SOMEWHAT FUZZY γ-IRRESOLUTE CONTINUOUS MAPPINGS

Abstract

We define and characterize a somewhat fuzzy ${\gamma}$-irresolute continuous mapping and a somewhat fuzzy irresolute ${\gamma}$-open mapping on a fuzzy topological space.

1. Introduction

The concept of fuzzy γ-continuous mappings on a fuzzy topological space was introduced and studied by I. M. Hanafy in [2]. Also, the concept of fuzzy γ-irresolute continuous mappings on a fuzzy topological space were introduced and studied by Y. B. Im et al. in [8] and fuzzy irresolute γ-open mappings on a fuzzy topological space was introduced and studied by Y. B. Im in [3].

Recently, somewhat fuzzy γ-continuous mappings on a fuzzy topological space were introduced and studied by G. Thangaraj and V. Seenivasan in [9].

In this paper, we define and characterize a somewhat fuzzy γ-irresolute con-tinuous mapping and a somewhat fuzzy irresolute γ-open mapping which are stronger than a somewhat fuzzy γ-continuous mapping and a somewhat fuzzy γ-open mapping respectively. Besides, some interesting properties of those map-pings are also given.

 

2. Preliminaries

A fuzzy set μ on a fuzzy topological space X is called fuzzy γ-open if μ ≤ ClIntμ ∨ IntClμ and μ is called fuzzy γ-closed if μc is a fuzzy γ-open set on X.

A mapping f : X → Y is called fuzzy γ-continuous if f−1(ν) is a fuzzy γ-open set on X for any fuzzy open set ν on Y and a mapping f : X → Y is called fuzzy γ-open if f(μ) is a fuzzy γ-open set on Y for any fuzzy open set μ on X. It is clear that every fuzzy continuous mapping is a fuzzy γ-continuous mapping. And every fuzzy open mapping is a fuzzy γ-open mapping from the above definitions. But the converses are not true in general [2].

A mapping f : X → Y is called fuzzy γ-irresolute continuous if f−1(ν) is a fuzzy γ-open set on X for any fuzzy γ-open set ν on Y and a mapping f : X → Y is called fuzzy irresolute γ-open if f(μ) is a fuzzy γ-open set on Y for any fuzzy γ-open set μ on X. It is clear that every fuzzy γ-irresolute continuous mapping is a fuzzy γ-continuous mapping. And every fuzzy irresolute γ-open mapping is a fuzzy open mapping from the above definitions. But the converses are not true in general [8] and [3].

A mapping f : X → Y is called somewhat fuzzy γ-continuous if there exists a fuzzy γ-open set μ ≠ 0X on X such that μ ≤ f−1(ν)≠ 0X for any fuzzy open set ν on Y . It is clear that every fuzzy γ-continuous mapping is a somewhat fuzzy γ-continuous mapping. But the converse is not true in general.

A mapping f : X → Y is called somewhat fuzzy γ-open if there exists a fuzzy γ-open set ν≠ 0Y on Y such that ν ≤ f(μ)≠0Y for any fuzzy open set μ on X. Every fuzzy open mapping is a somewhat fuzzy γ-open mapping but the converse is not true in general [9].

 

3. Somewhat fuzzy γ-irresolute continuous mappings

In this section, we introduce a somewhat fuzzy γ-irresolute continuous map-ping and a somewhat fuzzy irresolute γ-open mapping which are stronger than a somewhat fuzzy γ-continuous mapping and a somewhat fuzzy γ-open mapping respectively. And we characterize a somewhat fuzzy γ-irresolute continuous mapping and a somewhat fuzzy irresolute γ-open mapping.

Definition 3.1. A mapping f : X → Y is called somewhat fuzzy γ-irresolute continuous if there exists a fuzzy γ-open set μ≠ 0X on X such that μ ≤ f−1(ν) for any fuzzy γ-open set ν≠ 0Y on Y.

It is clear that every fuzzy γ-irresolute continuous mapping is a somewhat fuzzy γ-irresolute continuous mapping. And every somewhat fuzzy γ-irresolute continuous mapping is a fuzzy γ-continuous mapping from the above definitions. But the converses are not true in general as the following examples show.

Example 3.2. Let μ1, μ2 and μ3 be fuzzy sets on X = {a, b, c} and let ν1, ν2 and ν3 be fuzzy sets on Y = {x, y, z} with

Let be fuzzy topologies on X and let τ∗ = {0Y,ν1, ν2, 1Y} be fuzzy topologies on Y. Consider the mapping f : (X, τ) → (Y, τ∗) defined by f(a) = y, f(b) = y and f(c) = y. Then we have μ1 ≤ f−1(ν1) = μ2, f−1(ν2) = μ3 and μ1 ≤ f−1(ν3) = μ2. Since μ1 is a fuzzy γ-open set on (X, τ), f is somewhat fuzzy γ-irresolute continuous. But f−1(ν1) = μ2 and f−1(ν3) = μ2 are not fuzzy γ-open sets on (X, τ). Hence f is not a fuzzy γ-irresolute continuous mapping.

Example 3.3. Let μ1, μ2 and μ3 be fuzzy sets on X = {a, b, c} and let ν1, ν2 and ν3 be fuzzy sets on Y = {x, y, z} with

Let be fuzzy topologies on X and let τ∗ = {0Y, ν2, 1Y} be fuzzy topologies on Y. Consider the mapping f : (X, τ) → (Y, τ∗) defined by f(a) = y, f(b) = y and f(c) = y. Since f−1(ν2) = μ2 is fuzzy γ-open sets on (X, τ), f is fuzzy γ-continuous. But the inverse images 0X ≤ f−1(ν1) = μ1 of a fuzzy γ-open set ν1 on (Y, τ∗) is not fuzzy γ-open on (X, τ). Hence f is not a fuzzy somewhat γ-irresolute continuous mapping.

Definition 3.4 ([9]). A fuzzy set μ on a fuzzy topological space X is called fuzzy γ-dense if there exists no fuzzy γ-closed set ν such that μ < ν < 1.

Theorem 3.5. Let f : X → Y be a mapping. Then the following are equivalent:

Proof. (1) implies (2): Let ν be a fuzzy γ-closed set on Y such that f−1(ν)≠ 1X. Then νc is a fuzzy γ-open set on Y and f−1(νc) = (f−1(ν))c ≠ 0X. Since f is somewhat fuzzy γ-irresolute continuous, there exists a fuzzy γ-open set λ ≠ 0X on X such that λ ≤ f−1(νc). Let μ = λc. Then μ ≠ 1X is fuzzy γ-closed such that f−1(ν) = 1 − f−1(νc) ≤ 1 − λ = λc = μ.

(2) implies (3): Let μ be a fuzzy γ-dense set on X and suppose f(μ) is not fuzzy γ-dense on Y. Then there exists a fuzzy γ-closed set ν on Y such that f(μ) < ν < 1. Since ν < 1 and f−1(ν) ≠ 1X, there exists a fuzzy γ-closed set δ ≠ 1X such that μ ≤ f−1(f(μ)) < f−1(ν) ≤ δ. This contradicts to the assumption that μ is a fuzzy γ-dense set on X. Hence f(μ) is a fuzzy γ-dense set on Y.

(3) implies (1): Let ν ≠ 0Y be a fuzzy γ-open set on Y and f−1(ν) ≠ 0X. Suppose there exists no fuzzy γ-open μ ≠ 0X on X such that μ ≤ f−1(ν). Then (f−1(ν))c is a fuzzy set on X such that there is no fuzzy γ-closed set δ on X with (f−1(ν))c < δ < 1. In fact, if there exists a fuzzy γ-open set δc such that δc ≤ f−1(ν), then it is a contradiction. So (f−1(ν))c is a fuzzy γ-dense set on X. Then f((f−1(ν))c) is a fuzzy γ-dense set on Y. But f((f−1(ν))c) = f(f−1(νc))≠ νc < 1. This is a contradiction to the fact that f((f−1(ν))c) is fuzzy γ-dense on Y. Hence there exists a γ-open set μ ≠ 0X on X such that μ ≤ f−1(ν). Consequently, f is somewhat fuzzy γ-irresolute continuous.

A fuzzy topological space X is product related to a fuzzy topological space Y if for fuzzy sets μ on X and ν on Y whenever (in which case(γc × 1) ∨ (1 × δc) ≥ (μ × ν)) where γ is a fuzzy open set on X and δ is a fuzzy open set on Y, there exists a fuzzy open set γ1 on X and a fuzzy open set δ1 on Y such that and [1].

Theorem 3.6. Let X1 be product related to X2 and Y1 be product related to Y2. Then the product f1 × f2 : X1 × X2 → Y1 × Y2 of somewhat fuzzy γ-irresolute continuous mappings f1 : X1 → Y1 and f2 : X2 → Y2 is also somewhat fuzzy γ-irresolute continuous.

Proof. Let λ = ∨ i,j(μi×νj) be a fuzzy γ-open set on Y1×Y2 where are fuzzy γ-open sets on Y1 and Y2 respectively. Then Since f1 is somewhat fuzzy γ-irresolute continuous, there exists a fuzzy γ-open set such that And, since f2 is somewhat fuzzy γ-irresolute continuous, there exists a fuzzy γ-open set such that Now and is a fuzzy γ-open set on X1×X2. Hence is a fuzzy γ-open set on X1×X2 such that Therefore, f1 × f2 is somewhat fuzzy γ-irresolute continuous.

Theorem 3.7. Let f : X → Y be a mapping. If the graph g : X → X × Y of f is a somewhat fuzzy γ-irresolute continuous mapping, then f is also somewhat fuzzy γ-irresolute continuous.

Proof. Let ν be a fuzzy γ-open set on Y. Then f−1(ν) = 1∧f−1(ν) = g−1(1×ν). Since g is somewhat fuzzy γ-irresolute continuous and 1×ν is a fuzzy γ-open set on X×Y, there exists a fuzzy γ-open set μ ≠ 0X on X such that μ ≤ g−1(1×ν) = f−1(ν) ≠ 0X. Therefore, f is somewhat fuzzy γ-irresolute continuous.

Definition 3.8. A mapping f : X → Y is called somewhat fuzzy irresolute γ-open if there exists a fuzzy γ-open set ν ≠ 0Y on Y such that ν ≤ f(μ) for any fuzzy γ-open set μ ≠ 0X on X.

It is clear that every fuzzy irresolute γ-open mapping is a somewhat fuzzy ir-resolute γ-open mapping. And every somewhat fuzzy irresolute γ-open mapping is a fuzzy γ-open mapping. Also, every fuzzy γ-open mapping is a somewhat fuzzy γ-open mapping from the above definitions. But the converses are not true in general as the following examples show.

Example 3.9. Let μ1 and μ2 be fuzzy sets on X = {a, b, c} and let ν1 and ν2 be fuzzy sets on Y = {x, y, z} with

Let be fuzzy topologies on X and let be fuzzy topologies on Y. Consider the mapping f : (X, τ) → (Y, τ∗) defined by f(a) = y, f(b) = y and f(c) = y. Since and f is somewhat fuzzy irresolute γ-open. But f(μ2) = ν2 is not a fuzzy γ-open set on (Y, τ∗). Hence f is not a fuzzy irresolute γ-open mapping.

Example 3.10. Let μ1, μ2 and μ3 be fuzzy sets on X = {a, b, c} and let ν1 and ν2 be fuzzy sets on Y = {x, y, z} with

Let τ = {0X, μ1, μ2, 1X} be fuzzy topologies on X and let τ∗ = {0Y, ν2, 1Y} be fuzzy topologies on Y. Consider the mapping f : (X, τ) → (Y, τ∗) defined by f(a) = y, f(b) = y and f(c) = y. Since f(μ1) = ν1 and f(μ2) = ν2 are fuzzy γ-open sets on (Y, τ∗), f is fuzzy γ-open. But μ3 ≠ 0X is a fuzzy γ-open set on (X, τ) and f(μ3) = 0Y . Hence f is not a fuzzy somewhat irresolute γ-open mapping.

Example 3.11. Let μ1, μ2 and μ3 be fuzzy sets on X = {a, b, c} with

Let and τ∗ = {0X; μ1, μ3, 1X}be fuzzy topologies on X. Consider the identity mapping iX : (X, τ) → (X, τ∗). We have Since μ3 is a fuzzy γ-open set on (X, τ), iX is somewhat fuzzy γ-open. But is not a fuzzy γ-open set on (X; τ∗). Hence iX is not a fuzzy γ-open mapping.

Theorem 3.12. Let f : X → Y be a bijection. Then the following are equivalent:

(1) f is somewhat fuzzy irresolute γ-open.

(2) If μ is a fuzzy γ-closed set on X such that f(μ) ≠ 1Y, then there exists a fuzzy γ-closed set ν ≠ 1Y on Y such that f(μ) < ν.

Proof. (1) implies (2): Let μ be a fuzzy γ-closed set on X such that f(μ) ≠ 1Y. Since f is bijective and μc is a fuzzy γ-open set on X, f(μc) = (f(μ))c ≠ 0Y. And, since f is somewhat fuzzy irresolute γ-open, there exists a γ-open set δ ≠ 0Y on Y such that δ < f(μc) = (f(μ))c. Consequently, f(μ) < δc = ν ≠ 1Y and ν is a fuzzy γ-closed set on Y.

(2) implies (1): Let μ be a fuzzy γ-open set on X such that f(μ) ≠ 0Y. Then μc is a fuzzy γ-closed set on X and f(μc) ≠ 1Y. Hence there exists a fuzzy γ-closed set ν ≠ 1Y on Y such that f(μc) < ν. Since f is bijective, f(μc) = (f(μ))c < ν. Hence νc < f(μ) and νc ≠ 0X is a fuzzy γ-open set on Y. Therefore, f is somewhat fuzzy irresolute γ-open.

Theorem 3.13. Let f : X → Y be a surjection. Then the following are equiva-lent:

(1) f is somewhat fuzzy irresolute γ-open.

(2) If ν is a fuzzy γ-dense set on Y, then f−1(ν) is a fuzzy γ-dense set on X.

Proof. (1) implies (2): Let ν be a fuzzy γ-dense set on Y. Suppose f−1(ν) is not fuzzy γ-dense on X. Then there exists a fuzzy γ-closed set μ on X such that f−1(ν) < μ < 1. Since f is somewhat fuzzy irresolute γ-open and μc is a fuzzy γ-open set on X, there exists a fuzzy γ-open set δ ≠ 0Y on Y such that δ ≤ f(Intμc) ≤ f(μc). Since f is surjective, δ ≤ f(μc) < f(f−1(νc)) = νc. Thus there exists a γ-closed set δc on Y such that ν < δc < 1. This is a contradiction. Hence f−1(ν) is fuzzy γ-dense on X.

(2) implies (1): Let μ be a fuzzy open set on X and f(μ) ≠ 0Y . Suppose there exists no fuzzy γ-open ν ≠ 0Y on Y such that ν ≤ f(μ). Then (f(μ))c is a fuzzy set on Y such that there exists no fuzzy γ-closed set δ on Y with (f(μ))c < δ < 1. This means that (f(μ))c is fuzzy γ-dense on Y. Thus f−1((f(μ))c) is fuzzy γ-dense on X. But f−1((f(μ))c) = (f−1(f(μ)))c ≤ μc < 1. This is a contradiction to the fact that f−1((f(ν))c is fuzzy γ-dense on X. Hence there exists a γ-open set ν ≠ 0Y on Y such that ν ≤ f(μ). Therefore, f is somewhat fuzzy irresolute γ-open.