( ,)
is fuzzy pre-compact If and only if for every collection
{
: 𝝀∊ 𝚲
} of fuzzy pre-
closed set of (,)
having the finite intersection property
Min {
(x)} ≠
(x)
Proof:
Suppose that { : 𝝀∊ } be a collection
of fuzzy pre- closed with the finite intersection property
Let min {
(x)} =
(x)
and max {(x)} =
(x)
Since = { : 𝝀∊ 𝚲} is a
collection of fuzzy pre- open set cover of it
follows that there exists a finite subset μ ⊂
𝚲
Such that max {(x) =(x)
Then min {(x)} =
(x)
where 𝚲 ∊ μ
Which the contradiction
And therefore min {(x)} ≠
(x)
Conversely
Obvious
Theorem 2.16: [4] Let is
fuzzy open subset of ( ,
),
then 
is
fuzzy pre- compact if and only if sub
space to .
Proof:
Suppose that is
fuzzy pre-compact to
Suppose that { (x): 𝝀 ∊𝚲} is covering to
Such that is
fuzzy pre- open set in
Thus min {
(x), (x)}
is fuzzy pre- open set in
Then { (x): 𝝀 ∊ } is fuzzy pre- open cover
to
∵ is
fuzzy pre- compact
Then 
(x)
< max {min { (x ) ,
(x) : 𝝀 ∊
}
such that Ì 𝚲
And (x)
< max { (x) : 𝝀 ∊ }
∴ is
fuzzy pre- compact sub space to ■
Theorem 2.17: [4] Every fuzzy pre- compact space in fuzzy Housdorff space is fuzzy pre-closed
Proof:
Suppose that ( ,) is fuzzy
pre- compact in fuzzy Housdorff space.
∴ is
fuzzy compact space
∵ is
fuzzy Housdorff space
∴ is
fuzzy closed
∴ is
fuzzy pre- closed in ■
Proposition 2.18: [4] Let , be
two fuzzy subset of ( , ), ⊂ and is
fuzzy open set of ,
then is
fuzzy pre- compact relative to subspace if
and only if is
fuzzy pre- compact relative to .
Proof:
Suppose that is
fuzzy pre-compact subspace in
∴ is
fuzzy pre-compact relative to
And is
fuzzy pre-compact relative to
Hence is
fuzzy pre-compact in
Conversely
Let is
fuzzy pre-compact in
∴ is
fuzzy pre-compact relative to
Then is
fuzzy pre-compact relative to
Hence is
fuzzy pre-compact in ■
3. Fuzzy pre- continuous and fuzzy pre*-continuous
Definition 3.1: [3] A function f: ( 
) → ( ,
) is
fuzzy continuous (f- continuous) if and only if the inverse image of any fuzzy
open set in is
fuzzy open set in.
Definition 3.2: A function f: ( , ) 
( , ) is
said to be a fuzzy pre- continuous if and only if the invers image of any fuzzy
open set in is
fuzzy pre-open set in.
Definition 3.3: A function f: ( , ) ( , ) is
said to be a fuzzy pre*- continuous if and only if the invers image of any
fuzzy pre- open set in is
fuzzy pre-open set in .
Proposition 3.5: [3] If f: ( , ) ( , ) is
fuzzy continuous function, then f is fuzzy pre*- continuous
Proof:
Suppose that () is a
fuzzy pre-open fuzzy in
Then 
(x) ≤ 
x) and
(x) ≤
x) ≤
x)
Since f is fuzzy continuous
Then x) ≤ 
x) ≤ 
x)
Thus (x) ≤x) ≤x)
Then f is fuzzy pre* - continuous ■
Theorem 3.6: [3] Let f:
be
a function, and then the following are equivalent
1-f is fuzzy
pre*-continuous 2-f (p-cl () ⊆ p (cl(f())
,for every fuzzy set in
Proof:
2 →1Suppose that be a
fuzzy set of
Then p-cl-(f ( ) is
fuzzy pre-closed
By the (1) 
(p-cl
(f () is
pre-closed
And (
(x)) =
(
(x)
Since 
(x) ≤ 
(x)
And 
(x) ≤
(
(x) =
(x)
Hence 
(x) ≤ 
(x)
2←1 suppose that 
be a
fuzzy pre-closed set in
And If (x) =
(x)
Then
(x) ≤
(x)
≤ 
(x) =
(x)
Since (x) ≤
(x)
Then (x) =
(x)
Hence () is
fuzzy pre-closed set in and f
is fuzzy pre*-continuous ■
Corollary 3.7: [3] Let f:
be a
function, then the following are equivalent
1) f is fuzzy pre-continuous
2)
f (p-cl () ⊆ cl-(f ()), for
every fuzzy set in
Proof:
Obvious ■
Proposition 3.8: [3] If f: ( , ) ( ,) is
fuzzy open and fuzzy continuous function and is a
fuzzy pre- compact Then f () is
fuzzy pre- compact space.
Proof:
Obvious ■
Theorem 3.9: [8] The fuzzy pre- continuous image of a fuzzy pre- compact space is fuzzy compact space.
Proof:
Suppose that ( , ) be
a fuzzy pre- compact space
And f: ( ,
) ( ,
) be a
fuzzy pre continuous function
To prove (,) is a
fuzzy compact space
Let { : 𝝀∊ 𝚲} is a fuzzy open
cover of
.
Then {
(x): 𝝀∊𝚲} is fuzzy
pre-open cover of
Since f is fuzzy pre – continuous function a finite sub cover
{
(x)
i=1, 2, 3… n} which covering
Then is a
fuzzy compact space ■
Remark 3.10: [4] The fuzzy continuous image of fuzzy pre-compact need not be a fuzzy pre-compact space.
Theorem 3.11: [3] If a function f: ( , ) ( ,
)
is fuzzy pre*- continuous and is
a fuzzy pre- compact relative to
then
so is f ()
is fuzzy pre- compact.
Proof:
Suppose that { : 𝝀∊
} be a
fuzzy pre- open cover of
Since f is fuzzy pre*- continuous and {
(x): 𝝀∊ 𝚲} is a fuzzy pre-
open set cover of Ѕ () in
Since is a
fuzzy pre- open compact relative to
There is a finite subfamily { (x): 𝝀∊ 𝚲}
Such that 
(x) ≤
max {
(x)} =max { (x): 𝝀∊ 𝚲}
(x) =
(x) ≤
f max {
(x): 𝝀∊ } ≤ max{ (x): 𝝀∊ 𝚲 }
Therefore f ( ) is a
fuzzy pre- compact relative to
Propositions 3.12: [3]
1)
If f: 
is a
fuzzy pre*- open and bijective function and be
fuzzy pre- compact then is a
fuzzy pre- compact.
Proof:
Suppose that { : 𝝀∊
} be a
family of a fuzzy pre- open covering of
Let {
(x): 𝝀∊
} be
a fuzzy pre-open set covering of
Since is
fuzzy pre- compact then there exist a finite family ⊆ 
covers
Such that { (x): 𝝀∊
}
covers
Since f is bijective
Then 
(x) =
(x) =
max { (x): 𝝀∊ } =max { (x): 𝝀∊ }
Hence is a
fuzzy pre- compact ■
2)
Let f: 
be a
fuzzy pre- continuous surjective function and is a
fuzzy pre- closed compact then is a
fuzzy pre- closed compact.
Proof:
Obvious ■
3)
Let if f: be a
fuzzy -pre continues surjective function of a fuzzy pre- compact a space onto
a space then is
fuzzy pre- compact.
Proof:
Obvious ■
4)
Let if f: be a
fuzzy -pre continues bijective function and be a
fuzzy pre- compact space then is
fuzzy Pre- compact.
Proof
Obvious ■
Remark 3.13: [7] The following diagram explains the relationships among the different types of fuzzy continuous function.
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4. Conclusion
It is an interesting exercise to work on fuzzy, pre-continuous and fuzzy pre*-continuous; function of fuzzy pre-compact space in fuzzy topological space similarly other forms of fuzzy pre-open set can be applied to difine different forms of fuzzy pre-compact space.
Acknowledgements
The authors are grateful to the colleges' education department of mathematics. Al- Mustansiriya University, Baghdad, Iraq for its financial support.
References
[1] Chakraborty M.K. and T.M.G .Ahsanullah ˝ Fuzzy topology on fuzzy sets and tolerance topology ˝ fuzzy sets and systems, 45103-1o8 (1992)
[2] Chauldhuri M .K and P. Das ˝ some results on fuzzy topology on fuzzy sets ˝ fuzzy sets and systems, 56pp.331-336(1993).
[3] Ganesan Balasubramanian ˮ on fuzzy β –compact and fuzzy β- extermally disconnected space ˮ vol.33 (1997), No 3,271-277
[4] Neeran Tahir AlKhafaji ˝on certain Types of-compact spaces ˝ (2002)
[5] I.M.Hanafy ˝ fuzzy β- compactness and Fuzzy β- closed spaces ˝ 28(2004), 281-293
[6] Ahmed Ibrahem Nasir ˝ Some Kinde Of strongly compact and pair- wise compact space˝ Baghdad University (2005)
[7] Suaad Gedaan Gasim ˝ on semi –p-compact space ˝ Baghdad University (2006)
[8] Saleem yaseen mageed ˝on fuzzy compact space ˝ AL Mustansirya University (2012)
[9] V.Seenivasan, K, Kamala ˝ fuzzy e-continuity and fuzzy e-open sets ˝ volume x, No, x, (mm201y) pp.1-xx (2014)