Fuzzy precontinuous and fuzzy pre*continuous function of fuzzy precompact space in fuzzy topological space 

Munir Abdul Khalik AlKhafaji*, Marwah Flayyih Hasan 

Department of Mathematics, College of Education, AlMustansiriya University, Baghdad, Iraq *Corresponding author Email: mnraziz@yahoo.com 
Copyright © 2015 Munir Abdul Khalik AlKhafaji, 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.
The aim of this paper is to introduce and study the notion of a fuzzy precontinuous function, fuzzy pre* continuous function, fuzzy precompact 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 βcompact and fuzzy β extermally disconnected space Also in (2004) I.M.Hanafy Introduced fuzzy β compactness and fuzzy βclosed space. Also, in (2007) M.K.uma, EROJA and G.BALASUBAL MANINA Introduced on fuzzy βcompact * spaces and fuzzy filters in (2009) Atallah Th.ALAni Introduced Zcompact space. Also in (2011) J.Karnel introduced fuzzy, bcompact and fuzzy bclosed spaces.
2. Fuzzy precompact 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 preopen relative to if for every fuzzy pre open cover { : 𝝀∊ } such that is fuzzy preopen 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 preopen sets and such that and
Remark 2.6: [8] Every fuzzy open cover is a fuzzy preopen cover.
Proposition 2.7: [4] Every fuzzy precompact is fuzzy compact space.
Proof:
Let ( ,) is fuzzy precompact space
And { : 𝝀 ∊𝚲} is open cover to
∴ { : 𝝀 ∊𝚲} is preopen cover to
∵ is fuzzy precompact 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 precompact space Proposition (2.9) [7]
If every fuzzy preopen subset of a fuzzy topological space ( ,) is a fuzzy precompact, then every subset of is a fuzzy precompact.
Proof:
Obvious ■
Proposition (2.10) [4]
A fuzzy preclosed subset of fuzzy precompact space is fuzzy precompact
Proof:
Suppose that ( ,) be a fuzzy precompact space
And be a fuzzy preclosed subset of ( ,)
And {(x): 𝝀 ∊𝚲} is preopen cover to
Since is a fuzzy preopen cover.
Such that max {{(x): 𝝀 ∊𝚲}, (x)} is fuzzy preopen cover of .
Since ( ,) is fuzzy precompact space.
Hence is fuzzy precompact space ■
Corollary (2.11) [4]
A fuzzy closed subset of a fuzzy precompact is fuzzy compact.
Proof:
Suppose that (x) = max { (x): 𝝀 ∊ 𝚲} be fuzzy preopen cover to
Suppose that (x) = max {(x) (x)}
∴ is fuzzy open cover to ( ,)
∵ is a fuzzy precompact.
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 precompact subsets of a fuzzy topological space ( ,) then ⋂ is need not to be fuzzy precompact space.
Theorem 2.15: [4] Every fuzzy pre closed off ( ,) is fuzzy precompact if and only if ( ,) is fuzzy pre compact.
Proof:
Let { : 𝝀 ∊ 𝚲 } is fuzzy preopen 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 precompact.
Conversely ←
Let is fuzzy precompact
Suppose that is fuzzy closed set in
Then is fuzzy precompact ■
Theorem 2.16: [3] A fuzzy topological space ( ,) is fuzzy precompact 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 precompact 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 preclosed
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 precompact subspace in
∴ is fuzzy precompact relative to
And is fuzzy precompact relative to
Hence is fuzzy precompact in
Conversely
Let is fuzzy precompact in
∴ is fuzzy precompact relative to
Then is fuzzy precompact relative to
Hence is fuzzy precompact 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 preopen 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 preopen set in .
Proposition 3.5: [3] If f: ( , ) ( , ) is fuzzy continuous function, then f is fuzzy pre* continuous
Proof:
Suppose that () is a fuzzy preopen 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
1f is fuzzy pre*continuous 2f (pcl () ⊆ p (cl(f()) ,for every fuzzy set in
Proof:
2 →1Suppose that be a fuzzy set of
Then pcl(f ( ) is fuzzy preclosed
By the (1) (pcl (f () is preclosed
And ((x)) = ((x)
Since (x) ≤ (x)
And (x) ≤ ((x) =(x)
Hence (x) ≤ (x)
2←1 suppose that be a fuzzy preclosed set in
And If (x) = (x)
Then (x) ≤ (x)
≤ (x) = (x)
Since (x) ≤ (x)
Then (x) = (x)
Hence () is fuzzy preclosed 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 precontinuous
2) f (pcl () ⊆ 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 preopen 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 precompact need not be a fuzzy precompact 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 preopen 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.
4. Conclusion
It is an interesting exercise to work on fuzzy, precontinuous and fuzzy pre*continuous; function of fuzzy precompact space in fuzzy topological space similarly other forms of fuzzy preopen set can be applied to difine different forms of fuzzy precompact space.
Acknowledgements
The authors are grateful to the colleges' education department of mathematics. Al Mustansiriya University, Baghdad, Iraq for its financial support.
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