����������� ������ 55 (2003), 1–13

UDK 515.122.2

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r-FUZZY STRONGLY PREOPEN SETS IN FUZZY TOPOLOGICAL SPACES

Y. C. Kim, A. A. Ramadan and S. E. Abbas

Abstract. We introduce r-fuzzy strongly preopen and r-fuzzy strongly preclosed sets in fuzzy topological spaces in the sense of the definition of Šostak [8] and investigate some of their properties. Fuzzy strongly precontinuous, fuzzy strongly preopen and fuzzy strongly preclosed mappings between fuzzy topological spaces are defined. Their properties and the relationship between these mappings and other mappings introduced previously are investigated.

1. Introduction and preliminaries

A. P. Šostak [8] introduced a fuzzy topology as an extension of Chang’s fuzzy topology [2]. It has been developed in many directions [3,4,7]. B. Krsteska [1] introduced fuzzy strongly preopen and fuzzy strongly preclosed sets in the Chang’s fuzzy topology.

In this paper we define r-fuzzy strongly preopen and r-fuzzy strongly preclosed sets in a fuzzy topological space in view of the definition of Šostak [8] and investigate some of their properties. We show that fuzzy strong precontinuity implies fuzzy precontinuity, but the converse is not true. We obtain some properties of fuzzy strongly precontinuous mappings.

Throughout this paper, let X be a non-empty set, I = [0, 1] and I_{0} = (0, 1]. For α ∈ I, ᾱ(x) = α, for all x ∈ X. All other notations and definitions are standard in the fuzzy set theory.

Definition 1.1. [8] A function τ: I^{X} → I is called a fuzzy topology on X if it satisfies the following conditions: (O1) τ(¯0) = τ(¯1) = 1, (O2) τ(μ_{1} ∧ μ_{2}) ⩾ τ(μ_{1}) ∧ τ(μ_{2}), for any μ_{1}, μ_{2} ∈ I^{X}, (O3) τ(⋁_{i∈Γ} μ_{i}) ⩾ ⋀_{i∈Γ} τ(μ_{i}), for any {μ_{i}}_{i∈Γ} ⊂ I^{X}.

The pair (X, τ) is called a fuzzy topological space (fts, for short).

AMS Subject Classification: 54 A 40

Keywords and phrases: r-fuzzy strongly preopen, r-fuzzy strongly preclosed, fuzzy strong precontinuity, fuzzy strongly preopen mapping, fuzzy strongly preclosed mapping.

Remark 1.2. Let (X, τ) be an fts. Then, for each α ∈ I, τ_{α} = {μ ∈ I^{X} : τ(μ) ⩾ α} is a Chang’s fuzzy topology on X.

Theorem 1.1. [7] Let (X, τ) be an fts. For each r ∈ I_{0}, λ ∈ I^{X}, define an operator C_{τ}: I^{X} × I_{0} → I^{X} as follows

C_{τ} (λ, r) = ⋀{ μ ∈ I^{X} : λ ⩽ μ, τ(¯1 − μ) ⩾ r }.

For each λ, μ ∈ I^{X} and r, s ∈ I_{0} it satisfies the following conditions:

(1) C_{τ} (¯0, r) = ¯0.

(2) λ ⩽ C_{τ} (λ, r).

(3) C_{τ} (λ, r) ∨ C_{τ} (μ, r) = C_{τ} (λ ∨ μ, r).

(4) C_{τ} (λ, r) ⩽ C_{τ} (λ, s) if r ⩽ s.

(5) C_{τ} (C_{τ} (λ, r), r) = C_{τ} (λ, r).

Theorem 1.2. [6] Let (X, τ) be an fts. For each r ∈ I_{0}, λ ∈ I^{X}, define an operator I_{τ}: I^{X} × I_{0} → I^{X} as follows

I_{τ} (λ, r) = ⋁{ μ ∈ I^{X} : λ ⩾ μ, τ(μ) ⩾ r }.

For each λ, μ ∈ I^{X} and r, s ∈ I_{0} it satisfies the following conditions:

(1) I_{τ} (¯1 − λ, r) = ¯1 − C_{τ} (λ, r) and C_{τ} (¯1 − λ, r) = ¯1 − I_{τ} (λ, r).

(2) I_{τ} (¯1, r) = ¯1.

(3) I_{τ} (λ, r) ⩽ λ.

(4) I_{τ} (λ, r) ∧ I_{τ} (μ, r) = I_{τ} (λ ∧ μ, r).

(5) I_{τ} (λ, r) ⩾ I_{τ} (λ, s) if r ⩽ s.

(6) I_{τ} (I_{τ} (λ, r), r) = I_{τ} (λ, r).

Definition 1.2. [6] Let (X, τ) be an fts. For λ ∈ I^{X} and r ∈ I_{0}:

(1) λ is called an r-fuzzy semi-open (r-fso, for short) set if there exists ν ∈ I^{X} with τ(ν) ⩾ r such that ν ⩽ λ ⩽ C_{τ} (ν, r). Equivalently, λ ⩽ C_{τ} (I_{τ} (λ, r), r).

(2) λ is called an r-fuzzy semi-closed (r-fss, for short) set if there exists ν ∈ I^{X} with τ(¯1 − ν) ⩾ r such that I_{τ} (ν, r) ⩽ λ ⩽ ν. Equivalently, I_{τ} (C_{τ} (λ, r), r) ⩽ λ.

Definition 1.3. [6] Let (X, τ) and (Y, τ^{∗}) be fts’s. A mapping f: X → Y is said to be:

(1) fuzzy continuous iff τ^{∗}(μ) ⩽ τ(f^{−1}(μ)) for each μ ∈ I^{Y} .

(2) fuzzy open iff τ^{∗}(f(μ)) ⩾ τ(μ) for each μ ∈ I^{X}.
(3) fuzzy closed iff τ^{∗}(¯1 − f(μ)) ⩾ τ(¯1 − μ) for each μ ∈ I^{X}.

r-fuzzy strongly preopen sets 3

Definition 1.4. [6] Let (X, τ) and (Y, τ^{∗}) be fts’s. A mapping f: X → Y is said to be:

(1) fuzzy semi-continuous iff f^{−1}(μ) is r-fso for each μ ∈ I^{Y}, r ∈ I_{0} with τ^{∗}(μ) ⩾ r.

(2) fuzzy semi-open iff f(μ) is r-fso for each μ ∈ I^{X}, r ∈ I_{0} with τ(μ) ⩾ r.

(3) fuzzy semi-closed iff f(μ) is r-fsc for each μ ∈ I^{X}, r ∈ I_{0} with τ(¯1−μ) ⩾ r.

2. r-fuzzy strongly preopen and r-fuzzy strongly preclosed sets

Definition 2.1. [6] Let (X, τ) be an fts. For λ ∈ I^{X} and r ∈ I_{0}:

(1) λ is called r-fuzzy preopen (r-fpo, for short) iff

λ ⩽ I_{τ} (C_{τ} (λ, r), r).

(2) λ is called r-fuzzy preclosed (r-fpc, for short) iff

C_{τ} (I_{τ} (λ, r), r) ⩽ λ.

(3) λ is called r-fuzzy strongly semi-open (r-fsso, for short) iff

λ ⩽ I_{τ} (C_{τ} (I_{τ} (λ, r), r), r).

(4) λ is called r-fuzzy strongly semi-closed (r-fssc, for short) iff

C_{τ} (I_{τ} (C_{τ} (λ, r), r), r) ⩽ λ.

Definition 2.2. [6] Let (X, τ) be an fts. For λ ∈ I^{X} and r ∈ I_{0}:

(1) the r-fuzzy preinterior of λ, denoted by P I_{τ} (λ, r), is defined by

P I_{τ} (λ, r) = ⋁{ ν ∈ I^{X} : ν ⩽ λ, ν is r-fpo },

(2) the r-fuzzy preclosure of λ, denoted by P C_{τ} (λ, r), is defined by

P C_{τ} (λ, r) = ⋀{ ν ∈ I^{X} : ν ⩾ λ, ν is r-fpc },

(3) the r-fuzzy strongly semi-interior of λ, denoted by SSI_{τ} (λ, r), is defined by

SSI_{τ} (λ, r) = ⋁{ ν ∈ I^{X} : ν ⩽ λ, ν is r-fsso },

(4) the r-fuzzy strongly semi-closure of λ, denoted by SSC_{τ} (λ, r), is defined by

SSC_{τ} (λ, r) = ⋀{ ν ∈ I^{X} : ν ⩾ λ, ν is r-fssc }.

4 Y.C. Kim, A.A. Ramadan, S.E. Abbas

Theorem 2.1. Let (X, τ) be an fts, λ ∈ I^{X} and r ∈ I_{0}. Then

(1) λ ∨ C_{τ} (I_{τ} (λ, r), r) ⩽ P C_{τ} (λ, r),

(2) P I_{τ} (λ, r) ⩽ λ ∧ I_{τ} (C_{τ} (λ, r), r),

(3) I_{τ} (P C_{τ} (λ, r), r) ⩽ I_{τ} (C_{τ} (λ, r), r),

(4) I_{τ} (C_{τ} (I_{τ} (λ, r), r), r) ⩽ I_{τ} (P C_{τ} (λ, r), r),

(5) P C_{τ} (¯1 − λ, r) = ¯1 − P I_{τ} (λ, r) and P I_{τ} (¯1 − λ, r) = ¯1 − P C_{τ} (λ, r). Proof. (1) Since P C_{τ} (λ, r) is an r-fpc set, we have

C_{τ} (I_{τ} (λ, r), r) ⩽ C_{τ} (I_{τ} (P C_{τ} (λ, r), r), r) ⩽ P C_{τ} (λ, r).

Thus, λ ∨ C_{τ} (I_{τ} (λ, r), r) ⩽ P C_{τ} (λ, r).

(2) It can be shown as (1).

(3) It follows from the relation P C_{τ} (λ, r) ⩽ C_{τ} (λ, r).

(4) From (1) we have

I_{τ} (P C_{τ} (λ, r), r) ⩾ I_{τ} (λ ∨ C_{τ} (I_{τ} (λ, r), r), r) ⩾ I_{τ} (C_{τ} (I_{τ} (λ, r), r), r).

(5) Straightforward.

Definition 2.3. Let (X, τ) be an fts. For λ ∈ I^{X} and r ∈ I_{0}:

(1) λ is called r-fuzzy strongly preopen (r-fspo, for short) iff

λ ⩽ I_{τ} (P C_{τ} (λ, r), r).

(2) λ is called r-fuzzy strongly preclosed (r-fspc, for short) iff

C_{τ} (P I_{τ} (λ, r), r) ⩽ λ.

(3) The r-fuzzy strong preinterior of λ, denoted by SP I_{τ} (λ, r), is defined by

SP I_{τ} (λ, r) = ⋁{ ν ∈ I^{X} : ν ⩽ λ, ν is r-fspo }.

(4) The r-fuzzy strong preclosure of λ, denoted by SP C_{τ} (λ, r), is defined by

SP C_{τ} (λ, r) = ⋀{ ν ∈ I^{X} : ν ⩾ λ, ν is r-fspc }.

Theorem 2.2. Let (X, τ) be an fts, λ ∈ I^{X} and r ∈ I_{0}. Then

(1) if τ(λ) ⩾ r, then λ is an r-fspo set;

(2) if λ is r-fsso, then λ is r-fspo;

(3) if l is r-fspo, then λ is r-fpo. Proof. It follows from Theorem 2.1.

r-fuzzy strongly preopen sets 5

The following examples show that the converses in the above theorem are not true in general.

Example 2.1. Let X = {a, b, c}. Define fuzzy sets λ_{1}, λ_{2}, μ ∈ I^{X} as follows:

λ_{1}(a) = 0.3,  λ_{1}(b) = 0.2,  λ_{1}(c) = 0.7,
λ_{2}(a) = 0.8,  λ_{2}(b) = 0.8,  λ_{2}(c) = 0.4,
μ(a) = 0.8,  μ(b) = 0.7,  μ(c) = 0.6.

Define a fuzzy topology τ: I^{X} → I as follows:

τ(λ) = \{ \begin{matrix}1, & if λ = ¯0 or ¯1, \\ \frac{1}{3}, & if λ = λ_{1}, \\ \frac{1}{2}, & if λ = λ_{2}, \\ \frac{2}{3}, & if λ = λ_{1} ∨ λ_{2}, \\ \frac{2}{3}, & if λ = λ_{1} ∧ λ_{2}, \\ 0, & otherwise. \end{matrix}

For the fts (X, τ) with 0 < r ⩽ \frac{1}{3}, μ is an r-fspo set from

μ ⩽ I_{τ} (P C_{τ} (μ, r), r) = ¯1.

For 0 < r ⩽ \frac{1}{3}, μ is neither r-fsso nor τ(μ) ⩾ r from

μ ⩽̸ I_{τ} (C_{τ} (I_{τ} (μ, r), r), r) = λ_{1} ∧ λ_{2}.

Example 2.2. Let X be a non-empty set. We define fuzzy topology τ: I^{X} → I as follows:

τ(λ) = \{ \begin{matrix}1, & if λ = ¯0 or ¯1, \\ \frac{2}{3}, & if λ = \overline{0.2}, \\ \frac{1}{3}, & if λ = \overline{0.4}, \\ \frac{1}{2}, & if λ = \overline{0.5}, \\ \frac{3}{4}, & if λ = \overline{0.6}, \\ 0, & otherwise. \end{matrix}

For 0 < r ⩽ \frac{1}{3}, \overline{0.7} is an r-fpo set but it is not r-fspo, from the following:

\overline{0.7} ⩽ I_{τ} (C_{τ} (\overline{0.7}, r), r) = ¯1,
\overline{0.7} > I_{τ} (P C_{τ} (\overline{0.7}, r), r) = I_{τ} (\overline{0.7}, r) = \overline{0.4}.

For \frac{1}{3} < r ⩽ \frac{2}{3}, \overline{0.3} is r-fso but it is not r-fspo, from the following:

\overline{0.3} ⩽ C_{τ} (I_{τ} (\overline{0.3}, r), r) = \overline{0.4},
\overline{0.3} > I_{τ} (P C_{τ} (\overline{0.3}, r), r) = I_{τ} (\overline{0.4}, r) = \overline{0.2}.

Remark 2.1. From Example 2.1, for 0 < r ⩽ \frac{1}{3}, μ is an r-fspo set but not an r-fso set because

μ ⩽̸ C_{τ} (I_{τ} (μ, r), r) = ¯1 − λ_{1} ∧ λ_{2}.

Also, in Example 2.2, \overline{0.3} is an r-fso but not r-fspo set for \frac{1}{3} < r ⩽ \frac{2}{3}. From the above discussion there is no relation between the concepts of r-fso sets and r-fspo sets.

Theorem 2.3. Let (X, τ) be an fts and r ∈ I_{0}.

(1) Any union of r-fspo sets is an r-fspo set.

(2) Any intersection of r-fspc sets is an r-fspc set.

Proof. (1) Let {λ_{α} : α ∈ Γ} be a family of r-fspo sets. For each α ∈ Γ, λ_{α} ⩽ I_{τ} (P C_{τ} (λ_{α}, r), r). Hence we have

⋁_{α∈Γ} λ_{α} ⩽ ⋁_{α∈Γ} I_{τ} (P C_{τ} (λ_{α}, r), r) ⩽ I_{τ} (P C_{τ} ( ⋁_{α∈Γ} λ_{α}, r), r).

So, ⋁_{α∈Γ} λ_{α} is an r-fspo set.

(2) It is easily proved in the same manner.

Remark 2.2. The intersection of two r-fspo sets need not be an r-fspo set. And the union of two r-fspc sets need not be an r-fspc set. We will show it in the next example.

Example 2.3. Consider the fts (X, τ) form Example 2.1. The fuzzy set ρ defined as:

ρ(a) = 0.4, ρ(b) = 0.2, ρ(c) = 0.8,

is a \frac{1}{3}-fspo set, but λ_{2} ∧ ρ is not a \frac{1}{3}-fspo set. Also, (¯1 − λ_{2}) ∨ (¯1 − ρ) is not a \frac{1}{3}-fspc set in (X, τ).

Theorem 2.4. Let (X, τ) be an fts. For λ ∈ I^{X} and r ∈ I_{0} the following statements hold:

(1) C_{τ} (λ, r) is r-fspc.

(2) λ is r-fspo iff λ = SP I_{τ} (λ, r).

(3) λ is r-fspc iff λ = SP C_{τ} (λ, r).

(4) I_{τ} (λ, r) ⩽ SP I_{τ} (λ, r) ⩽ P I_{τ} (λ, r) ⩽ λ ⩽ P C_{τ} (λ, r) ⩽ SP C_{τ} (λ, r) ⩽ C_{τ} (λ, r).

(5) SP I_{τ} (¯1 − λ, r) = ¯1 − SP C_{τ} (λ, r) and SP C_{τ} (¯1 − λ, r) = ¯1 − SP I_{τ} (λ, r).

(6) C_{τ} (SP C_{τ} (λ, r), r) = SP C_{τ} (C_{τ} (λ, r), r) = C_{τ} (λ, r).

Proof. (1), (2), (3) and (4) follow from the definitions.

(5) For λ ∈ I^{X}, r ∈ I_{0} we have the following:

¯1 − SP I_{τ} (λ, r) = ¯1 − ⋁{ ν : ν ⩽ λ, ν is r-fspo }
= ⋀{ ¯1 − ν : ¯1 − λ ⩽ ¯1 − ν, ¯1 − ν is r-fspc }
= SP C_{τ} (¯1 − λ, r).

r-fuzzy strongly preopen sets 7

(6) From (1) and (3), SP C_{τ} (C_{τ} (λ, r), r) = C_{τ} (λ, r). We only show that C_{τ} (SP C_{τ} (λ, r), r) = C_{τ} (λ, r).

Since λ ⩽ SP C_{τ} (λ, r), C_{τ} (λ, r) ⩽ C_{τ} (SP C_{τ} (λ, r), r). Suppose that

C_{τ} (λ, r) ⩾̸ C_{τ} (SP C_{τ} (λ, r), r).

There exist x ∈ X and r ∈ I_{0} such that

C_{τ} (λ, r)(x) < C_{τ} (SP C_{τ} (λ, r), r)(x).

By the definition of C_{τ}, there exists ρ ∈ I^{X} with λ ⩽ ρ and τ(¯1 − ρ) ⩾ r such that

C_{τ} (SP C_{τ} (λ, r), r)(x) > ρ(x) ⩾ C_{τ} (λ, r)(x).

On the other hand, since ρ = C_{τ} (ρ, r), λ ⩽ ρ implies

SP C_{τ} (λ, r) ⩽ SP C_{τ} (ρ, r) = SP C_{τ} (C_{τ} (ρ, r), r) = C_{τ} (ρ, r) = ρ.

Thus,

C_{τ} (SP C_{τ} (λ, r), r) ⩽ ρ.

It is a contradiction. Hence, C_{τ} (SP C_{τ} (λ, r), r) ⩽ C_{τ} (λ, r).

3. Fuzzy strong precontinuity Definition 3.1. Let (X, τ) and (Y, τ^{∗}) be fts’s and f: X → Y be a mapping.

Then, f is called fuzzy strongly precontinuous (resp. fuzzy strongly semi-continuous, fuzzy precontinuous) if f^{−1}(μ) is an r-fspo (resp. r-fsso, r-fpo) set in X for each μ ∈ I^{Y} and r ∈ I_{0} with τ^{∗}(μ) ⩾ r.

The implications contained in the following diagram are true.

fuzzy continuity  fuzzy strong semi-continuity
⇓ ⇓

fuzzy strong precontinuity ⇓ fuzzy precontinuity

The following examples show that the reverse may not be true in general.

Example 3.1. Let X = {a, b}. Define λ_{1}, λ_{2}, λ_{3}, λ_{4} ∈ I^{X} as follows:

λ_{1}(a) = 0.2, λ_{1}(b) = 0.4, λ_{2}(a) = 0.6, λ_{2}(b) = 0.5;
λ_{3}(a) = 0.3, λ_{3}(b) = 0.4, λ_{4}(a) = 0.6, λ_{4}(b) = 0.7.

Let τ, τ^{∗}: I^{X} → I be fuzzy topologies on X defined as:

τ(λ) = \{ \begin{matrix}1, & if λ = ¯0 or ¯1, \\ \frac{1}{2}, & if λ = λ_{1} or \overline{0.4}, \\ \frac{2}{3}, & if λ = λ_{2} or \overline{0.7}, \\ 1, & otherwise. \end{matrix} τ^{∗}(λ) = \{ \begin{matrix}1, & if λ = ¯0 or ¯1, \\ \frac{1}{2}, & if λ = λ_{3}, \\ \frac{1}{2}, & if λ = λ_{4}, \\ 0, & otherwise. \end{matrix}

8 Y.C. Kim, A.A. Ramadan, S.E. Abbas

Then the identity mapping id_{X}: (X, τ) → (X, τ^{∗}) is fuzzy precontinuous but not fuzzy strongly precontinuous because: for 0 < r ⩽ \frac{1}{2}, λ_{3} is r-fpo and λ_{4} is r-fpo but not r-fspo because

λ_{3} ⩽ I_{τ} (C_{τ} (λ_{3}, r), r) = I(τ)(¯1 − λ_{2}, r) = \overline{0.4},
λ_{4} ⩽ I_{τ} (C_{τ} (λ_{4}, r), r) = ¯1,
λ_{4} > I_{τ} (P C_{τ} (λ_{4}, r), r) = I_{τ} (λ_{4}, r) = λ_{2}.

Example 3.2. We consider Example 2.1 and put

τ^{∗}(λ) = \{ \begin{matrix}1, & if λ = ¯0 or ¯1, \\ \frac{1}{3}, & if λ = μ, \\ 0, & otherwise. \end{matrix}

For fts (X, τ^{∗}) the identity mapping id_{X}: (X, τ) → (X, τ^{∗}) is fuzzy strongly precontinuous but it is neither fuzzy continuous nor fuzzy strongly semi-continuous.

Theorem 3.1. Let (X, τ) and (Y, τ^{∗}) be fts’s and f: X → Y. The following statements are equivalent:

(1) The mapping f is fuzzy strongly precontinuous.

(2) f^{−1}(μ) is r-fspc in X for each μ ∈ I^{Y}, r ∈ I_{0} with τ^{∗}(¯1 − μ) ⩾ r.

(3) f(SP C_{τ} (λ, r)) ⩽ C_{τ^{∗}}(f(λ), r), ∀λ ∈ I^{X}, r ∈ I_{0}.

(4) SP C_{τ} (f^{−1}(μ), r) ⩽ f^{−1}(C_{τ^{∗}}(μ, r)), ∀μ ∈ I^{Y}, r ∈ I_{0}.

(5) f^{−1}(I_{τ^{∗}}(μ, r)) ⩽ SP I_{τ} (f^{−1}(μ), r), ∀μ ∈ I^{Y}, r ∈ I_{0}.

Proof. (1) ⟺ (2). Clear.

(1) ⟹ (3). For all λ ∈ I^{X}, r ∈ I_{0}, since τ^{∗}(¯1 − C_{τ^{∗}}(f(λ), r)) ⩾ r from the definition of C_{τ^{∗}}, and Definition 1.1 (O3),

f^{−1}(¯1 − C_{τ^{∗}}(f(λ), r)) = ¯1 − f^{−1}(C_{τ^{∗}}(f(λ), r))

is r-fspo. By Definition 2.3.(2), f^{−1}(C_{τ^{∗}}(f(λ), r)) is an r-fspc set in X. Since

λ ⩽ f^{−1}(f(λ)) ⩽ f^{−1}(C_{τ^{∗}}(f(λ), r)),

we have SP C_{τ} (λ, r) ⩽ f^{−1}(C_{τ^{∗}}(f(λ), r)). Hence

f(SP C_{τ} (λ, r)) ⩽ f(f^{−1}(C_{τ^{∗}}(f(λ), r))) ⩽ C_{τ^{∗}}(f(λ), r).
(3) ⟹ (4). For μ ∈ I^{Y}, r ∈ I_{0}, let λ = f^{−1}(μ). By (3)
f(SP C_{τ} (f^{−1}(μ), r)) ⩽ C_{τ^{∗}}(f(f^{−1}(μ)), r) ⩽ C_{τ^{∗}}(μ, r).

It implies SP C_{τ} (f^{−1}(μ), r) ⩽ f^{−1}(C_{τ^{∗}}(μ, r)).

r-fuzzy strongly preopen sets 9

(4) ⟹ (5). For μ ∈ I^{Y}, r ∈ I_{0}, by (4) we have
f^{−1}(C_{τ^{∗}}(¯1 − μ, r)) ⩾ SP C_{τ} (f^{−1}(¯1 − μ), r) = SP C_{τ} (¯1 − f^{−1}(μ), r).

Then by Theorem 2.4.(5) we have

f^{−1}(C_{τ^{∗}}(¯1 − μ, r)) ⩾ ¯1 − SP I_{τ} (f^{−1}(μ), r),
¯1 − f^{−1}(¯1 − I_{τ^{∗}}(μ, r)) ⩽ SP I_{τ} (f^{−1}(μ), r),
f^{−1}(I_{τ^{∗}}(μ, r)) ⩽ SP I_{τ} (f^{−1}(μ), r).

(5) ⟹ (1). For each μ ∈ I^{Y}, r ∈ I_{0} with τ^{∗}(μ) ⩾ r, since I_{τ^{∗}}(μ, r) = μ,

f^{−1}(μ) = f^{−1}(I_{τ^{∗}}(μ, r)) ⩽ SP I_{τ} (f^{−1}(μ), r).

Hence, by definition of SP I_{τ} (f^{−1}(μ), r), f^{−1}(μ) = SP I_{τ} (f^{−1}(μ), r). By Theorem 2.4.(2), f^{−1}(μ) is r-fspo. Therefore f is a fuzzy strongly precontinuous mapping.

Theorem 3.2. Let (X, τ) and (Y, τ^{∗}) be fts’s and f: X → Y. The following statements are equivalent:

(1) The mapping f is fuzzy strongly precontinuous.

(2) C_{τ} (P I_{τ} (f^{−1}(μ), r), r) ⩽ f^{−1}(C_{τ^{∗}}(μ, r)), ∀μ ∈ I^{Y}, r ∈ I_{0}.

(3) f(C_{τ} (P I_{τ} (λ, r), r)) ⩽ C_{τ^{∗}}(f(λ), r), ∀λ ∈ I^{X}, r ∈ I_{0}.

Proof. The proof is standard and therefore omitted.

Theorem 3.3. Let (X, τ) and (Y, τ^{∗}) be fts’s and f: X → Y. The following statements are equivalent:

(1) The mapping f is fuzzy precontinuous.

(2) f^{−1}(μ) is r-fpc in X for each μ ∈ I^{Y}, r ∈ I_{0} with τ^{∗}(¯1 − μ) ⩾ r.

(3) f(P C_{τ} (λ, r)) ⩽ C_{τ^{∗}}(f(λ), r), ∀λ ∈ I^{X}, r ∈ I_{0}.

(4) P C_{τ} (f^{−1}(μ), r) ⩽ f^{−1}(C_{τ^{∗}}(μ, r)), ∀μ ∈ I^{Y}, r ∈ I_{0}.

(5) f^{−1}(I_{τ^{∗}}(μ, r)) ⩽ P I_{τ} (f^{−1}(μ), r), ∀μ ∈ I^{Y}, r ∈ I_{0}.

Proof. Similar to the proof of Theorem 3.1.

Theorem 3.4. Let (X, τ) and (Y, τ^{∗}) be fts’s and f: X → Y be a bijective mapping. The following statements are equivalent:

(1) The mapping f is fuzzy strongly precontinuous.

(2) I_{τ^{∗}}(f(λ), r) ⩽ f(SP I_{τ} (λ, r)), ∀λ ∈ I^{X}, r ∈ I_{0}.

Proof. (1) ⟹ (2). Let f be a fuzzy strongly precontinuous mapping and λ ∈ I^{X}, r ∈ I_{0}. Then, f^{−1}(I_{τ^{∗}}(f(λ), r)) is an r-fspo set in X. By Theorem 3.1.(5), and the fact that f is injective, we have

f^{−1}(I_{τ^{∗}}(f(λ), r)) ⩽ SP I_{τ} (f^{−1}(f(λ)), r) = SP I_{τ} (λ, r).

Since f is surjective, we have

I_{τ^{∗}}(f(λ), r) = f(f^{−1}(I_{τ^{∗}}(f(λ), r))) ⩽ f(SP I_{τ} (λ, r)).

(2) ⟹ (1). Let μ ∈ I^{Y}, r ∈ I_{0} with τ^{∗}(μ) ⩾ r. Then I_{τ^{∗}}(μ, r) = μ and by (2) we have

μ = I_{τ^{∗}}(μ, r) ⩽ f(SP I_{τ} (f^{−1}(μ), r)).

Since f is injective we have

f^{−1}(μ) ⩽ f^{−1}f(SP I_{τ} (f^{−1}(μ), r)) = SP I_{τ} (f^{−1}(μ), r).

Then, by the definition of SP I_{τ} (f^{−1}(μ), r) we have f^{−1}(μ) = SP I_{τ} (f^{−1}(μ), r). By Theorem 2.4.(2) we have f^{−1}(μ) is an r-fspo set in X. Thus, f is a fuzzy strongly precontinuous mapping.

4. Fuzzy strongly preopen and fuzzy strongly preclosed mappings

Definition 4.1. Let (X, τ) and (Y, τ^{∗}) be fts’s and f: X → Y be a mapping. Then f is called:

(1) fuzzy strongly preopen (resp. fuzzy strongly semi-open, fuzzy preopen) if f(λ) is an r-fspo (resp. r-fsso, r-fpo) set in Y for each λ ∈ I^{X}, r ∈ I_{0} with τ(λ) ⩾ r.

(2) fuzzy strongly preclosed (resp. fuzzy strongly semi-closed, fuzzy preclosed) if f(λ) is an r-fspc (resp. r-fssc, r-fpc) set in Y for each λ ∈ I^{X}, r ∈ I_{0} with τ(¯1 − λ) ⩾ r.

The implications contained in the following diagram are true.

fuzzy open (closed)  fuzzy strongly semi-open (semi-closed)
⇓ ⇓

fuzzy strongly preopen (preclosed) ⇓ fuzzy preopen (preclosed)

The following examples show that the reverse may not be true.

Example 4.1. In Example 3.1, the identity mapping id_{X}: (X, τ^{∗}) → (X, τ) is a fuzzy preopen mapping but not a fuzzy strongly preopen mapping.

Example 4.2. In Example 3.2, the identity mapping id_{X}: (X, τ^{∗}) → (X, τ) is a fuzzy strongly preopen mapping but it is neither fuzzy open nor fuzzy strongly semi-open mapping.

r-fuzzy strongly preopen sets 11

Theorem 4.1. Let f: (X, τ) → (Y, τ^{∗}) be a mapping. Then the following statements are equivalent:

(1) f is a fuzzy strongly preopen mapping.

(2) f(I_{τ} (λ, r)) ⩽ SP I_{τ^{∗}}(f(λ), r) for each λ ∈ I^{X}, r ∈ I_{0}.

(3) I_{τ} (f^{−1}(μ), r) ⩽ f^{−1}(SP I_{τ^{∗}}(μ, r)) for each μ ∈ I^{Y}, r ∈ I_{0}. Proof. (1) ⟹ (2). For all λ ∈ I^{X}, r ∈ I_{0}, since τ(I_{τ} (λ, r)) ⩾ r, f(I_{τ} (λ, r))

is r-(τ^{∗}, τ^{∗})-fspo. From Theorem 2.4.(2),

f(I_{τ} (λ, r)) = SP I_{τ^{∗}}(f(I_{τ} (λ, r), r)) ⩽ SP I_{τ^{∗}}(f(λ), r).

(2) ⟹ (1). For all λ ∈ I^{X}, r ∈ I_{0}, with τ(λ) ⩾ r we have I_{τ} (λ, r) = λ. From (2), we have

f(λ) = f(I_{τ} (λ, r)) ⩽ SP I_{τ^{∗}}(f(λ), r) ⩽ f(λ).

Then, f(λ) = SP I_{τ^{∗}}(f(λ), r), and by Theorem 2.4.(2) we have that f(λ) is an r-fspo set in Y. Therefore f is a fuzzy strongly preopen mapping.

(2) ⟹ (3). For all μ ∈ I^{Y}, r ∈ I_{0}, by (2) we have

f(I_{τ} (f^{−1}(μ), r)) ⩽ SP I_{τ^{∗}}(ff^{−1}(μ), r) ⩽ SP I_{τ^{∗}}(μ, r).

It implies that I_{τ} (f^{−1}(μ), r) ⩽ f^{−1}f(I_{τ} (f^{−1}(μ), r)) ⩽ f^{−1}(SP I_{τ^{∗}}(μ, r)). (3) ⟹ (2). For all λ ∈ I^{X}, r ∈ I_{0}, by (3) we have

I_{τ} (λ, r) ⩽ I_{τ} (f^{−1}(f(λ)), r) ⩽ f^{−1}(SP I_{τ^{∗}}(f(λ), r)).

It implies that f(I_{τ} (λ, r)) ⩽ f(f^{−1}(SP I_{τ^{∗}}(f(λ), r))) ⩽ SP I_{τ^{∗}}(f(λ), r).

Theorem 4.2. Let f: (X, τ) → (Y, τ^{∗}) be a mapping. Then the following statements are equivalent:

(1) f is a fuzzy strongly preclosed mapping.

(2) SP C_{τ^{∗}}(f(λ), r) ⩽ f(C_{τ} (λ, r)) for each λ ∈ I^{X}, r ∈ I_{0}. Proof. Similar to the proof of Theorem 4.1.

Theorem 4.3. Let f: (X, τ) → (Y, τ^{∗}) be a bijective mapping. Then the following statements are equivalent:

(1) f is a fuzzy strongly preclosed mapping.

(2) f^{−1}(SP C_{τ^{∗}}(μ, r)) ⩽ C_{τ} (f^{−1}(μ), r) for each μ ∈ I^{Y}, r ∈ I_{0}. Proof. (1) ⟹ (2). For all μ ∈ I^{Y}, r ∈ I_{0}, since τ(¯1 − C_{τ} (f^{−1}(μ), r)) ⩾ r,

from (1) we have f(C_{τ} (f^{−1}(μ), r)) is an r-fspc set in Y. Then by Theorem 2.4.(3) we have

f(C_{τ} (f^{−1}(μ), r)) = SP C_{τ^{∗}}(f(C_{τ} (f^{−1}(μ), r)), r) ⩾ SP C_{τ^{∗}}(ff^{−1}(μ), r).

By the surjectivity of f we have

f(C_{τ} (f^{−1}(μ), r)) ⩾ SP C_{τ^{∗}}(μ, r).

Also, by injectivity of f we have

f^{−1}(SP C_{τ} (μ, r)) ⩽ f^{−1}(f(C_{τ} (f^{−1}(μ), r))) = C_{τ} (f^{−1}(μ), r).
(2) ⟹ (1). For all λ ∈ I^{X}, r ∈ I_{0} with τ(¯1 − λ) ⩾ r, from (2) we have
f^{−1}(SP C_{τ^{∗}}(f(λ), r)) ⩽ C_{τ} (f^{−1}(f(λ)), r).

By the injectivity of f we have

f^{−1}(SP C_{τ^{∗}}(f(λ), r)) ⩽ C_{τ} (λ, r) = λ

and by surjectivity of f we have

SP C_{τ^{∗}}(f(λ), r) ⩽ f(λ) ⩽ SP C_{τ^{∗}}(f(λ), r).

Then f(λ) = SP C_{τ^{∗}}(f(λ), r). Hence, f(λ) is an r-fspc set in Y. So, f is a fuzzy strongly preclosed mappimng.

Theorem 4.4. Let f: (X, τ) → (Y, τ^{∗}) be a mapping. Then f is fuzzy strongly preopen iff for each ν ∈ I^{Y} and each λ ∈ I^{X}, r ∈ I_{0} with τ(¯1 − λ) ⩾ r, when f^{−1}(ν) ⩽ λ, there exists an r-fspc set μ in Y such that ν ⩽ μ and f^{−1}(μ) ⩽ λ.

Proof. Suppose that f is a fuzzy strongly preopen mapping, ν ∈ I^{Y} and λ ∈ I^{X}, r ∈ I_{0} with τ(¯1 − λ) ⩾ r such that f^{−1}(ν) ⩽ λ. Then, f(¯1 − λ) ⩽ f(f^{−1}(¯1 − ν)) ⩽ ¯1 − ν. Since τ(¯1 − λ) ⩾ r and f is a fuzzy strongly preopen mapping, then f(¯1 − λ) is an r-fspo set in Y. So,

f(¯1 − λ) = SP I_{τ^{∗}}(f(¯1 − λ), r) ⩽ SP I_{τ^{∗}}(¯1 − ν, r),
¯1 − λ ⩽ f^{−1}(f(¯1 − λ)) ⩽ f^{−1}(SP I_{τ^{∗}}(¯1 − ν, r)),
¯1 − f^{−1}(SP I_{τ^{∗}}(¯1 − ν, r)) = f^{−1}(¯1 − SP I_{τ^{∗}}(¯1 − ν, r)) ⩽ λ,
f^{−1}(SP C_{τ^{∗}}(ν, r)) ⩽ λ.

Let μ = SP C_{τ^{∗}}(ν, r). Then, μ is an r-fspc set in Y, ν ⩽ SP C_{τ^{∗}}(ν, r) = μ and f^{−1}(μ) ⩽ λ.

Conversely, for all η ∈ I^{X}, r ∈ I_{0} with τ(η) ⩾ r, take λ = ¯1 − η ∈ I^{X} and ν = ¯1 − f(η) ∈ I^{Y}. Then, τ(¯1 − λ) ⩾ r and since η ⩽ f^{−1}f(η) we have

¯1 − f^{−1}(f(η)) = f^{−1}(¯1 − f(η)) ⩽ ¯1 − η,

which implies that f^{−1}(ν) ⩽ λ. Then there exists an r-fspc set in Y such that ν = ¯1 − f(η) ⩽ μ and f^{−1}(μ) ⩽ ¯1 − η. Then, ¯1 − μ = f(η). But μ is an r-fspc set in Y and so ¯1 − μ = f(η) is an r-fspo set in Y. Hence, f is a fuzzy strongly preopen mapping.

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(received 25.09.2000, in revised form 06.06.2003)

Y.C. Kim, Dept. of Mathematics, Kangnung National University, Kangwondo, Korea A.A. Ramadan, Dept. of Mathematics, Faculty of Science, Beni-Suef, Egypt S.E. Abbas, Dept. of Mathematics, Faculty of Science, Sohag, Egypt