Fuzzy ideal topological spaces

Abstract

In this paper, it is introduced the notion of r-fuzzy ideal separation axioms T_i, i = 0, 1, 2 based on a fuzzy ideal $I$ on a fuzzy topological space (X, τ). An r-fuzzy ideal connectedness related to the fuzzy ideal $I$ is introduced which has relations with a previous r-fuzzy connectedness. An r-fuzzy ideal compactness related to $I$ is introduced which has also relations with many other types of fuzzy compactness.

Keywords

fuzzy ideal fuzzy separation axioms fuzzy compactness fuzzy connectedness

1 Introduction and Preliminaries

This is a way to use a fuzzy ideal $I$ defined on a fuzzy topological space (X, τ) giving generalizations of many notions and results in fuzzy topological spaces. r-fuzzy ideal T_i, i = 0, 1, 2 separation axioms are new types of fuzzy separation axioms related with the fuzzy ideal $I$ on X. It is proved many implications between these r-fuzzy ideal T_i, i = 0, 1, 2 spaces and the previous r-fuzzy T_i, i = 0, 1, 2 defined in [7] and studied in [4 –6], and also the preimage and the image of r-fuzzy ideal T_i, i = 0, 1, 2 spaces are r-fuzzy ideal T_i, i = 0, 1, 2 spaces as well. r-fuzzy ideal connectedness is introduced related with $I$ giving a generalization of the r-fuzzy connectedness notion ([9, 10]). The image of r-fuzzy ideal connected is r-fuzzy ideal connected as well. r-fuzzy ideal compactness is introduced using the fuzzy ideal $I$ on X giving a generalization of many other fuzzy compactness notions [1, 11]. The image of r-fuzzy compact is r-fuzzy ideal compact, and many special cases are deduced.

Note that: In [3], the author used the ideal notion to reduce the soft boundary region in ordinary soft rough topological space but here we joined the fuzzy ideal notion to fuzzy topology in sense of $\overset{ˇ}{S}$ ostak without concerning soft roughness. In [12], the authors introduced fuzzy soft separation axioms and fuzzy soft connectedness for fuzzy soft topological spaces in sense of Chang but in this paper we used the fuzzy ideal notion in defining fuzzy ideal separation axioms, fuzzy ideal connectedness and fuzzy ideal compactness in sense of $\overset{ˇ}{S}$ ostak.

Throughout the paper, X refers to an initial universe, I^X is the set of all fuzzy sets on X (where I = [0, 1] , I₀ = (0, 1], λ^c (x) =1 - λ (x) ∀ x ∈ X and for all t ∈ I, $\bar{t} (x) = t \forall x \in X$ ). A fuzzy point x_t is defined by x_t (y) = t at y = x and x_t (y) =0 otherwise.

(X, τ) is a fuzzy topological space ([14]), if τ : I^X → I satisfies the following conditions:

$τ (\bar{0}) = τ (\bar{1}) = 1$ ,

τ (λ₁ ∧ λ₂) ≥ τ (λ₁) ∧ τ (λ₂) for all λ₁, λ₂ ∈ I^X,

τ (⋁ _j∈Jλ_j) ≥ ⋀ _j∈J τ (λ_j) for all {λ_j} _j∈J ⊆ I^X.

A map $I : I^{X} \to I$ is called a fuzzy ideal ([13]) on X if it satisfies:

$I (\bar{0}) = 1$ ,

$λ \leq μ \Rightarrow I (λ) \geq I (μ)$ for all λ, μ ∈ I^X,

$I (λ \lor μ) \geq I (λ) \land I (μ)$ for all λ, μ ∈ I^X.

If $I_{1}$ and $I_{2}$ are fuzzy ideals on X, we have $I_{1}$ is finer than $I_{2}$ ( $I_{2}$ is coarser than $I_{1}$ ), denoted by $I_{1} \leq I_{2}$ iff $I_{1} (λ) \leq I_{2} (λ) \forall λ \in I^{X}$ . The triple $(X, τ, I)$ is called a fuzzy ideal topological space.

Define the fuzzy ideal $I^{\circ}$ by $\begin{matrix} I^{\circ} (μ) = {\begin{matrix} 1 & at μ = \bar{0}, \\ 0 & otherwise \end{matrix} \end{matrix}$

Recall that the fuzzy difference between two fuzzy sets is defined as follows ([8]): $\begin{matrix} (λ \bar{\land} μ) = {\begin{matrix} \bar{0} & if λ \leq μ, \\ λ \land μ^{c} & otherwise . \end{matrix} \end{matrix}$

Definition 1. [8] Let $(X, τ, I)$ be a fuzzy ideal topological space and λ ∈ I^X. Then, the r-fuzzy open local function $λ_{r}^{*} (τ, I)$ of λ is defined by $λ_{r}^{*} (τ, I) = ⋀ {μ \in I^{X} : I (λ μ) \geq r, τ (μ^{c}) \geq r} .$ Occasionally, we will write $λ_{r}^{*}$ or $λ_{r}^{*} (I)$ for $λ_{r}^{*} (τ, I)$ and it will be no ambiguity.

If $I = I^{\circ}$ then, for each λ ∈ I^X, r ∈ I₀, we have $λ_{r}^{*} = {cl}_{τ} (λ, r)$ ([8]).

Proposition 1. [8] Let $(X, τ, I)$ be a fuzzy ideal topological space and $I_{1}, I_{2}$ be fuzzy ideals on X. Then,

λ ≤ μ implies $λ_{r}^{*} \leq μ_{r}^{*}$ .

If $I_{1} \leq I_{2}$ , then $λ_{r}^{*} (I_{1}) \geq λ_{r}^{*} (I_{2})$ .

$λ_{r}^{*} = {cl}_{τ} (λ_{r}^{*}, r) \leq {cl}_{τ} (λ, r)$ and $(λ_{r}^{*})_{r}^{*} \leq λ_{r}^{*}$ .

$λ_{r}^{*} \lor μ_{r}^{*} \leq (λ \lor μ)_{r}^{*}$ , and $λ_{r}^{*} \land μ_{r}^{*} \geq (λ \land μ)_{r}^{*}$ .

In the following example, it is shown that: $λ_{r}^{*} ≰ (λ_{r}^{*})_{r}^{*}$ .

Example 1. Let τ be a fuzzy topology and $I$ a fuzzy ideal defined on X such that $\begin{matrix} τ (λ) = {\begin{matrix} 1 & at λ = \bar{0}, \bar{1} \\ \frac{1}{2} & at λ = \bar{0.5} \\ \frac{1}{2} & at λ = \bar{0.6} \\ 0 & otherwise, \end{matrix} \end{matrix}$ $\begin{matrix} I (λ) = {\begin{matrix} 1 & at λ = \bar{0} \\ \frac{1}{2} & at λ = \bar{0.5} \\ \frac{2}{3} & at \bar{0} < λ < \bar{0.5} \\ 0 & otherwise . \end{matrix} \end{matrix}$ $(\bar{0.6})_{\frac{1}{2}}^{*} = ⋀ {μ : I (\bar{0.6} μ) \geq \frac{1}{2}, τ (μ^{c}) \geq \frac{1}{2}} = \bar{0.5},$ but $((\bar{0.6})_{\frac{1}{2}}^{*})_{\frac{1}{2}}^{*} = (\bar{0.5})_{\frac{1}{2}}^{*} = ⋀ {μ I (\bar{0.5} μ) \geq \frac{1}{2}, τ (μ^{c}) \geq \frac{1}{2}} = \bar{0} .$

Proposition 2. [8] Let $(X, τ, I)$ be a fuzzy ideal topological space and {μ_j : j ∈ J} ⊆ I^X a family. Then,

$⋁ ((μ_{j})_{r}^{*} j \in J) \leq (⋁ (μ_{j}) : j \in J)_{r}^{*}$ .

$⋀ ((μ_{j})_{r}^{*} : j \in J) \geq (⋀ (μ_{j}) : j \in J)_{r}^{*}$ .

Definition 2. [8] Let $(X, τ, I)$ be a fuzzy ideal topological space and μ ∈ I^X. Then, ${cl}_{τ}^{*} (μ, r) = μ ⋁ μ_{r}^{*} and {int}_{τ}^{*} (μ, r) = μ ⋀ ((μ^{c})_{r}^{*})^{c} .$ If $I = I^{\circ}$ , then for each μ ∈ I^X, r ∈ I₀, ${cl}_{τ}^{*} (μ, r) = μ \lor μ_{r}^{*} = μ \lor {cl}_{τ} (μ, r) = {cl}_{τ} (μ, r)$ .

Proposition 3. [8] Let $(X, τ, I)$ be a fuzzy ideal topological space and λ, μ ∈ I^X, r ∈ I₀. Then,

${int}_{τ}^{*} (λ \lor μ, r) \geq {int}_{τ}^{*} (λ, r) \lor {int}_{τ}^{*} (μ, r)$ .

${int}_{τ} (λ, r) \leq {int}_{τ}^{*} (λ, r) \leq λ \leq {cl}_{τ}^{*} (λ, r) \leq {cl}_{τ} (λ, r)$ .

${cl}_{τ}^{*} (λ^{c}, r) = ({int}_{τ}^{*} (λ, r))^{c} and {int}_{τ}^{*} (λ^{c}, r) = ({cl}_{τ}^{*} (λ, r))^{c}$ .

${int}_{τ}^{*} (λ \land μ, r) \leq {int}_{τ}^{*} (λ, r) \land {int}_{τ}^{*} (μ, r)$ .

Corollary 1. [8] Let $(X, τ_{1}, I)$ , $(X, τ_{2}, I)$ be fuzzy ideal topological spaces and τ₁ ≤ τ₂. Then, for each λ ∈ I^X, r ∈ I₀, we have $λ_{r}^{*} (τ_{2}, I) \leq λ_{r}^{*} (τ_{1}, I)$ .

Corollary 2. [8] Let $(X, τ, I_{1})$ , $(X, τ, I_{2})$ be fuzzy ideal topological spaces and $I_{1} \leq I_{2}$ . Then, for each λ ∈ I^X, r ∈ I₀, we have $λ_{r}^{*} (τ, I_{1}) \geq λ_{r}^{*} (τ, I_{2})$ .

Proposition 4. [8] Let (X, τ) be a fuzzy topological space and $I_{1}, I_{2}$ fuzzy ideals on X. Then, for each λ ∈ I^X, r ∈ I₀, we have $λ_{r}^{*} (τ, I_{1} \land I_{2}) = λ_{r}^{*} (τ, I_{1}) \lor λ_{r}^{*} (τ, I_{2})$ .

2 Fuzzy ideal r-(t, s)-T_i separation axioms

Here, we introduce fuzzy separation axioms in fuzzy ideal topological spaces.

Definition 3.

A fuzzy ideal topological space $(X, τ, I)$ is called r-(t, s)-FI-T₀ if for t, s ∈ I₀, then x ≠ y in X implies that there exists λ ∈ I^X, r ∈ I₀ with $t \leq {int}_{τ}^{*} (λ, r) (x)$ such that t > λ (y) or there exists μ ∈ I^X, r ∈ I₀ with $s \leq {int}_{τ}^{*} (μ, r) (y)$ such that s > μ (x).

A fuzzy ideal topological space $(X, τ, I)$ is called r-(t, s)-FI-T₁ if for t, s ∈ I₀, then x ≠ y in X implies that there exist λ, μ ∈ I^X, r ∈ I₀ with $t \leq {int}_{τ}^{*} (λ, r) (x)$ , $s \leq {int}_{τ}^{*} (μ, r) (y)$ such that t > λ (y) and s > μ (x).

A fuzzy ideal topological space $(X, τ, I)$ is called r-(t, s)-FI-T₂ if for t, s ∈ I₀, then x ≠ y in X implies that there exist λ, μ ∈ I^X, r ∈ I₀ with $t \leq {int}_{τ}^{*} (λ, r) (x)$ , $s \leq {int}_{τ}^{*} (μ, r) (y)$ such that (t ∧ s) > sup(λ ∧ μ).

Remark 1. Consider a fuzzy ideal topological space $(X, τ, I)$ with $I = I^{\circ}$ . Then, the graded fuzzy separation axioms defined in [7] and the r-(t, s)-FI-T_i separation axioms are identical, i = 0, 1, 2.

Any fuzzy topological space (X, τ) satisfying (t, s)-T_i separation axiom as defined in [7] will be r-(t, s)-FI-T_i with respect to some fuzzy ideal on X as well but not converse, i = 0, 1, 2. It is coming from that: ${int}_{τ} (μ, r) \leq {int}_{τ}^{*} (μ, r) \forall μ \in I^{X}, r \in I_{0}$ .

Proposition 5. Every r-(t, s)-FI-T_i fuzzy ideal topological space $(X, τ, I)$ is an r-(t, s)-FI-T_i-1 space, i = 1, 2.

Proof. r-(t, s)-FI-T₂ ⇒ r-(t, s)-FI-T₁: Let $(X, τ, I)$ be an r-(t, s)-FI-T₂ space, and suppose that $(X, τ, I)$ is not r-(t, s)-FI-T₁. That is, for all x ≠ y in X and for all λ ∈ I^X, r ∈ I₀ with $t \leq {int}_{τ}^{*} (λ, r) (x)$ , suppose that λ (y) ≥ t ; t ∈ I₀. Now, for μ ∈ I^X with $s \leq {int}_{τ}^{*} (μ, r) (y); s \in I_{0}$ , we get that s ≤ μ (y), and thus sup(λ ∧ μ) ≥ (λ ∧ μ) (y) ≥ (t ∧ s), which is a contradiction to $(X, τ, I)$ is an r-(t, s)-FI-T₂ space. Hence, $(X, τ, I)$ is an r-(t, s)-FI-T₁ space.

r-(t, s)-FI-T₁ ⇒ r-(t, s)-FI-T₀: It is clear. □

Recall that: a mapping f : (X, τ) → (Y, σ) is said to be fuzzy continuous ([14]) if ${int}_{τ} (f^{- 1} (ν), r) \geq f^{- 1} ({int}_{σ} (ν, r)) \forall ν \in I^{Y} r \in I_{0} .$ It is equivalent to satisfy the following ${cl}_{τ} (f^{- 1} (ν), r) \leq f^{- 1} ({cl}_{σ} (ν, r)) \forall ν \in I^{Y} r \in I_{0} .$

Now, let us call a mapping $f : (X, τ) \to (Y, σ, I^{'})$ fuzzy ideal continuous provided that ${int}_{τ} (f^{- 1} (ν), r) \geq f^{- 1} ({int}_{σ}^{*} (ν, r)) \forall ν \in I^{Y} r \in I_{0} .$ It is easily shown that it is equivalent to

${cl}_{τ} (f^{- 1} (ν), r) \leq f^{- 1} ({cl}_{σ}^{*} (ν, r)) \forall ν \in I^{Y} r \in I_{0}$ .

Also, let us call $f : (X, τ, I) \to (Y, σ)$ a fuzzy ideal open mapping provided that $f ({int}_{τ}^{*} (λ, r)) \leq {int}_{σ} (f (λ), r) \forall λ \in I^{X} r \in I_{0} .$ It is easily shown that it is equivalent to

${cl}_{σ} (f (λ), r) \leq f ({cl}_{τ}^{*} (λ, r)) \forall λ \in I^{X} r \in I_{0}$ .

It is clear that: any map f satisfying condition (IC) (or fuzzy ideal continuous) will be a fuzzy continuous mapping f : (X, τ) → (Y, σ), but not every fuzzy continuous mapping f : (X, τ) → (Y, σ) will satisfy the condition (IC) (or fuzzy ideal continuous) with respect to a fuzzy ideal $I^{'}$ on Y. Also, a map f satisfying condition (IO) (or fuzzy ideal open) will be a fuzzy open mapping f : (X, τ) → (Y, σ), but not every fuzzy open mapping f : (X, τ) → (Y, σ) will satisfy the condition (IO) (or fuzzy ideal open) with respect to a fuzzy ideal $I$ on X.

Theorem 1. Let $(X, τ, I), (Y, σ, I^{'})$ be fuzzy ideal topological spaces and f : (X, τ) → (Y, σ, $I^{'}$ ) be an injective fuzzy ideal continuous mapping. Then, $(X, τ, I)$ is an r-(t, s)-FI-T_i space if $(Y, σ, I^{'})$ is an r-(t, s)-FI-T_i space, i = 0, 1, 2.

Proof. Since x ≠ y in X implies that f (x) ≠ f (y) in Y and for $(Y, σ, I^{'})$ is r-(t, s)-FI-T₂, then there exist ν, ρ ∈ I^Y, r ∈ I₀ with $t \leq {int}_{σ}^{*} (ν, r) (f (x)), s \leq {int}_{σ}^{*} (ρ, r) (f (y)); t, s \in I_{0}$ so that t ∧ s > sup(ν ∧ ρ), that is, $t \leq f^{- 1} ({int}_{σ}^{*} (ν, r)) (x)$ , $s \leq f^{- 1} ({int}_{σ}^{*} (ρ, r)) (y);$ t, s ∈ I₀ .Since f is fuzzy ideal continuous, $t \leq {int}_{τ} (f^{- 1} (ν), r) (x) \leq {int}_{τ}^{*} (f^{- 1} (ν), r) (x),$ s ≤ int_τ $(f^{- 1} (ρ), r) (y) \leq {int}_{τ}^{*} (f^{- 1} (ρ), r) (y); t, s \in I_{0} .$ That is, there exist λ = f^-1 (ν) , μ = f^-1 (ρ) ∈ I^X with $t \leq {int}_{τ}^{*} (λ, r) (x), s \leq {int}_{τ}^{*} (μ, r) (y); t, s \in I_{0} .$ Now, f is injective implies that sup(λ ∧ μ) = sup(f^-1 (ν) ∧ f^-1 (ρ)) ≤ sup(ν ∧ ρ) < t ∧ s . Hence, $(X, τ, I)$ is an r-(t, s)-FI-T₂ space.

For the cases of $(X, τ, I)$ is r-(t, s)-FI-T₀ and r-(t, s)-FI-T₁, it is similar. □

Theorem 2. Let $(X, τ, I), (Y, σ, I^{'})$ be fuzzy ideal topological spaces and $f : (X, τ, I) \to (Y, σ)$ is a surjective fuzzy ideal open mapping. Then, $(Y, σ, I^{'})$ is an r-(t, s)-FI-T_i space if $(X, τ, I)$ is an r-(t, s)-FI-T_i space, i = 0, 1, 2.

Proof. Since p ≠ q in Y implies that there are x ≠ y in X where x ∈ f^-1 (p) , y ∈ f^-1 (q) and for $(X, τ, I)$ is r-(t, s)-FI-T₂, then there exist λ, μ ∈ I^X, r ∈ I₀ with $t \leq {int}_{τ}^{*} (λ, r) (x), s \leq {int}_{τ}^{*} (μ, r) (y); t, s \in I_{0}$ so that t ∧ s > sup(λ ∧ μ). Now, t ≤ ⋁ _x∈f^-1(p) ${int}_{τ}^{*} (λ, r) (x) = f ({int}_{τ}^{*} (λ, r)) (p), s \leq ⋁_{y \in f^{- 1} (q)} {int}_{τ}^{*} (μ,$ $r) (y) = f ({int}_{τ}^{*} (μ, r)) (q) .$ From f is fuzzy ideal open, then t≤ int_σ (f (λ) , r) (p) ≤ ${int}_{σ}^{*} (f (λ), r) (p)$ , $s \leq {int}_{σ} (f (μ), r) (q) \leq {int}_{σ}^{*} (f (μ), r) (q),$ which means that there exist ν = f (λ) , ρ = f (μ) ∈ I^Y, r ∈ I₀ with $t \leq {int}_{σ}^{*} (ν, r) (p)$ , $s \leq {int}_{σ}^{*} (ρ, r) (q);$ t, s ∈ I₀. Since f is surjective, $\sup (ν \land ρ) = \sup (f (λ) \land f (μ)) \leq \sup (λ \land μ) < t \land s .$ Hence, $(Y, σ, I^{'})$ is an r-(t, s)-FI-T₂ space.

For the cases of $(Y, σ, I^{'})$ is r-(t, s)-FI-T₀ and r-(t, s)-FI-T₁, it is similar. □

Example 2. Let X = {x, y}, τ be a fuzzy topology on X defined by $τ (λ) = {\begin{matrix} 1 & at λ = \bar{0}, \bar{1} \\ 0.5 & at λ = x_{1} \\ 0.7 & at λ = y_{1} \\ 0 & otherwise, \end{matrix}$ and $I$ a fuzzy ideal on X defined by $I (λ) = {\begin{matrix} 1 & at λ = \bar{0} \\ 0.4 & at λ = \bar{0.6} \\ 0.7 & at \bar{0} < λ < \bar{0.6} \\ 0 & otherwise . \end{matrix}$

Then, $(X, τ, I)$ is a 0.6-(t, s)-FI-T_i space, i = 0, 1, 2 but (X, τ) is not 0.6-(t, s)-F-T_i space, i = 1, 2 because:

For r = 0.6 and any μ ∈ I^X, we have $μ_{0.6}^{*} = ⋀ {ν : I (μ ν) \geq 0.6, τ (ν^{c}) \geq 0.6}$ (where ν may be $\bar{1}$ , $\bar{0}$ or x₁). That is, $μ_{0.6}^{*} = {\begin{matrix} \bar{0} & at \bar{0} \leq μ < \bar{0.6} \\ x_{1} & at μ = (x_{k} \lor y_{m}), m < 0.6 < k \\ \bar{1} & otherwise, \end{matrix}$ which means $\begin{matrix} {int}_{τ}^{*} (μ, 0.6) & = & {\begin{matrix} μ & at \bar{0.4} < μ \leq \bar{1} \\ y_{q} & at μ = (x_{p} \lor y_{q}), \\ p < 0.4 < q \\ \bar{0} & otherwise \end{matrix} \end{matrix}$ (1) while ${int}_{τ} (μ, 0.6) = {\begin{matrix} μ & at μ \in {y_{1}, \bar{1}} \\ \bar{0} & otherwise \end{matrix}$ (2)

For λ = x_0.6 ∨ y_0.5, then ${int}_{τ}^{*} (λ, 0.6) (x) = λ (x) = 0.6, λ (y) = 0.5 < 0.6$ , and for μ = x_0.5 ∨ y_0.6, then ${int}_{τ}^{*} (μ, 0.6) (y) = μ (y) = 0.6, μ (x) = 0.5 < 0.6$ , which means that for t = s = 0.6, we get $t \leq {int}_{τ}^{*} (λ, r) (x), s \leq {int}_{τ}^{*} (μ, r) (y)$ such that λ (y) < t, μ (x) < s. Moreover, sup(λ ∧ μ) =0.5 < 0.6 = (t ∧ s). Hence, $(X, τ, I)$ is satisfying the r-(t, s)-FI-T_i axioms, i = 0, 1, 2, and we can not find λ, μ ∈ I^X satisfying the fuzzy r-(t, s)-T₁ or r-(t, s)-T₂ (From Equation 2.2, ${int}_{τ} (μ, 0.6) = \bar{0} \forall μ \notin {y_{1}, \bar{1}}$ ).

Example 3. Let X = {x, y} and τ be a fuzzy topology on X defined by $τ (λ) = {\begin{matrix} 1 & at λ = \bar{0}, \bar{1} \\ 0.5 & at λ = x_{1} \\ 0.6 & at λ = y_{1} \\ 0 & otherwise, \end{matrix}$ and $I$ a fuzzy ideal on X defined by $I (λ) = {\begin{matrix} 1 - m & at λ \leq x_{m}, 0 \leq m \leq 0.5 \\ 0 & otherwise . \end{matrix}$ Then

$(X, τ, I)$ is a 0.6-(t, s)-FI-T₀ space but it is neither 0.6-(t, s)-FI-T₁ space nor 0.6-(t, s)-FI-T₂ space because:

For r = 0.6 and any choice for μ ∈ I^X as μ = (x_k ∨ y_m) , k ≤ 0.5, 0 ≤ m < 1, we get that μ^c = (x_p ∨ y_q) , p > 0.5, q > 0, and then $(μ^{c})_{r}^{*} = \bar{1}$ , and for μ = (x_k ∨ y_m) , k > 0.5, 0 ≤ m < 1, we get μ^c = (x_p ∨ y_q) , p ≤ 0.5, q > 0, and then $(μ^{c})_{r}^{*} = \bar{1}$ . That is, ${int}_{τ}^{*} (μ, r) (x) = 0$ for any choice of μ with y₁notleμ, and thus we can not find λ, μ ∈ I^X satisfying any of the r-(t, s)-FI-T₁ axiom or the r-(t, s)-FI-T₂ axiom, while $(X, τ, I)$ could be only an r-(t, s)-FI-T₀ space (by choosing μ = x_0.8 ∨ y₁, then ${int}_{τ}^{*} (μ, r) (y) = μ (y) = 1 \geq t > 0.8 = μ (x); t = 0.9$ ). Moreover, (X, τ) is also satisfying the fuzzy 0.6-(t, s)-T₀ axiom (taking μ = y₁, then int_τ (μ, r) (y) = μ (y) =1 ≥ t > 0 = μ (x) ; t = 0.6).

In case of r = 0.8, we deduce that: (X, τ) is not fuzzy r-(t, s)-T_i, i = 0, 1, 2, and $(X, τ, I)$ is not r-(t, s)-FI-T_i, i = 1, 2 while the r-(t, s)-FI-T₀ axiom is satisfied (by taking μ = x_0.8 ∨ y₁).

3 Connectedness in fuzzy ideal topological spaces

Here, we introduce the r-fuzzy ideal connectedness of a fuzzy ideal topological space $(X, τ, I)$ .

Definition 4. Let $(X, τ, I)$ be a fuzzy ideal topological space. Then,

the fuzzy sets λ, μ ∈ I^X are called r-fuzzy ideal separated (r-FI-separated for short) if ${cl}_{τ}^{*} (λ, r) \land μ = λ \land {cl}_{τ}^{*} (μ, r) = \bar{0} r \in I_{0} .$

$(X, τ, I)$ is called r-fuzzy ideal connected space (r-FI-connected for short) if it could not be found r-FI-separated sets λ, μ ∈ I^X, $λ \neq \bar{0}, μ \neq \bar{0}$ such that $λ \lor μ = \bar{1}$ . That is, there are no r-FI-separated sets λ, μ ∈ I^X except $λ = \bar{0}$ or $μ = \bar{0}$ .

Definition 5. Let λ, μ ∈ I^X, $λ \neq \bar{0}, μ \neq \bar{0}$ such that:

λ, μ are r-FI-separated and $λ \lor μ = \bar{1}$ . Then $(X, τ, I)$ is called an r-FI-disconnected space.

λ, μ are r-FI-separated and λ ∨ μ = ν. Then ν is called r-FI-disconnected fuzzy set in I^X.

λ, μ are r-FI-separated and λ ∨ μ = χ_A, A ⊆ X. Then A is called r-FI-disconnected crisp set in I^X.

Remark 2. Consider a fuzzy ideal topological space $(X, τ, I)$ .

Any two r-fuzzy separated sets ([9]) λ, μ in I^X are r-FI-separated as well from that: ${cl}_{τ}^{*} (ν, r) \leq {cl}_{τ} (ν, r) \forall ν \in I^{X}; r \in I_{0}$ .

That is, r-fuzzy disconnectedness ([9]) implies r-FI-disconnectedness and thus, r-FI-connectedness implies r-fuzzy connectedness ([9]).

Example 4. Let X = {x, y}. Define $τ, I : I^{X} \to I$ as in Example 2.1. Then:

For 0.5 < r ≤ 0.7 and for any μ ∈ I^X, we have ${cl}_{τ}^{*} (μ, r) = {\begin{matrix} μ & at \bar{0} \leq μ < \bar{0.6} \\ (μ \lor x_{1}) & at μ = (x_{k} \lor y_{m}), \\ m < 0.6 < k \\ \bar{1} & otherwise . \end{matrix}$ (3) while ${cl}_{τ} (μ, r) = {\begin{matrix} μ & at μ \in {x_{1}, \bar{0}} \\ \bar{1} & otherwise . \end{matrix}$ (4)

So, for 0.5 < r ≤ 0.7, we can easily find λ = x_0.7, μ = y_0.5 ∈ I^X with ${cl}_{τ}^{*} (λ, r) = x_{1}, {cl}_{τ}^{*} (μ, r) = y_{0.5}$ , and then ${cl}_{τ}^{*} (λ, r) \land μ = {cl}_{τ}^{*} (μ, r) \land λ = \bar{0}$ , which means there are r-FI-separated sets. But for all possible choices of such r-FI-separated sets, we have $λ \lor μ \neq \bar{1}$ . Hence, $(X, τ, I)$ is not an r-FI-disconnected space, and hence $(X, τ, I)$ is an r-FI-connected space. Note that: not every r-FI-separated sets are r-fuzzy separated sets, where ${cl}_{τ} (λ, 0.7) = {cl}_{τ} (x_{0.7}, 0.7) = {cl}_{τ} (μ, 0.7) = {cl}_{τ} (y_{0.5}, 0.7) = \bar{1}$ , which means that cl_τ (λ, 0.7) ∧ μ = y_0.5, cl_τ (μ, 0.7) ∧ λ = x_0.7. Hence, the result in Remark 3.1 is true.

Lemma 1. Let $(X, τ, I)$ be a fuzzy ideal topological space. Then, for any μ ∈ I^X, r ∈ I₀ with τ (μ^c) ≥ r, we get that: $μ_{r}^{*} \leq μ$ .

Proof. From that: $μ_{r}^{*} = {cl}_{τ} (μ_{r}^{*}, r) \leq {cl}_{τ}^{*} (μ, r) \leq {cl}_{τ} (μ, r) = μ$ . □

Proposition 6. Let $(X, τ, I)$ be a fuzzy ideal topological space. Then the following are equivalent.

$(X, τ, I)$ is r-FI-connected.

$λ \land μ = \bar{0}$ , τ (λ) ≥ r, τ (μ) ≥ r ; r ∈ I₀, and $λ \lor μ = \bar{1}$ imply $λ = \bar{0}$ or $μ = \bar{0}$ .

$λ \land μ = \bar{0}$ , τ (λ^c) ≥ r, τ (μ^c) ≥ r ; r ∈ I₀, and $λ \lor μ = \bar{1}$ imply $λ = \bar{0}$ or $μ = \bar{0}$ .

Proof. (1) ⇒ (2): Let λ, μ ∈ I^X with τ (λ) ≥ r, τ (μ) ≥ r ; r ∈ I₀ such that $λ \land μ = \bar{0}$ and $λ \lor μ = \bar{1}$ . Then, λ = μ^c and μ = λ^c, which means (from Lemma 3.1) that $λ_{r}^{*} \leq λ$ and $μ_{r}^{*} \leq μ$ , and then ${cl}_{τ}^{*} (λ, r) = λ \lor λ^{*} = λ$ and ${cl}_{τ}^{*} (μ, r) = μ \lor μ^{*} = μ$ , which means that $\bar{0} = λ \land μ = {cl}_{τ}^{*} (μ, r) \land λ = {cl}_{τ}^{*} (λ, r) \land μ$ . That is, λ, μ are r-FI-separated sets in I^X so that $λ \lor μ = \bar{1}$ . But $(X, τ, I)$ is r-FI-connected implies that $λ = \bar{0}$ or $μ = \bar{0}$ .

(2) ⇒ (3): Clear.

(3) ⇒ (1): Let λ, μ ∈ I^X with τ (λ^c) ≥ r, τ (μ^c) ≥ r ; r ∈ I₀ such that $λ \land μ = \bar{0}$ and $λ \lor μ = \bar{1}$ . Then, λ = μ^c and μ = λ^c, and moreover ${cl}_{τ}^{*} (λ, r) = λ$ and ${cl}_{τ}^{*} (μ, r) = μ$ , which implies that $\bar{0} = λ \land μ = {cl}_{τ}^{*} (μ, r) \land λ = {cl}_{τ}^{*} (λ, r) \land μ$ . That is, λ, μ are r-FI-separated sets with $λ \lor μ = \bar{1}$ . From (3), we have $λ = \bar{0}$ or $μ = \bar{0}$ . Hence, $(X, τ, I)$ is an r-FI-connected space. □

Proposition 7. Let $(X, τ, I)$ be a fuzzy ideal topological space and λ ∈ I^X. Then, the following are equivalent.

λ is r-FI-connected.

If μ, ρ are r-FI-separated sets with λ ≤ μ ∨ ρ, then $λ \land μ = \bar{0}$ or $λ \land ρ = \bar{0}$ .

If μ, ρ are r-FI-separated sets with λ ≤ μ ∨ ρ, then λ ≤ μ or λ ≤ ρ.

Proof. (1) ⇒ (2): Let μ, ρ be r-FI-separated with λ ≤ μ ∨ ρ. That is, ${cl}_{τ}^{*} (μ, r) \land ρ = {cl}_{τ}^{*} (ρ, r) \land μ = \bar{0}; r \in I_{0}$ so that λ ≤ μ ∨ ρ. Since ${cl}_{τ}^{*} (λ \land μ, r) \leq {cl}_{τ}^{*} (λ, r) \land {cl}_{τ}^{*} (μ, r) and {cl}_{τ}^{*} (λ \land ρ, r) \leq {cl}_{τ}^{*} (λ, r) \land {cl}_{τ}^{*} (ρ, r),$ we get that ${cl}_{τ}^{*} (λ$ $\land μ, r) \land (λ \land ρ) \leq ({cl}_{τ}^{*} (λ, r) \land λ) \land ({cl}_{τ}^{*} (μ, r) \land ρ) = λ \land \bar{0} = \bar{0},$ ${cl}_{τ}^{*} (λ \land ρ, r) \land (λ \land μ) \leq ({cl}_{τ}^{*} (λ, r) \land λ) \land ({cl}_{τ}^{*} (ρ, r) \land μ) = λ \land \bar{0} = \bar{0},$ and thus λ ∧ μ, λ ∧ ρ are r-FI-separated sets with λ = (λ ∧ μ) ∨ (λ ∧ ρ). But λ is r-FI-connected implies that $λ \land μ = \bar{0}$ or $λ \land ρ = \bar{0}$ .

(2) ⇒ (3): If $λ \land μ = \bar{0}, λ \leq μ \lor ρ$ means that λ = λ ∧ (μ ∨ ρ) = λ ∧ ρ, and thus λ ≤ ρ. Also, if $λ \land ρ = \bar{0}$ , then λ = λ ∧ μ, and then λ ≤ μ.

(3) ⇒ (1): Let μ, ρ be r-FI-separated sets such that λ = μ ∨ ρ. Then, from (3), λ ≤ μ or λ ≤ ρ. If λ ≤ μ, then $ρ = (μ \lor ρ) \land ρ = λ \land ρ \leq μ \land ρ \leq {cl}_{τ}^{*} (μ, r) \land ρ = \bar{0}$ . Also, if λ ≤ ρ, then $μ = (μ \lor ρ) \land μ = λ \land μ \leq ρ \land μ \leq {cl}_{τ}^{*} (ρ, r) \land μ = \bar{0}$ . Hence, λ is r-FI-connected. □

Theorem 3. Let $(X, τ, I), (Y, σ, I^{'})$ be fuzzy ideal topological spaces and f : (X, τ) → (Y, σ, $I^{'}$ is a mapping satisfying the condition (IC). Then, f (λ) ∈ I^Y is r-FI-connected if λ ∈ I^X is r-FI-connected.

Proof. Let μ, ρ ∈ I^Y be r-FI-separated with f (λ) = μ ∨ ρ. That is, ${cl}_{σ}^{*} (μ, r) \land ρ = {cl}_{σ}^{*} (ρ, r) \land μ = \bar{0}; r \in I_{0} .$ Then, λ ≤ f^-1 (μ) ∨ f^-1 (ρ), and from condition (IC), we get that $\begin{matrix} {cl}_{τ}^{*} (f^{- 1} (μ), r) \land f^{- 1} (ρ) & \leq & {cl}_{τ} (f^{- 1} (μ), r) \land f^{- 1} (ρ) \\ \leq & f^{- 1} ({cl}_{σ}^{*} (μ, r)) \land f^{- 1} (ρ) \\ = & f^{- 1} ({cl}_{σ}^{*} (μ, r) \land ρ) \\ = & f^{- 1} (\bar{0}) = \bar{0}, \end{matrix}$ $\begin{matrix} {cl}_{τ}^{*} (f^{- 1} (ρ), r) \land f^{- 1} (μ) & \leq & {cl}_{τ} (f^{- 1} (ρ), r) \land f^{- 1} (μ) \\ \leq & f^{- 1} ({cl}_{σ}^{*} (ρ, r)) \land f^{- 1} (μ) \\ = & f^{- 1} ({cl}_{σ}^{*} (ρ, r) \land μ) \\ = & f^{- 1} (\bar{0}) = \bar{0} . \end{matrix}$ Hence, f^-1 (μ) and f^-1 (ρ) are r-FI-separated sets in X so that λ ≤ f^-1 (μ) ∨ f^-1 (ρ). But λ is r-FI-connected means, from (3) in Proposition 3.2, that λ ≤ f^-1 (μ) or λ ≤ f^-1 (ρ), which means that f (λ) ≤ μ or f (λ) ≤ ρ. Thus, again from (3) in Proposition 3.2, we get that f (λ) is r-FI-connected. □

Corollary 3. If λ is r-fuzzy connected in (X, τ) or λ is r-FI-connected in $(X, τ, I)$ with respect to a fuzzy ideal $I$ on X, f : (X, τ) → (Y, σ) is fuzzy continuous mapping, then f (λ) is r-fuzzy connected in (Y, σ), and it is not necessary that f (λ) is r-FI-connected with respect to a fuzzy ideal $I^{'}$ on Y. With condition (IC), f (λ) is r-FI-connected whenever λ is r-fuzzy connected or λ is r-FI-connected. Moreover, f (λ) is r-fuzzy connected whenever λ is r-fuzzy connected or λ is r-FI-connected.

Proof. Clear from fuzzy continuity, (IC) and Theorem 3.1. □

The implications in the following diagram are satisfied whenever f satisfies condition (IC).

Only the implications in the following diagram are satisfied whenever f is fuzzy continuous.

Proposition 8. Any fuzzy point x_t, t ∈ I₀ is r-FI-connected, and consequently x₁ ∀ x ∈ X is r-FI-connected.

Proof. Clear. □

Definition 6. Let X be a non-empty set and λ ∈ I^X. Then, λ is r-FI-component if λ is maximal r-FI-connected set in X, that is, if μ ≥ λ and μ is r-FI-connected set, then λ = μ.

Proposition 9. Let $λ \neq \bar{0}$ be r-FI-connected in X and $λ \leq μ \leq {cl}_{τ}^{*} (λ, r); r \in I_{0}$ . Then, μ is r-FI-connected as well.

Proof. Let ν, ρ be r-FI-separated sets in I^X such that μ = ν ∨ ρ. That is, ${cl}_{τ}^{*} (ν, r) \land ρ = {cl}_{τ}^{*} (ρ, r) \land ν = \bar{0}; r \in I_{0} .$ Since λ ≤ μ implies that λ ≤ (ν ∨ ρ) and λ is r-FI-connected, then from (3) in Proposition 3.2, we have λ ≤ ν or λ ≤ ρ. From $μ \leq {cl}_{τ}^{*} (λ, r)$ we get that

If λ ≤ ν, then $ρ = (ν \lor ρ) \land ρ = μ \land ρ \leq {cl}_{τ}^{*} (λ, r) \land ρ \leq {cl}_{τ}^{*} (ν, r) \land ρ = \bar{0}$ . If λ ≤ ρ, then $ν = (ν \lor ρ) \land ν = μ \land ν \leq {cl}_{τ}^{*} (λ, r) \land ν \leq {cl}_{τ}^{*} (ρ, r) \land ν = \bar{0}$ . Hence, μ is r-FI-connected. □

4 Compactness in fuzzy ideal topological spaces

This section is devoted to introduce the notion of r-fuzzy ideal compact spaces.

Definition 7. Let $(X, τ, I)$ be a fuzzy ideal topological space, λ ∈ I^X, r ∈ I₀. Then,

λ is said to be r-fuzzy I-compact (r-FI-compact, for short) if for every family {μ_j ∈ I^X : τ (μ_j) ≥ r j ∈ J} with λ ≤ ⋁ _j∈Jμ_j, there exists a finite subset J₀ of J such that $I (λ \bar{\land} (⋁_{j \in J_{0}} μ_{j})) \geq r .$

λ is said to be r-fuzzy almost I-compact (r-FAI-compact, for short) if for every family {μ_j ∈ I^X : τ (μ_j) ≥ r j ∈ J} with λ ≤ ⋁ _j∈Jμ_j, there exists a finite subset J₀ of J such that $I (λ \bar{\land} (⋁_{j \in J_{0}} {cl}_{τ}^{*} (μ_{j}, r))) \geq r .$

λ is said to be r-fuzzy nearly I-compact (r-FNI-compact for short) if for every family {μ_j ∈ I^X : τ (μ_j) ≥ r j ∈ J} with λ ≤ ⋁ _j∈Jμ_j, there exists a finite subset J₀ of J such that $I (λ \bar{\land} (⋁_{j \in J_{0}} {int}_{τ} ({cl}_{τ}^{*} (μ_{j}, r), r))) \geq r .$

It is clear that: r-FI-compactness ⇒ r-FAI-compactness ⇒ r-FNI-compactness.

Remark 3. If $I = I^{\circ}$ , then the concepts of:

r-fuzzy compact and r-FI-compact are equivalent.

r-fuzzy almost compact and r-FAI-compact are equivalent.

r-fuzzy nearly compact and r-FNI-compact are equivalent.

Definition 8. Let $(X, τ, I)$ be a fuzzy ideal topological space. Then, X is said to be r-fuzzy I-regular space if for each λ ∈ I^X, r ∈ I₀ with τ (λ) ≥ r, $λ = ⋁_{j \in J} {λ_{j} : τ (λ_{j}) \geq r, {cl}_{τ}^{*} (λ_{j}, r) \leq λ} .$ It is clear that every r-fuzzy regular space is an r-fuzzy I-regular space. But if $I = I^{\circ}$ , then the concept of r-fuzzy I-regular space and r-fuzzy regular space are equivalent.

Theorem 4. Let $(X, τ, I)$ be r-FAI-compact and r-fuzzy I-regular. Then, X is an r-FI-compact space.

Proof. For every family {μ_j ∈ I^X : τ (μ_j) ≥ r, j ∈ J} with $\bar{1} = ⋁_{j \in J} μ_{j}$ . By r-fuzzy I-regularity of X, then for each τ (μ_j) ≥ r, we have $μ_{j} = ⋁_{j_{k} \in J_{K}} {μ_{j_{k}} : τ (μ_{j_{k}}) \geq r, {cl}_{τ}^{*} (μ_{j_{k}}, r) \leq μ_{j}} .$ Hence, $\bar{1} = ⋁_{j \in J} (⋁_{j_{k} \in J_{K}} μ_{j_{k}})$ . Since X is r-FAI-compact, then there exists a finite index subset J₀ × J_K of J such that $I (\bar{1} \bar{\land} (⋁_{j \in J_{0}} (⋁_{j_{k} \in J_{K}} {cl}_{τ}^{*} (μ_{j_{k}}, r)))) \geq r .$

For each j ∈ J₀, $⋁_{j_{k} \in J_{K}} {cl}_{τ}^{*} (μ_{j_{k}}, r) \leq μ_{j}$ , which implies that $\bar{1} \bar{\land} (\underset{j \in j_{0}}{\lor} (\underset{j_{k} \in j_{k}}{\lor} {cl}_{τ}^{*} (μ_{j k,} r))) \geq \bar{1} \bar{\land} (\underset{j \in j_{0}}{\lor} μ_{j})$ Therefore, $I (\bar{1} (⋁_{j \in J_{0}} μ_{j})) \geq r$ , and thus $(X, τ, I)$ is r-FI-compact. □

Theorem 5. Let $(X, τ, I)$ be r-FNI-compact and r-fuzzy I-regular. Then, X is an r-FI-compact space.

Proof. Similar to the proof of Theorem 4.1. □

Theorem 6. Let $f : (X, τ, I_{1}) \to (Y, σ, I_{2})$ be injective fuzzy continuous mapping, λ ∈ I^X is an r-FI-compact and $I_{1} (ν) \leq I_{2} (f (ν)) \forall ν \in I^{X}$ . Then, f (λ) is r-FI-compact as well.

Proof. Let {μ_j ∈ I^Y : σ (μ_j) ≥ r j ∈ J} be a family with f (λ) ≤ ⋁ _j∈Jμ_j. By fuzzy continuity of f, τ (f^-1 (μ_j)) ≥ r and λ ≤ ⋁ _j∈Jf^-1 (μ_j). By r-FI-compactness of λ, there exists a finite subset J₀ of J such that $I_{1} (λ \bar{\land} (⋁_{j \in J_{0}} (f^{- 1} (μ_{j})))) \geq r .$

Since $I_{1} (ν) \leq I_{2} (f (ν)) \forall ν \in I^{X}$ , then $I_{2} [f (λ \bar{\land} (⋁_{j \in J_{0}} (f^{- 1} (μ_{j}))))] \geq r .$ From f is injective, then f (λ $\bar{\land}$ (⋁ _j∈J₀ (f^-1 (μ_j)))) = f (λ) $\bar{\land}$ (⋁ _j∈J₀ (μ_j)). Thus, $I_{2} (f (λ) \bar{\land} (⋁_{j \in J_{0}} (μ_{j}))) \geq r .$ Hence, f (λ) is r-FI-compact. □

The concept of a fuzzy operation, associated with a fuzzy topology τ, on a set X is a map α : I^X × I₀ → I^X so that int_τ ≤ α ≤ cl_τ. This type of maps is called an expansion on X or a fuzzy operator on (X, τ). Let (X, τ₁) and (Y, τ₂) be two fuzzy topological spaces, α and β are fuzzy operators on X, θ and δ are fuzzy operators on Y, respectively [2].

Definition 9. Let $(X, τ, I)$ be a fuzzy ideal topological space, α a fuzzy operator on X and λ ∈ I^X, r ∈ I₀. Then, λ is called r-fuzzy ideal α-compact (r-FIα-compact for short) if for each family {μ_j ∈ I^X : τ (μ_j) ≥ r, j ∈ J} with λ ≤ ⋁ _j∈Jμ_j, there exists a finite subset J₀ of J such that $I (λ \bar{\land} (⋁_{j \in J_{0}} α (μ_{j}, r))) \geq r .$ It is clear that for α = identity operator (resp. $α = {cl}_{τ}^{*}$ and $α = {int}_{τ} {cl}_{τ}^{*}$ ), we get the r-FI-compact (resp. r-FAI-compact and r-FNI-compact).

Definition 10. [2] A mapping $f : (X, τ, I_{1}) \to (Y, σ, I_{2})$ is said to be fuzzy ideal $(α, β, θ, δ, I)$ -continuous if for every μ ∈ I^Y, $I_{1} [α (f^{- 1} (δ (μ, r)), r) \bar{\land} β (f^{- 1} (θ (μ, r)), r)] \geq σ (μ);$ r ∈ I₀, where α, β are fuzzy operators on X and θ, δ are fuzzy operators on Y.

The concept of fuzzy almost ideal continuous (FAIC for short) mapping is defined by: for every μ ∈ I^Y, r ∈ I₀ with σ (μ) ≥ r, then $f^{- 1} (μ) \leq {int}_{τ} (f^{- 1} ({int}_{σ} ({cl}_{σ}^{*} (μ, r), r)), r) .$ Here, $α = {id}_{X}, β = {int}_{τ}, δ = {id}_{Y}, θ = {int}_{σ} {cl}_{σ}^{*}$ and $I_{1} = I^{\circ}$ .

The concept of fuzzy weakly ideal continuous (FWIC for short) mapping is defined by: for every μ ∈ I^Y, r ∈ I₀ with σ (μ) ≥ r, then $f^{- 1} (μ) \leq {int}_{τ} (f^{- 1} ({cl}_{σ}^{*} (μ, r)), r) .$ Here, $α = {id}_{X}, β = {int}_{τ}, δ = {id}_{Y}, θ = {cl}_{σ}^{*}$ and $I_{1} = I^{\circ}$ .

The concept of fuzzy almost weakly ideal continuous (FAWIC for short) mapping is defined by: for every μ ∈ I^Y, r ∈ I₀ with σ (μ) ≥ r, then $f^{- 1} (μ) \leq {int}_{τ} ({cl}_{τ} (f^{- 1} ({cl}_{σ}^{*} (μ, r)), r), r) .$ Here, $α = {id}_{X}, β = {int}_{τ} {cl}_{τ}, δ = {id}_{Y}, θ = {cl}_{σ}^{*}$ and $I_{1} = I^{\circ}$ .

The following diagram commutes.

For $I_{2} = I^{\circ}$ , we get that:

(FAC ⇔ FAIC), (FWC ⇔ FWIC) and (FAWC ⇔ FAWIC).

Theorem 7. Let $(X, τ, I_{1})$ and $(Y, σ, I_{2})$ be fuzzy ideal topological spaces, α a fuzzy operator on X and δ, θ are fuzzy operators on Y such that $ν \leq α (ν, r) \forall ν \in I^{X}, ρ \leq δ (ρ, r) \forall ρ \in I^{Y}, r \in I_{0},$ and f : X → Y is an injective mapping with α (f^-1 (δ (ρ, r)) , r) ≤ int_τ (f^-1 (θ (ρ, r)) , r) ∀ ρ ∈ I^Y, r ∈ I₀ . If μ ∈ I^X is r-FI-compact and $I_{1} (μ) \leq I_{2} (f (μ))$ . Then, f (μ) ∈ I^Y is r-FIθ-compact.

Proof. Let {λ_j ∈ I^Y : σ (λ_j) ≥ r, j ∈ J} be a family with f (μ) ≤ ⋁ _j∈Jλ_j. Then, take μ_j = int_τ (f^-1 (θ (λ_j, r)) , r) with τ (μ_j) ≥ r such that α (f^-1 (δ (λ_j, r)) , r) ≤ μ_j ≤ f^-1 (θ (λ_j, r)) . Also, since f^-1 (δ (λ_j, r)) ≤ α (f^-1 (δ (λ_j, r)) , r), f^-1 (λ_j) ≤ f^-1 (δ (λ_j, r)), then $f^{- 1} (λ_{j}) \leq f^{- 1} (δ (λ_{j}, r)) \leq μ_{j} \leq f^{- 1} (θ (λ_{j}, r)),$ which means that $μ \leq ⋁_{j \in J} f^{- 1} (λ_{j}) \leq ⋁_{j \in J} μ_{j} \leq f^{- 1} (⋁_{j \in J} θ (λ_{j}, r)),$ that is, μ ≤ ⋁ _j∈Jμ_j and μ is r-FI-compact. Then, there exists a finite set J₀ ⊆ J such that $I_{1} (μ \bar{\land} (⋁_{j \in J_{0}} μ_{j})) \geq r .$ From f is injective, then f (μ $\bar{\land}$ (⋁ _j∈J₀μ_j)) = f (μ) $\bar{\land}$ (⋁ _j∈J₀ (f (μ_j))) ≥ f (μ) $\bar{\land}$ (⋁ _j∈J₀θ (λ_j, r)). Thus, $I_{2} (f (λ) \bar{\land} (⋁_{j \in J_{0}} (θ (μ_{j}, r)))) \geq r .$ Hence, f (λ) is r-FIθ-compact. □

Corollary 4. Let $(X, τ, I_{1})$ and $(Y, σ, I_{2})$ be fuzzy ideal topological spaces. Let f : X → Y be an injective FWIC mapping, $I_{1} (ν) \leq I_{2} (f (ν)) \forall ν \in I^{X}$ , and μ ∈ I^X is an r-FI-compact. Then, f (μ) ∈ I^Y is an r-FAI-compact.

Proof. Let α = id_X, $θ = {cl}_{σ}^{*}$ , δ = id_Y, and $I = I^{\circ}$ . Then, the result follows from Theorem 4.4. □

Corollary 5. Let $(X, τ, I_{1})$ and $(Y, σ, I_{2})$ be fuzzy ideal topological spaces. Let f : X → Y be an injective FAIC mapping, $I_{1} (ν) \leq I_{2} (f (ν)) \forall ν \in I^{X}$ , and μ ∈ I^X is an r-FI-compact. Then, f (μ) ∈ I^Y is an r-FNI-compact.

Proof. Let α = id_X, $θ = {int}_{σ} {cl}_{σ}^{*}$ , δ = id_Y, and $I = I^{\circ}$ . Then, the result follows from Theorem 4.4. □

5 Conclusion

Joining the concept of fuzzy ideal to the concept of fuzzy topology on an underlying set X has some effects as we shown in the paper. Fuzzy separation axioms, fuzzy connectedness and fuzzy compactness defined in a fuzzy ideal topological space were different from those defined in a fuzzy topological space. Although the concept of fuzzy ideal is independent from the concept of fuzzy topology, but studying the fuzzy ideal topological spaces added some results new and different.

Footnotes

Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions helping in improving this paper. The authors are grateful to the Deanship of Scientific Research of Jazan University for supporting financially this work under the Waed grant No. (7-39).

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Fuzzy ideal topological spaces

Abstract

Keywords

1 Introduction and Preliminaries

2 Fuzzy ideal r-(t, s)-T i separation axioms

5 Conclusion

Footnotes

Acknowledgments

References

2 Fuzzy ideal r-(t, s)-T_i separation axioms