Fuzzy pre-continuous and fuzzy pre*-continuous function of fuzzy pre-compact space in fuzzy topological space

 
 
 
  • Abstract
  • Keywords
  • References
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  • 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.


  • Keywords


    Fuzzy Pre-Continuous .Fuzzy Pre*-Continuous; Fuzzy Compact Space in Fuzzy Topological Space.

  • 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)


  • The Format of the IJOPCM, first submission

     

    Fuzzy pre-continuous and fuzzy pre*-continuous function of fuzzy pre-compact space in fuzzy topological space

     

    Munir Abdul Khalik Al-Khafaji*, Marwah Flayyih Hasan

     

    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:

     

    21Suppose 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.

    Fuzzy continuous 
Function
Fuzzy pre -continuous function Fuzzy pre* - continuous function
     

     

     

     

     

     

     

     

     

     

     


    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)

 

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Article ID: 4520
 
DOI: 10.14419/ijams.v3i1.4520




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