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

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.

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.

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