t-syntopogenous structures compatible with fuzzy t-uniformities and fuzzy t-neighbourhoods...

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Fuzzy Sets and Systems ••• (••••) •••–•••www.elsevier.com/locate/fss

T -syntopogenous structures compatible with fuzzyT -uniformities and fuzzy T -neighbourhoods structures

Nehad N. Morsi a, Khaled A. Hashem b,∗

a Department of Basic Science, Arab Academy for Sciences, Technology and Maritime Transport, P.O. Box 2033,Al-Horraya, Heliopolis, Cairo, Egypt

b Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt

Received 2 February 2012; received in revised form 7 February 2014; accepted 12 February 2014

Abstract

In this paper, we continue the discussion of T -syntopogenous spaces, which were introduced by the second author in 2012. Thismanuscript shows that the T -syntopogenous spaces agree well with Hashem–Morsi fuzzy T -neighbourhood spaces (2003), fuzzyT -proximity spaces (2002) and Höhle fuzzy T -uniform spaces (1982). Also, we show that each syntopogenous space in the senseof Csàszàr (1963), generates a T -syntopogenous space. Finally, we deduce the concept of fuzzy T -neighbourhood syntopogenousstructures and some related results.© 2014 Elsevier B.V. All rights reserved.

Keywords: T -syntopogenous spaces; Fuzzy T -neighbourhood spaces; Fuzzy T -uniform spaces

0. Introduction

The concept of a T -syntopogenous structure was introduced and studied in [7] which is closely connected withLowen I -topological space [12]. A.K. Katsaras [11] deduced the notion of fuzzy syntopogenous spaces and showedthat there are correspondences between those spaces and each of Lowen fuzzy neighbourhood spaces [14], fuzzyuniform spaces [13], Artico–Moresco fuzzy proximity spaces [1] and Csàszàr syntopogenous spaces [2]. In thismanuscript, we continue our study of T -syntopogenous spaces. We show how the T -syntopogenous spaces agreewell with the fuzzy T -neighbourhood spaces [4], fuzzy T -uniform spaces [9], fuzzy T -proximity spaces [3] andsyntopogenous spaces [2]. More precisely, it is shown that there is a one-to-one correspondence between the fuzzyT -neighbourhood spaces and the so-called perfect T -syntopogenous spaces. We show that every T -syntopogenousstructure induces an I -topology which is given by some fuzzy T -neighbourhood structure. Also, there exists aone-to-one correspondence between the Höhle fuzzy T -uniform spaces and the biperfect T -syntopogenous spaces.We prove that there is a one-to-one correspondence between the fuzzy T -proximity spaces and the symmetrical

* Corresponding author.E-mail addresses: [email protected], [email protected] (K.A. Hashem).

http://dx.doi.org/10.1016/j.fss.2014.02.0080165-0114/© 2014 Elsevier B.V. All rights reserved.

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2 N.N. Morsi, K.A. Hashem / Fuzzy Sets and Systems ••• (••••) •••–•••

T -syntopogenous spaces. These correspondences generalize the correspondences between A.K. Katsaras fuzzysyntopogenous spaces (1990) and each of Lowen fuzzy neighbourhood spaces, fuzzy uniform spaces and Artico–Moresco fuzzy proximity spaces (see [11]). We also show that the functional fuzzy T -separatedness [6] may generatea T -topogenous order [7]. Moreover, we show that every syntopogenous space (in the sense of Csàszàr [2]) inducesa T -syntopogenous space. Finally, we deduce the notion of fuzzy T -neighbourhood syntopogenous structure within afuzzy T -neighbourhood structure.

We proceed as follows:In Section 1, we supply some basic definitions and ideas of fuzzy sets, I -topological spaces and T -syntopogenous

spaces which will be needed in the sequel.In Section 2, we show that there is a one-to-one correspondence between the fuzzy T -neighbourhood structures

and the perfect T -syntopogenous structures.In Section 3, we prove that there exists a one-to-one correspondence between the fuzzy T -uniform spaces and the

biperfect T -syntopogenous spaces, together with an illustrative example of this notion.In Section 4, we show that there is a one-to-one correspondence between the fuzzy T -proximity spaces and the

symmetrical T -syntopogenous spaces.In Section 5, we generate a T -syntopogenous structure from a functional fuzzy T -separatedness.In Section 6, we show that the ordinary syntopogenous structure can induce a T -syntopogenous structure.In Section 7, we introduce the notion of fuzzy T -neighbourhood syntopogenous structures and we show there is

a one-to-one correspondence between the fuzzy T -neighbourhood structures and the fuzzy T -neighbourhood synto-pogenous structures.

1. Preliminaries

In this section, we will recall some of the definitions and results related to fuzzy sets, I -topological spaces andfuzzy T -syntopogenous spaces.

A triangular norm (cf. [16]) is a binary operation on the unit interval I = [0,1] that is associative, symmetric,isotone in each argument and has the neutral element 1. The triangular conorm of a triangular norm T is the binaryoperation T ∗ on the unit interval I given by:

α T ∗ β = 1 − [(1 − α) T (1 − β)], α,β ∈ I.

For a continuous triangular norm T , the following binary operation on I

J(α, γ ) = sup{θ ∈ I : α T θ � γ }, α, γ ∈ I ;is called the residual implication of T [10,15].

The residuated implication J, enjoys the following properties:

α T β � γ iff α � J(β, γ ) ∀α,β, γ ∈ I. (1)

α T β > γ iff α > J(β, γ ) ∀α,β, γ ∈ I. (2)

A fuzzy set λ in a universe set X, introduced by H.Y. Li and J.G. Peng [17], is a function λ : X −→ I . The collectionof all fuzzy sets of X is denoted by IX. The height of a fuzzy set λ is the following real number:

hgtλ = sup{λ(x): x ∈ X

}.

If H is a subset of X, then we shall denote its characteristic function by the symbol 1H, said to be crisp fuzzy subsetof X. The collection of all crisp fuzzy subsets of X is denoted by 2X. We also denote the constant fuzzy set of X withvalue α ∈ I by α.

Given a fuzzy set λ ∈ IX and a real number α ∈ I1 = [0,1[, the strong α-cut of λ is the following subset of X:

λα = {x ∈ X: λ(x) > α};

and for a real number α ∈ I , the weak α-cut of λ, is the subset of X:

λα∗ = {x ∈ X: λ(x) � α}.


N.N. Morsi, K.A. Hashem / Fuzzy Sets and Systems ••• (••••) •••–••• 3

It is directly verified that every λ ∈ IX has the following formulation

λ =∨α∈I

[α T 1λα∗ ]. (3)

Given two fuzzy sets μ,λ ∈ IX, we denote by μT λ the following fuzzy set of X: (μT λ)(x) = μ(x)T λ(x), x ∈ X.

Lemma 1.1. (See [3].) For every μ,λ ∈ IX and α ∈ I , we have

(i) hgtλ < α ⇒ λα∗ = Ø;(ii) (μ T λ)α∗ =⋃θT β�α(μθ∗ ∩ λβ∗).

Lemma 1.2. (See [5].) For every λ ∈ IX and α,γ ∈ I , we have

(α T λ)γ = λJ(α,γ ).

Lemma 1.3. For every λ ∈ IX and α,γ ∈ I , we have[(1 − α) T ∗ λ

]γ ∗ = λ(1−J(α,1−γ ))∗ .

Proof. Let λ ∈ IX and α,γ ∈ I . Then[(1 − α) T ∗ λ

]γ ∗ = {x ∈ X: (1 − α) T ∗ λ(x) � γ

}= {x ∈ X: 1 − [α T (1 − λ)(x)

]� γ}

= {x ∈ X: α T (1 − λ)(x) � 1 − γ}

= {x ∈ X: (1 − λ)(x) � J(α,1 − γ )}, by (1)

= {x ∈ X: λ(x)� 1 − J(α,1 − γ )}

= λ(1−J(α,1−γ ))∗ .

This proves our assertion. �Here, we will recall the notion of T -syntopogenous spaces.

Definition 1.1. (See [7].) A T -topogenous order on a set X is a function ζ : IX × IX −→ I , which satisfies, for anyμ,λ, ν ∈ IX and α ∈ I , the following:

(TT1) ζ(1, α) = α and ζ(α,0) = 1 − α;

(TT2) ζ(μ ∨ λ, ν) = ζ(μ, ν) ∧ ζ(λ, ν) and ζ(μ,λ ∧ ν) = ζ(μ,λ) ∧ ζ(μ, ν);

(TT3) if ζ(μ,λ) > 1 − (θ T β) for some θ,β ∈ I0 = ]0,1], there is C ⊆ X such that ζ(μ,1C) � 1 − θ and ζ(1C, λ)

� 1 − β;

(TT4) ζ(μ,λ) � 1 − hgt[μ T (1 − λ)];(TT5) ζ(α T μ,λ) = (1 − α) T ∗ ζ(μ,λ) = ζ(μ, (1 − α) T ∗ λ).

Definition 1.2. (See [7].) A T -topogenous order ζ on a set X is said to be:

(i) perfect if ζ(∨

j∈J μj ,λ) =∧j∈J ζ(μj ,λ), μj ,λ ∈ IX;

(ii) biperfect if it is perfect and ζ(μ,∧

j∈J λj ) =∧j∈J ζ(μ,λj ), μ,λj ∈ IX;

(iii) symmetrical if ζ(μ,λ) = ζ(1 − λ,1 − μ), μ,λ ∈ IX.

Definition 1.3. (See [7].) A T -syntopogenous structure on a set X is a family ℘ of T -topogenous orders on X satisfyingthe following conditions



(TS1) ℘ is directed in the sense that, given ζ, η ∈ ℘ there is ξ ∈ ℘ such that ξ � ζ ∨ η;(TS2) for every ζ ∈ ℘ and ε ∈ I0, there is ζε ∈ ℘ such that (ζε oT ζε) + ε � ζ ,

where the T -composition of T -topogenous orders ζ and η define as:

(ζ oT η)(μ,λ) = supC⊆X

[η(μ,1C) T ζ(1C, λ)

], μ,λ ∈ IX.

The pair (X,℘) is said to be a T -syntopogenous space.If a T -syntopogenous structure ℘ on X consists of a single T -topogenous order, then ℘ is called a T -topogenous

structure and (X,℘) a T -topogenous space.

Definition 1.4. (See [7].) Let (X,℘) and (Y,L) be T -syntopogenous spaces. A function f : X −→ Y is calledsyntopogenously continuous, if for every η ∈ L there is ζ ∈ ℘ such that

η(ν,ρ) � ζ(f ←(ν), f ←(ρ)

), ν, ρ ∈ IY.

Equivalently, if for every η ∈ L there is ζ ∈ ℘ such that

η(f (μ),1 − f (λ)

)� ζ(μ,1 − λ), μ,λ ∈ IX.

For an I -topology τ on a set X we mean a subset of IX which contains all constant fuzzy sets α, α ∈ I and isclosed with respect to finite infima and arbitrary suprema [12]. The pair (X, τ ) is called an I -topological space.

Proposition 1.1. (See [7].) Let (X,℘) be a T -syntopogenous space. Then the I -topology τ(℘) induced by ℘, is givenby the fuzzy interior operator:

μo(x) = supζ∈℘

ζ(1x,μ), μ ∈ IX, x ∈ X;

or equivalently, by the fuzzy closure operator:

μ−(x) = infζ∈℘

[1 − ζ(1x,1 − μ)

], μ ∈ IX, x ∈ X.

2. Compatibility between T -syntopogenous spaces and fuzzy T -neighbourhood spaces

In this section, we show that there is a one-to-one correspondence between the family of perfect T -syntopogenousspaces and the family of fuzzy T -neighbourhood spaces.

The fuzzy T -neighbourhood spaces were introduced by Hashem and Morsi, for more definitions and properties,we refer to [4].

Definition 2.1. (See [4].) A (fuzzy) T -neighbourhood space is an I -topological space (X,− ) whose fuzzy closureoperator − is induced by some indexed family Σ = (Σ(x))x∈X of I -filters in IX, in the following manner:

μ−(x) = infν∈Σ(x)

hgt(μ T ν), for all μ ∈ IX and x ∈ X.

Theorem 2.1. (See [4].) An I -topological space (X,− ) is a T -neighbourhood space if and only if (α T μ)− = αT μ−,

for every α ∈ I and μ ∈ IX.

Lemma 2.1. (See [6].) If (X,− ) is a T -neighbourhood space, then for all μ ∈ IX and H ⊆ X, we have μ T (1H)− �(μ T 1H)−.

Now, we show how a T -syntopogenous structure can generate a T -neighbourhood structure.

Theorem 2.2. If (X,℘) is a T -syntopogenous space, then (X, τ (℘)) = (X,− ) is a T -neighbourhood space.



Proof. For every μ ∈ IX, α ∈ I and x ∈ X, we have(α T μ−)(x) = α T μ−(x)

= α T infζ∈℘

[1 − ζ(1x,1 − μ)

], by Proposition 1.1

= infζ∈℘

{α T[1 − ζ(1x,1 − μ)

]}, by continuity of T

= infζ∈℘

{1 − [(1 − α) T ∗ ζ(1x,1 − μ)

]}= inf

ζ∈℘

{1 − ζ(1x, (1 − α) T ∗ (1 − μ)

)}, by (TT5)

= infζ∈℘

[1 − ζ(1x,1 − (α T μ)

)]= (α T μ)−(x).

Therefore, by Theorem 2.1, (X, τ (℘)) is a T -neighbourhood space. �Next, we see how a T -neighbourhood structure may generate a T -syntopogenous structure.Take a T -neighbourhood space (X,− ) = (X, τ ) and define a mapping ζτ : IX × IX −→ I by:

ζτ (μ,λ) = 1 − hgt[μ T (1 − λ)−

], μ,λ ∈ IX. (4)

Theorem 2.3. Let (X,− ) = (X, τ ) be a T -neighbourhood space. Then the map ζτ defined above is a perfect T -topo-genous order on X, satisfying τ = τ({ζτ }).

Proof. In fact, it is easy to see that ζτ satisfies (TT1), (TT4) of Definition 1.1. Now, for all μ,λ, ν ∈ IX and α ∈ I ,we have

(TT2)

ζτ (μ ∨ λ, ν) = 1 − hgt[(μ ∨ λ) T (1 − ν)−

]= 1 − hgt

{[μ T (1 − ν)−

]∨ [λ T (1 − ν)−]}

= 1 − {hgt[μ T (1 − ν)−

]∨ hgt[λ T (1 − ν)−

]}= {1 − hgt

[μ T (1 − ν)−

]}∧ {1 − hgt[λ T (1 − ν)−

]}= ζτ (μ, ν) ∧ ζτ (λ, ν);

ζτ (μ,λ ∧ ν) = 1 − hgt{μ T[1 − (λ ∧ ν)

]−}= 1 − hgt

{μ T[(1 − λ) ∨ (1 − ν)

]−}= 1 − hgt

{μ T[(1 − λ)− ∨ (1 − ν)−

]}= {1 − hgt

[μ T (1 − λ)−

]}∧ {1 − hgt[μ T (1 − ν)−

]}= ζτ (μ,λ) ∧ ζτ (μ, ν).

(TT3) Suppose that ζτ (μ,λ) > 1 − (θ T β) for some θ,β ∈ I0. Then θ T β > hgt[μ T (1 − λ)−], which implies

Ø = [μ T (1 − λ)−](θT β)∗ , by Lemma 1.1(i)

=⋃

εT γ�θT β

[με∗ ∩ ((1 − λ)−

)γ ∗], by Lemma 1.1(ii)

⊇ μθ∗ ∩ ((1 − λ)−)β∗ , that is

μθ∗ ⊆ X − ((1 − λ)−)β∗ . (5)



By taking C = μθ∗ ⊆ X, we have

ζτ (μ,1C) = 1 − hgt[μ T (1 − 1C)−

]= 1 − hgt

[μ T (1 − 1μθ∗ )

−]� 1 − hgt

[μ T (1 − 1μθ∗ )

]−, by Lemma 2.1

� 1 − hgt(θ)−, clear

= 1 − θ, and

ζτ (1C, λ) = 1 − hgt[1C T (1 − λ)−

]= 1 − hgt

[1μθ∗ T (1 − λ)−

]� 1 − hgt

[1(X−((1−λ)−)β∗ ) T (1 − λ)−

], by (5)

� 1 − β.

(TT5)

ζτ (α T μ,λ) = 1 − hgt[(α T μ) T (1 − λ)−

]= 1 − {α T hgt

[μ T (1 − λ)−


= (1 − α) T ∗ {1 − hgt[μ T (1 − λ)−

]}= (1 − α) T ∗ ζτ (μ,λ), by (4)

= (1 − α) T ∗ {1 − hgt[μ T (1 − λ)−

]}= 1 − {α T hgt

[μ T (1 − λ)−

]}= 1 − hgt

{μ T[α T (1 − λ)−


= 1 − hgt{μ T[α T (1 − λ)

]−}, by Theorem 2.1

= 1 − hgt{μ T[1 − ((1 − α) T ∗ λ

)]−}= ζτ

(μ, (1 − α) T ∗ λ

), by (4) again.

This demonstrates that ζτ is a T -topogenous order on X.Now, we show that ζτ is a perfect, as follows

ζτ

(∨j∈J

μj ,λ

)= 1 − hgt

[(∨j∈J

μj

)T (1 − λ)−

]

= 1 −{∨

j∈J

hgt[μj T (1 − λ)−

]}

=∧j∈J

{1 − hgt

[μj T (1 − λ)−

]}

=∧j∈J

ζτ (μj ,λ).

Next, we prove that {ζτ }, is a T -topogenous structure.

(TS1) Obviously {ζτ } is directed.(TS2) Put α T α < ζτ (μ,λ), α ∈ I1. We need to show that there is C ⊂ X such that ζτ (μ,1C) T ζτ (1C, λ) � α T α.

Take C = {x ∈ X: (1 − λ)−(x) < 1 − α} ∪ {y ∈ X: (1 − μ)(y) < α}, we have



ζτ (1C, λ) = 1 − hgt[1C T (1 − λ)−

]= 1 − sup

x∈C(1 − λ)−(x)

� α and

ζτ (μ,1C) = 1 − hgt[μ T (1 − 1C)−

]= 1 − hgt

[μ T (1(X\C))

−]� 1 − hgt[μ T 1(X\C)]−, by Lemma 2.1

� 1 − hgt(1 − α)−

= α,

consequently (ζτ oT ζτ )(μ,λ) � ζτ (μ,1C) T ζτ (1C, λ) � α T α.Hence (ζτ oT ζτ ) � ζτ .

This proves our assertion.Finally, it is easy to see that the fuzzy closure operator induced by {ζτ } coincides with −.This completes the proof. �

Corollary 2.1. The mapping τ �−→ ℘τ from the family of all T -neighbourhood topologies on a set X, to the family ofall perfect T -topogenous structures on X, is a one-to-one correspondence.

Proof. By Theorem 2.3, the mapping defined there, between T -neighbourhood topologies and perfect T -topogenousstructures, is well defined. But, by Theorem 2.2, this mapping is evidently inverse to that given in Theorem 2.3. Hence,each of them is a one-to-one correspondence. �Proposition 2.1.

(i) If f : (X,℘) −→ (Y,L) is a syntopogenously continuous function, then f : (X, τ (℘)) −→ (Y, τ (L)) is continu-ous function.

(ii) If f : (X, τ ) −→ (Y, σ ) is a continuous function, then f : (X, {ζτ }) −→ (Y, {ζσ }) is syntopogenously continuousfunction.

Proof. (i) As [7, Theorem 4.3].(ii) Let f : (X, τ ) −→ (Y, σ ) be continuous function and μ,λ ∈ IX. Then

ζσ

(f (μ),1 − f (λ)

)=∧{1 − [f (μ)(y) ∨ (1 − clσ(f (λ))(y)): y ∈ Y

]}, by Proposition 1.1

�∧{

1 − [f (μ)(y) ∨ (1 − f(clσ (λ)

)(y)): y ∈ Y

]}, by hypothesis

=∧{

1 −∨{[

μ(x): x ∈ f ←(y)]∨ [1 −

∨clσ (λ)(x): x ∈ f ←(y)

]}: y ∈ Y

}�∧{

1 − [μ(x) ∨ (1 − clσ (λ)(x))]

: x ∈ X}

= ζτ (μ,1 − λ).

Thus, by Definition 1.4, f is syntopogenously continuous, and this completes the proof. �Corollary 2.1 and Proposition 2.1 entails the following

Proposition 2.2. The category of perfect T -topogenous spaces, with syntopogenously continuous functions as mor-phisms is isomorphic to the category of T -neighbourhood spaces and continuous functions as morphisms.



3. Compatibility between T -syntopogenous spaces and fuzzy T -uniform spaces

In this section, we are going to study the correspondence between our T -syntopogenous structures and fuzzyT -uniform structures defined by Höhle (1982).

In [9], Höhle defines for every ψ,ϕ ∈ IX×X and λ ∈ IX: the T -section of ψ over λ by ψ〈λ〉T(x) = supy∈X[λ(y) T

ψ(y,x)], x ∈ X; the T -composition of ψ,ϕ by (ψ oT ϕ)(x,y) = supz∈X[ϕ(x, z) T ψ(z,y)], x,y ∈ X; the symmetricof ψ by sψ(x,y) = ψ(y,x), x,y ∈ X.

Definition 3.1. (See [9].) (i) A (fuzzy) T -uniform base on a set X is a subset υ ⊆ IX×X which fulfils the followingproperties:

(TUB1) υ is an I -filterbase, i.e. 0 /∈ υ and the meets of two members of υ contain a member of υ;(TUB2) for all ψ ∈ υ and x ∈ X, ψ(x,x) = 1;(TUB3) for all ψ ∈ υ and ε ∈ I0, there is ψε ∈ υ such that

(ψε oT ψε) − ε � sψ.

(ii) A (fuzzy) T -uniformity on X is a saturated T -uniform base on X, where an I -filterbase υ is said to be satu-rated [13], when

υ = {ψ ∈ IX×X: ∀ε > 0 ∃ψε ∈ υ with ψε − ε � ψ}.

It follows that a T -uniformity μ will contain sψ for all ψ ∈ μ.

The pair (X,μ) consisting of a set X and a T -uniformity μ on X is called a (fuzzy) T -uniform space and theI -topology determined by μ, denoted by τ(μ), is given by the fuzzy closure operator:

λ− = infψ∈μ

sψ〈λ〉T, for all λ ∈ IX. (6)

Given T -uniform spaces (X,μ), (Y,W) and a function f : X −→ Y, we say that f is uniformly continuous if forevery ϕ ∈ W there is ψ ∈ μ such that ψ � (f × f )←(ϕ).

Lemma 3.1. (See [3].) For a T -uniformity μ on a set X and μ,λ ∈ IX, hgt(ψ〈μ〉T T λ) = hgt(ψ〈μ〉T T sψ〈λ〉T) =hgt(μ T sψ〈λ〉T), ∀ψ ∈ μ.

Now, we see how a T -uniformity structure can generate a T -syntopogenous structure.Take a T -uniform space (X,μ) and for any ψ ∈ μ, we define a mapping ζψ : IX × IX −→ I by:

ζψ(μ,λ) = 1 − hgt[ψ〈μ〉T T sψ〈1 − λ〉T

], μ,λ ∈ IX. (7)

Theorem 3.1. If (X,μ) is a T -uniform space and ψ ∈ μ, then the mapping ζψ defined above is a biperfect T -topo-genous order on X.

Proof. Let μ,λ, ν ∈ IX and α ∈ I . Then

(TT1) Trivially holds.

(TT2)

ζψ(μ ∨ λ, ν) = 1 − hgt[ψ〈μ ∨ λ〉T T sψ〈1 − ν〉T

]= 1 − hgt

[(μ ∨ λ) T sψ〈1 − ν〉T

], by Lemma 3.1

= 1 − hgt{[

μ T sψ〈1 − ν〉T]∨ [λ T sψ〈1 − ν〉T

]}= {1 − hgt

[μ T sψ〈1 − ν〉T

]}∧ {1 − hgt[λ T sψ〈1 − ν〉T

]}= ζψ(μ, ν) ∧ ζψ(λ, ν), by Lemma 3.1



By a similar way, we can prove

ζψ(μ,λ ∧ ν) = ζψ(μ,λ) ∧ ζψ(μ, ν).

(TT3) Let ζψ(μ,λ) > 1 − (θ T β) for some θ,β ∈ I0. Then

hgt[ψ〈μ〉T T sψ〈1 − λ〉T

]< θ T β

i.e. Ø = [ψ〈μ〉T T sψ〈1 − λ〉T](θT β)∗ , by Lemma 1.1(i)

=⋃

εT γ�θT β

[(ψ〈μ〉T

)ε∗ ∩ (sψ〈1 − λ〉T

)γ ∗], by Lemma 1.1(ii)

⊇ (ψ〈μ〉T)θ∗ ∩ (sψ〈1 − λ〉T

)β∗ , that is(

ψ〈μ〉T)θ∗ ⊆ X − (sψ〈1 − λ〉T

)β∗ . (8)

Now, by taking C = (ψ〈μ〉T)θ∗ ⊆ X, we have

ζψ(μ,1C) = 1 − hgt[ψ〈μ〉T T (1 − 1C)

], by Lemma 3.1

= 1 − hgt[ψ〈μ〉T T (1 − 1(ψ〈μ〉T )θ∗ )

]= 1 − hgt

[ψ〈μ〉T T 1(X−(ψ〈μ〉T )θ∗ )

]� 1 − θ and

ζψ(1C, λ) = 1 − hgt[1C T sψ〈1 − λ〉T

], by Lemma 3.1

= 1 − hgt[1(ψ〈μ〉T )θ∗ T sψ〈1 − λ〉T

]� 1 − hgt

[1(X−(sψ〈1−λ〉T )β∗ ) T sψ〈1 − λ〉T

], by (8)

� 1 − β.

(TT4) Since (ψ〈μ〉T) � μ, ∀ψ ∈ μ and μ ∈ IX, then obviously,

ζψ(μ,λ)� 1 − hgt[μ T (1 − λ)

].

(TT5)

ζψ(α T μ,λ) = 1 − hgt[(α T μ) T sψ〈1 − λ〉T

], by Lemma 3.1

= 1 − {α T hgt[μ T sψ〈1 − λ〉T

]}= (1 − α) T ∗ {1 − hgt

[μ T sψ〈1 − λ〉T

]}= (1 − α) T ∗ ζψ(μ,λ)

= (1 − α) T ∗ {1 − hgt[ψ〈μ〉T T (1 − λ)

]}, by Lemma 3.1

= 1 − {α T hgt[ψ〈μ〉T T (1 − λ)

]}= 1 − hgt

{ψ〈μ〉T T

[α T (1 − λ)

]}= 1 − hgt

{ψ〈μ〉T T

[1 − ((1 − α) T ∗ λ

)]}= ζψ

(μ, (1 − α) T ∗ λ

), by Lemma 3.1 again

This completes the proof of ζψ is a T -topogenous order on X.Now, we show that ζψ is a biperfect, as follows

ζψ

(∨j∈J

μj ,λ

)= 1 − hgt

[(∨j∈J

μj

)T sψ〈1 − λ〉T

]

= 1 −{∨

hgt[μj T sψ〈1 − λ〉T

]}
j∈J



=∧j∈J

{1 − hgt

[μj T sψ〈1 − λ〉T

]}

=∧j∈J

ζψ(μj ,λ).

This demonstrates that ζψ is a perfect.Moreover,

ζψ

(μ,∧j∈J

λj

)= 1 − hgt

[ψ〈μ〉T T

(1 −(∧

j∈J

λj

))]

= 1 − hgt

[ψ〈μ〉T T

(∨j∈J

(1 − λj )

)]

= 1 −{∨

j∈J

hgt[ψ〈μ〉T T (1 − λj )

]}

=∧j∈J

{1 − hgt

[ψ〈μ〉T T (1 − λj )

]}

=∧j∈J

ζψ(μ,λj ).

Which proves our assertion. �The following lemma help in the proof of Proposition 3.1, below, which shows that the T -uniformity structure can

induce a T -syntopogenous structure.

Lemma 3.2. Let (X,μ) be a T -uniform space and ψ,ϕ ∈ μ. Then the T -topogenous orders ζψ , ζϕ satisfy:

(i) ζψ(1x,1 − 1y) = 1 − (ψ oT ψ)(x,y), x,y ∈ X;

(ii) ζψ(μ,λ) = infx,y∈X{[1 − μ(x)] T ∗ λ(y) T ∗ [1 − (ψ oT ψ)(x,y)]}, μ,λ ∈ IX;(iii) ζψ oT ζψ � ζψoTψ ;(iv) For all ε ∈ I0, we have ζψ � ζϕ − ε if and only if ψ oT ψ � (ϕ oT ϕ) + ε.

Proof. Let μ,λ ∈ IX and x,y ∈ X. Then

(i) ζψ(1x,1 − 1y) = 1 − supz∈X

[ψ〈1x〉T T sψ〈1y〉T

](z)

= 1 − supz∈X

[ψ(x, z) T sψ(y, z)

]= 1 − sup

z∈X

[ψ(x, z) T ψ(z,y)

]= 1 − (ψ oT ψ)(x,y).

Rendering (i).

(ii) ζψ(μ,λ) = 1 − hgt[ψ〈μ〉T T sψ〈1 − λ〉T

]= 1 − sup

z∈X

[ψ〈μ〉T T sψ〈1 − λ〉T

](z)

= 1 − supz∈X

{supx∈X

[μ(x) T ψ(x, z)

]T sup

y∈X

[(1 − λ)(y) T sψ(y, z)

]}= 1 − sup sup

[μ(x) T (1 − λ)(y) T ψ(x, z) T sψ(y, z)

]
z∈X x,y∈X



= 1 − supx,y∈X

{[μ(x) T (1 − λ)(y)

]T sup

z∈X

[ψ(x, z) T ψ(z,y)

]}= 1 − sup

x,y∈X

[μ(x) T (1 − λ)(y) T (ψ oT ψ)(x,y)

]= inf

x,y∈X

{[1 − μ(x)

]T ∗ λ(y) T ∗ [1 − (ψ oT ψ)(x,y)

]}.

That is (ii) holds.(iii) Since ζψ is biperfect, then obviously ζψ oT ζψ is biperfect and since μ = ∨x∈X[μ(x) T 1x] and λ =∧

y∈X[λ(y) T ∗ (1 − 1y)], so we have

(ζψ oT ζψ)(μ,λ) =∧

x,y∈X

(ζψ oT ζψ)(μ(x) T 1x, λ(y) T ∗ (1 − 1y)

)=∧

x,y∈X

{[1 − μ(x)

]T ∗ λ(y) T ∗ (ζψ oT ζψ)(1x,1 − 1y)

}, by (TT5)

Since μ is an I -filter and ψ oT ψ � ψ , for any ψ ∈ μ, then ψ oT ψ ∈ μ, and hence ζψoTψ is a T -topogenous orderon X (by Theorem 3.1).

Thus by (ii), we get

ζψoTψ(μ,λ) = infx,y∈X

{[1 − μ(x)

]T ∗ λ(y) T ∗ [1 − (ψ oT ψ oT ψ oT ψ)(x,y)

]}.

It suffices to show that (ζψ oT ζψ)(1x,1 − 1y) � 1 − (ψ oT ψ oT ψ oT ψ)(x,y). For this, let θ ∈ I1 be such thatθ T θ < 1 − (ψ oT ψ oT ψ oT ψ)(x,y), thus for every z ∈ X, we get

1 − (θ T θ) > (ψ oT ψ)(x, z) T (ψ oT ψ)(z,y).

So by (1), we have

(ψ oT ψ)(x, z)� J((ψ oT ψ)(z,y),1 − (θ T θ)

), ∀z ∈ X. (9)

Take C = {z ∈ X: (ψ oT ψ)(x, z) > 1 − θ} ⊆ X, hence by (ii) we have

ζψ(1x,1C) = infy,z∈X

{[1 − 1x(y)

]T ∗ 1C(z) T ∗ [1 − (ψ oT ψ)(y, z)

]}= inf

z/∈C

[1 − (ψ oT ψ)(x, z)

]� θ,

ζψ(1C,1 − 1y) = infx,z∈X

{[1 − 1C(x)

]T ∗ (1 − 1y)(z) T ∗ [1 − (ψ oT ψ)(x, z)

]}= inf

x∈C

[1 − (ψ oT ψ)(x,y)

]� 1 − sup

x∈CJ((ψ oT ψ)(z,y),1 − (θ T θ)

), by (9)

= θ.

It follows that

(ζψ oT ζψ)(1x,1 − 1y)�[ζψ(1x,1C) T ζψ(1C,1 − 1y)

]� θ T θ,

hence (ζψ oT ζψ)(1x,1 − 1y) � 1 − (ψ oT ψ oT ψ oT ψ)(x,y), which proves (ζψ oT ζψ) � ζψoTψ .This renders (iii).(iv) If ζψ � ζϕ , then by (iii), we get ψ oT ψ � (ϕ oT ϕ) + ε.The converse follows from (ii), and this completes the proof. �

Proposition 3.1. If (X,μ) is a T -uniform space, then the family

℘μ = {ζψ : ψ ∈ μ}is a biperfect T -syntopogenous structure on X with τ(μ) = τ(℘μ).



Proof. Let ψ1,ψ2 ∈ μ and take ϕ ∈ μ such that ϕ � ψ1,ψ2. Then by the preceding lemma, we have ζϕ � ζψ1 ∨ ζψ2 ,that is ℘μ is directed.

Also, given ψ ∈ μ and ε ∈ I0, so sψ ∈ μ. By choosing ϕ ∈ μ for which (ϕ oT ϕ) − ε � s (sψ) = ψ , we have ζψ �ζϕoTϕ + ε � (ζϕ oT ζϕ)+ ε. This shows that ℘μ is a T -syntopogenous structure which is a biperfect ( by Theorem 3.1).

It remains to show τ(μ) = τ(℘μ). So that, we denote the fuzzy closure operators associated with τ(μ) and τ(℘μ)

by − and ∼, respectively. Let λ ∈ IX and x ∈ X, we have

λ−(x) = infψ∈μ

sψ〈λ〉T(x), by (6)

= infψ∈μ

hgt[1x T sψ〈λ〉T

]= inf

ψ∈μhgt[ψ〈1x〉T T sψ〈λ〉T

], by Lemma 3.1

= infψ∈μ

{1 − [1 − hgt

(ψ〈1x〉T T sψ〈λ〉T

)]}= inf

ζψ∈℘

[1 − ζψ(1x,1 − λ)

]= λ∼(x), by Proposition 1.1

Thus, λ− = λ∼, which implies τ(μ) = τ(℘μ) and this completes the proof. �Definition 3.2. Let μ and W be T -uniformities on a set X, we say that W ⊆ μ, if for any ϕ ∈ W and ε ∈ I0, there isψε ∈ μ such that ψε − ε � ϕ.

Theorem 3.2. Let μ and W be T -uniformities on a set X. Then ℘W is coarser than ℘μ if and only if W ⊆ μ.

Proof. Suppose that W ⊆ μ, then obviously ℘W is coarser than ℘μ. Conversely, let ℘W be coarser than ℘μ and letϕ ∈ W , so sϕ ∈ W . Now, by (TUB3) we have for every ε ∈ I0 there is ϕε ∈ W such that

(ϕε oT ϕε) − ε � s(sϕ) = ϕ.

By our assumption, there is ψ ∈ μ such that ζϕε � ζψ , hence by Lemma 3.2(ii), we get

ψ − ε � (ψ oT ψ) − ε � (ϕε oT ϕε) − ε � ϕ,

which proves ϕ ∈ μ. Consequently W ⊆ μ. �Theorem 3.3. Let (X,μ), (Y,W) be T -uniform spaces and ℘μ, ℘W the associated T -syntopogenous structures,respectively. If the function f : X −→ Y is uniformly continuous, then it is syntopogenously continuous.

Proof. Let f : X −→ Y be uniformly continuous and ν,ρ ∈ IY. Then for every ϕ ∈ W there is ψ ∈ μ such thatψ � (f × f )←(ϕ).

Consequently, for every ηϕ ∈ ℘W , there is ζψ ∈ ℘μ satisfying

ηϕ(ν,ρ) = 1 − hgt[ϕ〈ν〉T T sϕ〈1 − ρ〉T

]= 1 − sup

y∈Y

[ϕ〈ν〉T T (1 − ρ)

](y), by Lemma 3.1

= 1 − supy∈Y

{sups∈Y

[ν(s) T ϕ(z,y)

]T (1 − ρ)(y)

}� 1 − sup

x,z∈X

[ν(f (z))T ϕ(f (z), f (x)

)T (1 − ρ)

(f (x))]

, because rangef ⊆ Y

= 1 − supx,z∈X

[(f ←(ν)

)(z) T((f × f )←(ϕ)

)(z,x) T

(f ←(1 − ρ)

)(x)]

� 1 − sup{

sup[(

f ←(ν))(z) T ψ(z,x)

]T(1 − f ←(ρ)

)(x)}, by hypothesis

x∈X z∈X



= 1 − hgt[ψ⟨f ←(ν)

⟩T T(1 − f ←(ρ)

)]= ζψ

(f ←(ν), f ←(ρ)

), by Lemma 3.1 again

Hence, the syntopogenous continuity of f yields. �Now, we show how every symmetrical T -syntopogenous space induces a T -uniform space.

Theorem 3.4. (See [4].) Let (X,μ) be a T -uniform space. Then (X, τ (μ)) is a T -neighbourhood space.

Theorem 3.5. Let (X,℘) be a symmetrical T -syntopogenous space. For every ζ ∈ ℘, define ψζ ∈ IX×X by:

ψζ (x,y) = 1 − ζ(1x,1 − 1y), x,y ∈ X.

Then the family υ℘ = {ψζ : ζ ∈ ℘} is a T -uniform base on X. Moreover, τ(℘) = τ(υ℘) whenever ℘ is a biperfect.

Proof. (TUB2) For every ψζ ∈ υ℘ and x ∈ X, we get by (TT4) that

ψζ (x,x) = 1 − ζ(1x,1 − 1x)� hgt[1x T 1x] = 1.

(TUB1) We show that υ℘ is indeed an I -filterbase

(a) Obviously 0 /∈ υ℘ from (TUB2), which proved above.(b) The intersection of two members of υ℘ contains a member:

Given ψζ1,ψζ2 ∈ υ℘ , then for every x,y ∈ X, we have

(ψζ1 ∧ ψζ2)(x,y) = ψζ1(x,y) ∧ ψζ2(x,y)

= [1 − ζ1(1x,1 − 1y)]∧ [1 − ζ2(1x,1 − 1y)

]= 1 − [(ζ1 ∨ ζ2)(1x,1 − 1y)

]� 1 − ζ(1x,1 − 1y), for some ζ ∈ ℘, by (TS1)

= ψζ (x,y).

Therefore (ψζ1 ∧ ψζ2 ) contains a member ψζ of υ℘ .

(TUB3) Let ψζ ∈ υ℘ and ε ∈ I0. Then for every x,y ∈ X, we have

sψζ (x,y) = ψζ (y,x) = 1 − ζ(1y,1 − 1x)

= 1 − ζ(1x,1 − 1y), because ζ is symmetrical

� 1 − [(ζε oT ζε) + ε](1x,1 − 1y), by (TS2)

= 1 −{

supC⊆X

[ζε(1x,1C) T ζε(1C,1 − 1y)

]}− ε.

Hence, for every γ ∈ I0 with γ > sψζ (x,y) there is M ⊆ X such that

γ > 1 − [ζε(1x,1M) T ζε(1M,1 − 1y)]− ε,

thus, for every z ∈ X, we must have one of the following two cases:



Either z ∈ M and in this case we have

γ > 1 − ζε(1M,1 − 1y) − ε

� 1 − ζε(1z,1 − 1y) − ε, by (TT2).

Or z /∈ M (i.e. M ⊆ X \ {z}) and in this case we have

γ > 1 − ζε(1x,1M) − ε

� 1 − ζε(1x,1 − 1z) − ε, by (TT2) again.

Hence,

γ � supz∈X

{[1 − ζε(1x,1 − 1z) − ε

]∧ [1 − ζε(1z,1 − 1y) − ε]}

� supz∈X

{[1 − ζε(1x,1 − 1z)

]T[1 − ζε(1z,1 − 1y)

]}− ε

= supz∈X

[ψζε (x, z) T ψζε (z,y)

]− ε

= (ψζε oT ψζε )(x,y) − ε,

this proves sψζ � (ψζε oT ψζε ) − ε.Which renders (TUB3).Finally, we denote the fuzzy closure operators associated with τ(℘) and τ(υ℘) by − and ∼, respectively. Let μ ∈ IX

and x ∈ X, we have

μ−(x) =∨α∈I

[α T (1μα∗ )

−](x), by Theorem 2.2 and [4, Theorem 2.3]

=∨α∈I

{α T inf

ζ∈℘

[1 − ζ(1x,1 − 1μα∗ )

]}

=∨α∈I

{α T inf

ζ∈℘

[1 − ζ

(1x,

(1 −∨

y∈μα∗1y

))]}

=∨α∈I

{α T inf

ζ∈℘

[1 − ζ

(1x,∧

y∈μα∗(1 − 1y)

)]}

=∨α∈I

{α T inf

ζ∈℘

[1 −∧

y∈μα∗ζ(1x,1 − 1y)

]}, because ℘ is biperfect

=∨α∈I

{α T inf

ζ∈℘

[ ∨y∈μα∗

(1 − ζ(1x,1 − 1y)

)]}

=∨α∈I

{α T inf

ψζ ∈υ℘

[ ∨y∈μα∗

ψζ (x,y)

]}

=∨α∈I

{α T inf

ψζ ∈υ℘

[ ∨y∈μα∗

sψζ (y,x)

]}

=∨α∈I

infψζ ∈υ℘

{α T

[ ∨y∈μα∗

sψζ (y,x)


=∨α∈I

infψζ ∈υ℘

(sψζ 〈α T 1μα∗ 〉T

)(x)

=∨

(α T 1μα∗ )∼(x), by (4)

α∈I



=∨α∈I

[α T (1μα∗ )

∼](x), by Theorems 3.4, 2.1

= μ∼(x), by Theorem 3.4 and [4, Theorem 2.3]

Therefore μ− = μ∼, which proves τ(℘) = τ(υ℘), and this completes the proof. �Theorem 3.6. Let (X,℘), (Y,L) be T -syntopogenous spaces and let υ℘ , υL be the associated T -uniform bases,respectively. If f : X −→ Y is syntopogenously continuous function, then it is uniformly continuous: (X, υ℘) −→(Y, υL).

Proof. Let f : X −→ Y be syntopogenously continuous and x,y ∈ X. Then for every ϕη ∈ υL, there is ψζ ∈ υ℘

satisfies((f × f )←(ϕη)

)(x,y) = ϕη

(f (x), f (y)

)= 1 − η(1f (x),1 − 1f (y)), η ∈ L

� 1 − ζ(f ←(1f (x)), f

←(1 − 1f (y))), ζ ∈ ℘, by hypothesis

= 1 − ζ(f ←(f (1x)

),1 − f ←(f (1y)

)), ζ ∈ ℘

� 1 − ζ(1x,1 − 1y), ζ ∈ ℘, by (TT2)

= ψζ (x,y).

That is (f × f )←(ϕη) � ψζ , which proves the uniform continuity of f . �The preceding results show the category of symmetrical T -syntopogenous spaces, with syntopogenously continu-

ous functions as morphisms is isomorphic to the category of T -uniform spaces and uniformly continuous functions asmorphisms.

Example 1. Let X be a nonempty set and ℘ = {ζ , η} where ζ, η : IX × IX −→ I , defined for every μ,λ ∈ IX, by:

ζ(μ,λ) = 1 − hgt[μ ∧ (1 − λ)

]and η(μ,λ) = 1 − [(hgtμ) ∧ hgt(1 − λ)

].

Thus as [7, Example 3.11], (X,℘) is a biperfect symmetrical Min-syntopogenous space.Now, we verify that the family υ℘ = {ψζ ,ψη: ζ, η ∈ ℘} is a Min-uniform base on X and τ(℘) = τ(υ℘).It suffices to check (TUB3), since the other axioms trivially hold.Let ψζ ∈ υ℘ , ε ∈ I0 and x,y ∈ X, hence by (TS2) we have (ζ oT ζ ) + ε � ζ (cf. [7]). Therefore

sψζ (x,y) = ψζ (y,x)

= 1 − ζ(1y,1 − 1x), by definition of ψζ

� 1 − [(ζ oT ζ ) + ε](1y,1 − 1x)

= 1 − (ζ oT ζ )(1y,1 − 1x) − ε

= 1 − supC⊆X

[ζ(1y,1C) ∧ ζ(1C,1 − 1x)

]− ε, where T = Min.

Hence, for every γ ∈ I0 with γ > sψζ (x, y) there is H ⊆ X such that

γ > 1 − [ζ(1y,1H) ∧ ζ(1H,1 − 1x)]− ε,

thus, for every z ∈ X, we must have one of the following two cases:Either z ∈ H and in this case we get

γ > 1 − ζ(1H,1 − 1x) − ε

� 1 − ζ(1z,1 − 1x) − ε, by (TT2).

Or z /∈ H (i.e. H ⊆ X\{z}) and in this case we get



γ > 1 − ζ(1y,1H) − ε

� 1 − ζ(1y,1 − 1z) − ε, by (TT2) again.

Hence,

γ � supz∈X

{[1 − ζ(1z,1 − 1x) − ε

]∧ [1 − ζ(1y,1 − 1z) − ε]}

= supz∈X

{[1 − ζ(1x,1 − 1z)

]∧ [1 − ζ(1z,1 − 1y)]}− ε, by symmetry of ζ

= supz∈X

[ψζ (x, z) ∧ ψζ (z,y)

]− ε

= (ψζ oT ψζ )(x,y) − ε.

Which proves sψζ � (ψζ oT ψζ ) − ε.Also, since ζ � η, we have sψη � sψζ and hence sψη � (ψζ oT ψζ ) − ε.This renders (TUB3).Finally, it is easy to see τ(℘) = τ(υ℘), because the I -topology generated by υ℘ is the discrete one (i.e. every fuzzy

set is closed), since for every x ∈ X and μ ∈ IX, we have

μ−(x) = inf{(

sϕ〈μ〉T)(x): ϕ ∈ υ℘

}, by (6)

= (sψζ 〈μ〉T)(x), for sψζ � sψη in υ℘

= supy∈X

[μ(y) ∧ sψζ (y,x)

], since T = Min

= supy∈X

[μ(y) ∧ ψζ (x,y)

]= sup

y∈X

{μ(y) ∧ [1 − ζ(1x,1 − 1y)

]}= sup

y∈X

[μ(y) ∧ hgt(1x ∧ 1y)

]= μ(x).

On the other hand, since ζ � η in ℘, then from [7, Example 3.11] we have

τ(℘) = τ({ζ }) is the discrete I -topology on X.

This shows our assertion.

4. Compatibility between T -syntopogenous spaces and fuzzy T -proximity spaces

In this section, we discuss the correspondence between T -syntopogenous spaces and fuzzy T -proximity spaces.The fuzzy T -proximity spaces were introduced by Hashem and Morsi, for more definitions and properties, we refer

to [3].

Definition 4.1. (See [3].) A (fuzzy) T -proximity on a set X is a function δ : IX × IX −→ I which satisfies, for anyμ,λ, ν ∈ IX the following conditions:

(TP1) δ(0,1) = 0;(TP2) δ(μ ∨ λ, ν) = δ(μ, ν) ∨ δ(λ, ν) and δ(μ,λ ∨ ν) = δ(μ,λ) ∨ δ(μ, ν);(TP3) if δ(μ,λ) < θ T β for some θ,β ∈ I0 there is C ⊆ X such that δ(μ,1C)� θ and δ(1(X−C), λ)� β;(TP4) δ(μ,λ) � hgt(μ T λ);(TP5) δ(α T μ,λ) = α T δ(μ,λ) = δ(μ,λ T α), α ∈ I ;(TP6) δ(μ,λ) = δ(λ,μ).



The pair (X, δ) is said to be a (fuzzy) T -proximity space and the I -topology determined by δ, denoted by τ(δ), is givenby the fuzzy closure operator:

λ−(x) = δ(λ,1x) for all λ ∈ IX, x ∈ X.

Given T -proximity spaces (X, δ), (Y, σ ) and a function f : X −→ Y, we say that f is proximally continuous if itsatisfies

δ(f ←(ν), f ←(ρ)

)� σ(ν,ρ), for every ν,ρ ∈ IY.

We omit the proofs of the following two easily established theorems.

Theorem 4.1. Let (X, δ) be a T -proximity space. Define a mapping ζδ : IX × IX −→ I by:

ζδ(μ,λ) = 1 − δ(μ,1 − λ), μ,λ ∈ IX. (10)

Then ζδ is a symmetrical T -topogenous order on X, for which

τ(δ) = τ({ζδ}).

Theorem 4.2. Let ζ be a symmetrical T -topogenous order on a set X. Define a mapping δ : IX × IX −→ I , by

δ(μ,λ) = 1 − ζ(μ,1 − λ), μ,λ ∈ IX.

Then δ is a T -proximity on X, with τ({ζ }) = τ(δ).

Theorem 4.3. Let (X, δ), (Y, σ ) be T -proximity spaces and let ζδ , ζσ be the associated T -topogenous orders, respec-tively. Then f : X −→ Y is proximally continuous if and only if it is syntopogenously continuous.

Proof. Let ν,ρ ∈ IY. Then

f : (X, δ) −→ (Y, σ ) is proximally continuous

iff δ(f ←(ν), f ←(ρ)

)� σ(ν,ρ)

iff 1 − ζδ

(f ←(ν),1 − f ←(ρ)

)� 1 − ζσ (ν,1 − ρ), by (10)

iff ζσ (ν,1 − ρ) � ζδ

(f ←(ν), f ←(1 − ρ)

)iff f : (X, {ζδ}

)−→ (Y, {ζσ }) is syntopogenously continuous. �The above, shows that the category of symmetrical T -syntopogenous spaces, with syntopogenously continuous

functions as morphisms is isomorphic to the category of T -proximity spaces and proximally continuous functions asmorphisms.

Example 2. Let X be a nonempty set and ζ1, ζ2 : IX × IX −→ I , be the symmetrical T -topogenous orders on X,defined by, for all μ,λ ∈ IX (see [7, Example 3.10]):

ζ1(μ,λ) = 1 − hgt[μ T (1 − λ)

],

ζ2(μ,λ) = 1 − [(hgtμ) T hgt(1 − λ)].

Then from the preceding theorem, we have

δ1(μ,λ) = 1 − ζ1(μ,1 − λ) = hgt(μ T λ) and

δ2(μ,λ) = 1 − ζ2(μ,1 − λ) = (hgtμ) T (hgtλ),

which are T -proximities on X (see [3]). Moreover, as in [3,7], we get

τ(δ1) = τ({ζ1})

is the discrete I -topology on X and

τ(δ2) = τ({ζ2})

is the indiscrete I -topology on X.



5. Compatibility between T -syntopogenous spaces and functional fuzzy T -separatedness

In this section, we show how a functional fuzzy T -separatedness may generate a T -topogenous order.A distance distribution function (ddf) [16] is a function, from the set R+ of positive real numbers to the unit interval

I = [0,1], which is monotone, left continuous and has supremum 1. The set of all ddf’s is denoted by D+.The partial order � on D+ is the opposite of the partial order of ddf’s as real functions. It was shown that (D+,�)

is a lattice. We denote its join by �, and its meet by �. The set R∗ of nonnegative real numbers, can be embedded in(D+,�) by sending every r � 0 onto the crisp ddf εr given by

εr(s) ={

0, 0 < s � r,1, s > r.

In particular, ε0 = the constant function 1 on R+, is the bottom element of (D+,�).The T -addition ⊕T and scalar multiplication by positive real numbers are defined on D+ as follows, for λ– ,� ∈ D+

and s > 0:

(λ– ⊕T �)(s) = sup{λ– (b) T �(s − b): 0 < b < s

}. (11)

(bλ– )(s) = λ– (s − b), for any b > 0. (12)

Also, for λ– ∈ D+, we write λ– (0+) = infr>0 λ– (r).In consequence, (see [6]), if f,g : X −→ D+ are two (uniformly) continuous functions, then so will be f � g,

f � g, f ⊕T g, bf .For each triangular norm T , Höhle introduced in [8] a probabilistic T -metric on D+, which we denote by �,

as follows: for all λ– ,� ∈ D+,

�(λ– ,�) =�{£ ∈ D+: λ– � � ⊕T £ and � � λ– ⊕T £}. (13)

Obviously, it follows at once that [4]:

�(λ– , ε0) = λ– , ∀λ– ∈ D+. (14)

Theorem 5.1. (See [7].) A function ζ : IX × IX −→ I is a T -topogenous order on X if and only if it satisfies thefollowing five axioms, the first four of which are properties of its restriction ζ : 2X × 2X −→ I . For all H,M,N ∈ 2X:

(TT1/) ζ(1X,1X) = ζ(1Ø,1Ø) = 1 and ζ(1X,1Ø) = 0;(TT2/) ζ(1(H∪M),1N) = ζ(1H,1N) ∧ ζ(1M,1N) and ζ(1H,1(M∩N)) = ζ(1H,1M) ∧ ζ(1H,1N);(TT3/) if ζ(1H,1M) > 1 − (θ T β) for some θ,β ∈ I0, there is C ⊆ X such that ζ(1H,1C) � 1 − θ and ζ(1C,1M) �

1 − β;(TT4/) if H �⊂ M, then ζ(1H,1M) = 0;(TT5/) ζ(μ,λ) =∧θ,β∈I [θ T ∗ β T ∗ ζ(1μ(1−θ)∗ ,1λβ∗ )].

If (TT2/) is replaced by the condition

(TT2//) ζ(1(H∪M),1N) � ζ(1H,1N) ∧ ζ(1M,1N) and ζ(1H,1(M∩N)) � ζ(1H,1M) ∧ ζ(1H,1N).

Then ζ is called T -semi-topogenous order.

Definition 5.1. (See [6].) Let (X, τ ) be a T -neighbourhood space. For all nonempty subsets H,M ⊆ X, let �(H,M) =�τ (H,M) be the following set of functions:

�(H,M) = {f : (X, τ ) −→ (D+, τ (�)): f is continuous, f (H) = ε0 and f is constant on M

}.

(This set is nonempty, as it contains the constant function ε0.)A function Γ = Γτ : 2X × 2X −→ I is defined by:



Γ (1H,1M) = supf ∈�(H,M)

[1 − f (M)(0+)

], H,M ⊆ X (15)

and Γ (1H,1Ø) = Γ (1Ø,1H) = 1, H ⊆ X. (16)

The real number Γ (1H,1M) is called the degree of functional (fuzzy) T -separatedness of H and M in (X, τ ).

Theorem 5.2. Let (X, τ ) be a T -neighbourhood space and define ζΓ : 2X × 2X −→ I , by:

ζΓ (1H,1M) = Γ (1H,1(X\M)), H,M ⊆ X. (17)

Then ζΓ is a T -semi-topogenous order on X.

Proof. These are easily seen to hold whenever one of the entering sets is empty. So, suppose that H, M and N arenonempty.

(TT1/) It is clearly by (16), (15) that

ζΓ (1X,1X) = Γ (1X,1Ø) = 1, ζΓ (1Ø,1Ø) = Γ (1Ø,1X) = 1 and

ζΓ (1X,1Ø) = Γ (1X,1X) = supf ∈�(X,X)

[1 − f (X)(0+)

]= [1 − ε0(0+)]= 0.

(TT2//) Obviously

ζΓ (1(H∪M),1N) = Γ (1(H∪M),1(X\N)), by (17)

= suph∈�(H∪M,X\N)

[1 − h(X\N)(0+)

]�{

supf ∈�(H,X\N)

[1 − f (X\N)(0+)

]}∧{

supg∈�(M,X\N)

[1 − g(X\N)(0+)

]}= Γ (1H,1(X\N)) ∧ Γ (1M,1(X\N))

= ζΓ (1H,1N) ∧ ζΓ (1M,1N), by (17)

Indeed, the third inequality follows from the obvious fact that for every h ∈ �(H ∪ M,X\N), we have h(H ∪ M) = ε0and h is constant on X\N, that implies h(H) = ε0, h(M) = ε0 and h is constant on X\N. Therefore h ∈ �(H,X\N)

and h ∈ �(M,X\N).Which establishes one half of (TT2//).For the other half, we have: M ∩ N ⊆ M,N evidently implies that �(H,X\(M ∩ N)) ⊆ �(H,X\M) and

�(H,X\(M ∩ N)) ⊆ �(H,X\N), so Γ (1H,1(X\(M∩N))) � Γ (1H,1(X\M)) ∧ Γ (1H,1(X\N)). Hence

ζΓ (1H,1(M∩N)) � ζΓ (1H,1M) ∧ ζΓ (1H,1N).

(TT3/) Suppose that ζΓ (1H,1M) > 1 − (θ T β) for some θ,β ∈ I , hence Γ (1H,1(X\M)) > 1 − (θ T β), then thereare α,γ ∈ I0 and f1 ∈ �(H,X\M) such that [1 − f1(X\M)(0+)] = α > γ > 1 − (θ T β).

Let � ∈ D+ be the ddf defined by

�(s) ={0, s = 0,

1 − γ, 0 < s � 1,

1, s > 1

Let f : X −→ D+ defined by f (x) = f1(x) � � , x ∈ X.Then f (H) = ε0, f (X\M) = � and f is continuous.Take C = {x ∈ X: f (x)(1) � 1 − α}, and let λ– ∈ D+ be the ddf

λ– (s) ={0, s = 0,

1 − α, 0 < s � 1,

1, s > 1;Let h : X −→ D+ defined by h(x) = f (x) � λ– , x ∈ X. Then h is continuous, h(H) = ε0 and for all x ∈ X\C, we have,at s ∈ ]0,1]:



(h(x))(s) = (f (x)

)(s) ∨ λ– (s) � λ– (s) = 1 − α = (f (x)

)(1) ∨ (1 − α)

�(f (x))(s) ∨ λ– (s), because f (x) is monotone

= (h(x))(s).

Whenever s > 1:(h(x))(s) = (f (x)

)(s) ∨ λ– (s) = (f (x)

)(s) ∨ 1 = 1 = λ– (s),

also, (h(x))(0) = (f1(x))(0) ∨ ξ(0) ∨ λ– (0) = 0.This proves h(X\C) = λ– , which completes the proof of h is in �(H,X\C). In consequence

ζΓ (1H,1C) = Γ (1H,1(X\C)) � 1 − h(X\C)(0+)

= 1 − λ– (0+) = α > 1 − (θ T β) � 1 − θ,

which establishes one half of (TT3/).Define a function g : X −→ D+ by

g(x) = �(λ– , λ– � �(

�,� � 1

2f (x)

), x ∈ X.

By the continuity of f , �, � and �, this g is a continuous with respect to τ and τ(�).Now, we need the following identities, which easily follow from the definitions of �, λ– and � :

�(�,λ– ⊕T �) = λ– . (18)

�(

�,1

2�

)= �. (19)

�(λ– ,�) = �. (20)

For all y ∈ X\M, we have

g(y) = �(

λ– , λ– � �(

�,� � 1

2�

))

= �(

λ– , λ– � �(

�,1

2�

)), because

1

2� � �

= �(λ– , λ– � �), by (19)

= �(λ– ,�), because λ– � �

= �, by (20).

That is, g(X\M) = � .Now, for all s � 0, we have

(λ– ⊕T �)(s) =

⎧⎪⎨⎪⎩

0, s = 0(1 − α) T (1 − γ ), 0 < s � 11 − α, 1 < s � 21, s > 2

that is,

� � λ– ⊕T �. (21)

Hence, for all x ∈ C, we get

ε0 � g(x) = �(

λ– , λ– � �(

�,� � 1

2f (x)

))� �(λ– , λ– � �(�,λ– ⊕T �)

), by (21) and definition of �

= �(λ– , λ– � λ– ), by (18)

= �(λ– , λ– )

= ε0,



that is g(C) = ε0, which completes the proof of g is in �(C,X\M).Consequently,

ζΓ (1C,1M) = Γ (1C,1(X\M)) � 1 − g(X\M)(0+)

= 1 − ξ(0+) = γ > 1 − (θ T β)� 1 − β,

which establishes the other half of (TT3/).(TT4/) If H �⊂ M, then we have H ∩ (X\M) �= Ø.Hence, evidently every f in �(H,X\M) must be ε0 on (X\M). Thus

ζΓ (1H,1M) = Γ (1H,1(X\M))

= supf ∈�(H,X\M)

[1 − f (X\M)(0+)

]= 1 − ε0(0+)

= 0,

and this completes the proof. �6. T -syntopogenous structures induced by syntopogenous structures

In this section, we show how a syntopogenous structure Ω due to Csàszàr induces a T -syntopogenous structure ℘Ω

and the I -topology τ(℘Ω) coincides with the topologically generated space ω(T(Ω)) [12].Recall that an order relation 〈〈 on the subsets of a set X is called a topogenous order on X (Csàszàr [2]) if it satisfies

the following, for all H,M,N ⊆ X:

(T1) X 〈〈 X and Ø 〈〈 Ø;(T2) H 〈〈 M implies that H ⊆ M;(T3) H1 ⊆ H 〈〈 M ⊆ M1 implies that H1 〈〈 M1;(T4) H 〈〈 N and M 〈〈 N, imply that H ∪ M 〈〈 N. Also H 〈〈 M and H 〈〈 N, imply that H 〈〈 M ∩ N.(T5) if H 〈〈 M, then there is C ⊆ X such that H 〈〈 C and C 〈〈 M.

A topogenous order 〈〈 on a set X is said to be:

(i) symmetrical when H 〈〈 M if and only if X\M 〈〈 X\H.(ii) perfect if Hj 〈〈 M, (j ∈ J ) implies

⋃j∈J Hj 〈〈 M;

(iii) biperfect if it is perfect and if H 〈〈 Mj , (j ∈ J ) implies H 〈〈⋂j∈J Mj .

A syntopogenous structure on X is a nonempty set Ω of topogenous orders on X which has the following two prop-erties:

(S1) if 〈〈1, 〈〈2 are in Ω , then there exists 〈〈 ∈ Ω finer than 〈〈1 and 〈〈2;(S2) for every 〈〈 ∈ Ω there exists 〈〈1 ∈ Ω such that 〈〈1o〈〈1 is finer than 〈〈.

The syntopogenous structure Ω induces a topology T(Ω) on X. The T(Ω)-interior of a subset H of X is given by

intT(Ω)H =⋃

{C ⊆ X: C 〈〈 H for some 〈〈 ∈ Ω}.

Theorem 6.1. Let 〈〈 be a topogenous order on a set X and define a mapping ζ〈〈 : IX × IX −→ I by:

ζ〈〈(μ,λ) = supε,γ∈I0

{ε T γ : μ1−ε 〈〈 λγ ∗

}, μ,λ ∈ IX.

Then:

(i) ζ〈〈 is a T -topogenous order on X.(ii) if 〈〈 is symmetrical (resp. perfect, biperfect), then ζ〈〈 is symmetrical (resp. perfect, biperfect).



Proof. (i) We verify that ζ〈〈 satisfies the conditions (TT1)–(TT5). Let μ,λ, ν ∈ IX and α ∈ I . Then(TT1) Trivially holds.(TT2) We have

ζ〈〈(μ ∨ λ, ν) = supε,γ∈I0

{ε T γ : (μ ∨ λ)1−ε 〈〈 νγ ∗

}= sup

ε,γ∈I0

{ε T γ : μ1−ε ∪ λ1−ε 〈〈 νγ ∗

}� sup

ε,γ∈I0

{ε T γ : μ1−ε 〈〈 νγ ∗ and λ1−ε 〈〈 νγ ∗

}, by (T4)

= supε,γ∈I0

{[ε T γ : μ1−ε 〈〈 νγ ∗

]∧ [ε T γ : λ1−ε 〈〈 νγ ∗]}

� supε,γ∈I0

{ε T γ : μ1−ε 〈〈 νγ ∗

}∧ supθ,β∈I0

{θ T β: λ1−θ 〈〈 νβ∗

}= ζ〈〈(μ, ν) ∧ ζ〈〈(λ, ν).

For the opposite inequality,

ζ〈〈(μ ∨ λ, ν) = supε,γ∈I0

{ε T γ : (μ ∨ λ)1−ε 〈〈 νγ ∗

}= sup

ε,γ∈I0

{ε T γ : μ1−ε ∪ λ1−ε 〈〈 νγ ∗

}� sup

ε,γ∈I0

{ε T γ : μ1−ε 〈〈 νγ ∗

}, by (T3)

= ζ〈〈(μ, ν)

� ζ〈〈(μ, ν) ∧ ζ〈〈(λ, ν).

Consequently ζ〈〈(μ ∨ λ, ν) = ζ〈〈(μ, ν) ∧ ζ〈〈(λ, ν).In an analogous way we get ζ〈〈(μ,λ ∧ ν) = ζ〈〈(μ,λ) ∧ ζ〈〈(μ, ν).(TT3) Let ζ〈〈(μ,λ) > 1 − (θ T β), θ,β ∈ I0. Then we have μ1−ε 〈〈 λγ ∗ , for some ε, γ ∈ I0 such that ε T γ �

1 − (θ T β).Hence, by (T5), there is C ⊆ X such that μ1−ε 〈〈 C and C 〈〈 λγ ∗ ; i.e., μ1−ε 〈〈 1Cγ ∗ and (1C)1−ε 〈〈 λγ ∗ .Therefore,

ζ〈〈(μ,1C) = supα,δ∈I0

{α T δ: μ1−α 〈〈 (1C)δ∗

}� ε T γ � 1 − (θ T β) � 1 − θ;

ζ〈〈(1C, λ) = supα,δ∈I0

{α T δ: (1C)1−α 〈〈 λδ∗

}� ε T γ � 1 − (θ T β) � 1 − β.

(TT4) Let θ,β ∈ I1 be such that θ T β > 1 − hgt[μT (1 −λ)], that is 1 − (θ T β) < μ(x) T (1 −λ)(x),∀x ∈ X; i.e.(1 − θ) ∨ (1 − β) � (1 − θ) T ∗ (1 − β) < μ(x) T (1 − λ)(x),∀x ∈ X. Hence 1 − θ < μ(x) and 1 − β < (1 − λ)(x),

∀x ∈ X. Therefore x ∈ μ1−θ but x /∈ λβ∗ , so that μ1−θ 〈〈 λβ∗ cannot hold.Then ζ〈〈(μ,λ)� θ T β .Which proves ζ〈〈(μ,λ)� 1 − hgt[μ T (1 − λ)].(TT5)

ζ〈〈(α T μ,λ) = supε,γ∈I0

{ε T γ : (α T μ)1−ε 〈〈 λγ ∗

}= sup

ε,γ∈I0

{ε T γ : μJ(α,1−ε) 〈〈 λγ ∗

}, by Lemma 1.2

� supθ,γ∈I0

{[(1 − α) T ∗ θ

]T γ : μ1−θ 〈〈 λγ ∗

}, by putting J(α,1 − ε) = 1 − θ and by (1)

= sup{[

(1 − α) T γ]T ∗ (θ T γ ): μ1−θ 〈〈 λγ ∗

}
θ,γ∈I0



� supθ,γ∈I0

{(1 − α) T ∗ (θ T γ ): μ1−θ 〈〈 λγ ∗

}= (1 − α) T ∗ sup

θ,γ∈I0

{θ T γ : μ1−θ 〈〈 λγ ∗

}, by continuity of T ∗

= (1 − α) T ∗ ζ〈〈(μ,λ)

= (1 − α) T ∗ supε,γ∈I0

{ε T γ : μ1−ε 〈〈 λγ ∗

}= sup

ε,γ∈I0

{(1 − α) T ∗ (ε T γ ): μ1−ε 〈〈 λγ ∗

}, by continuity of T ∗

= supε,γ∈I0

{[(1 − α) T ∗ ε

]T[(1 − α) T ∗ γ

]: μ1−ε 〈〈 λγ ∗

},

� supθ,γ∈I0

{θ T[(1 − α) T ∗ γ

]: μJ(α,1−θ) 〈〈 λγ ∗

},

by putting 1 − ε > J(α,1 − θ) and by (2), (T3)

= supθ,γ∈I0

{θ T[(1 − α) T ∗ γ

]: (α T μ)1−θ 〈〈 λγ ∗

}, by Lemma 1.2

� supζψ

{θ T β: (α T μ)1−θ 〈〈 λβ∗

}, by putting (1 − α) T ∗ γ � β ⇒ γ � β and by (T3)

= ζ〈〈(α T μ,λ).

This demonstrates that ζ〈〈(α T μ,λ) = (1 − α) T ∗ ζ〈〈(μ,λ).On the other hand,

ζ〈〈(μ, (1 − α) T ∗ λ

)= supε,γ∈I0

{ε T γ : μ1−ε 〈〈 [(1 − α) T ∗ λ

]γ ∗}

= supε,γ∈I0

{ε T γ : μ1−ε 〈〈 λ(1−J(α,1−γ ))∗

}, by Lemma 1.3

� supε,θ∈I0

{ε T[θ T ∗ (1 − α)

]: μ1−ε 〈〈 λθ∗

}, by putting 1 − J(α,1 − γ ) = θ and by (1)

= supε,θ∈I0

{(ε T θ) T ∗ [ε T (1 − α)

]: μ1−ε 〈〈 λθ∗

}� sup

ε,θ∈I0

{(ε T θ) T ∗ (1 − α): μ1−ε 〈〈 λθ∗

}= (1 − α) T ∗ sup

ε,θ∈I0

{ε T θ : μ1−ε 〈〈 λθ∗

}= (1 − α) T ∗ ζ〈〈(μ,λ)

= (1 − α) T ∗ supε,γ∈I0

{ε T γ : μ1−ε 〈〈 λγ ∗

}= sup

ε,γ∈I0

{(1 − α) T ∗ (ε T γ ): μ1−ε 〈〈 λγ ∗

}= sup

ε,γ∈I0

{[(1 − α) T ∗ ε

]T[(1 − α) T ∗ γ

]: μ1−ε 〈〈 λγ ∗

}� sup

ε,θ∈I0

{[(1 − α) T ∗ ε

]T θ : μ1−ε 〈〈 λ(1−J(α,1−θ))∗

},

by putting (1 − α) T ∗ γ < θ and by (2), (T3)

= supε,θ∈I0

{[(1 − α) T ∗ ε

]T θ : μ1−ε 〈〈 [(1 − α) T ∗ λ

]θ∗}, by Lemma 1.3

� supβ,θ∈I0

{β T θ : μ1−β 〈〈 [(1 − α) T ∗ λ

]θ∗},

by putting (1 − α) T ∗ ε � β ⇒ ε � β and by (T3)

= ζ〈〈(μ, (1 − α) T ∗ λ

).



That is ζ〈〈(μ, (1 − α) T ∗ λ) = (1 − α) T ∗ ζ〈〈(μ,λ).Which proves that ζ〈〈, is a T -topogenous order on X.(ii) Let 〈〈 be symmetrical. Then

ζ〈〈(1 − μ,1 − λ) = supε,γ∈I0

{ε T γ : (1 − μ)1−ε 〈〈 (1 − λ)γ ∗

}= sup

ε,γ∈I0

{ε T γ : X − με∗ 〈〈 X − λ1−γ

}= sup

ε,γ∈I0

{ε T γ : λ1−γ 〈〈 με∗

}, by hypothesis

= ζ〈〈(λ,μ).

Hence ζ〈〈 is a symmetrical.Moreover, let 〈〈 be a perfect and μj ∈ IX, j ∈ J . Then for every j ∈ J we have

ζ〈〈(μj ,λ) = supε,γ∈I0

{ε T γ : (μj )

1−ε 〈〈 λγ ∗}

� supε,γ∈I0

{ε T γ :

⋃j∈J

(μj )1−ε 〈〈 λγ ∗

}, by hypothesis

= supε,γ∈I0

{ε T γ :

(∨j∈J

μj

)1−ε

〈〈 λγ ∗}

= ζ〈〈(∨

j∈J

μj ,λ

).

Therefore,∧

j∈J ζ〈〈(μj ,λ) � ζ〈〈(∨

j∈J μj ,λ).For the opposite inequality;

ζ〈〈(∨

j∈J

μj ,λ

)= sup

ε,γ∈I0

{ε T γ :

(∨j∈J

μj

)1−ε

〈〈 λγ ∗}

= supε,γ∈I0

{ε T γ :

⋃j∈J

(μj )1−ε 〈〈 λγ ∗

}

� supε,γ∈I0

{ε T γ : (μj )

1−ε 〈〈 λγ ∗}, by (T3)

= ζ〈〈(μj ,λ)

�∧j∈J

ζ〈〈(μj ,λ).

Thus, ζ〈〈(∨

j∈J μj ,λ) =∧j∈J ζ〈〈(μj ,λ), which shows that ζ〈〈 is a perfect.Similarly, we can show that ζ〈〈 is a biperfect and this completes the proof. �

Lemma 6.1. If 〈〈1, 〈〈2 are topogenous orders on a set X, then

(i) ζ〈〈1 � ζ〈〈2 if and only if 〈〈1 finer than 〈〈2;(ii) ζ〈〈1o〈〈2 = ζ〈〈1 oT ζ〈〈2 .

Proof. (i) Follows immediately from the definitions.(ii) If θ T β < (ζ〈〈1 oT ζ〈〈2)(μ,λ), then there is C ⊆ X such that θ T β < ζ〈〈2(μ,1C) T ζ〈〈1(1C, λ). Therefore μ1−θ 〈〈2

1C = 1C1∗ and 11−1C = 1C 〈〈1 λβ∗ , that is μ1−θ 〈〈2 1C 〈〈1 λβ∗ and so μ1−θ 〈〈1 o 〈〈2 λβ∗ , which implies that θ T β �

ζ〈〈 o〈〈 (μ,λ).
1 2



Thus ζ〈〈1 oT ζ〈〈2 � ζ〈〈1o〈〈2 .On the other hand, if θ T β < ζ〈〈1o〈〈2(μ,λ), then μ1−θ 〈〈1 o 〈〈2 λβ∗ and so μ1−θ 〈〈21C 〈〈1 λβ∗, for some C ⊆ X, that

is μ1−θ 〈〈2 1C1∗ and 11−1C 〈〈1 λβ∗ . Hence

θ T β � ζ〈〈2(μ,1C) T ζ〈〈1(1C, λ) � (ζ〈〈1 oT ζ〈〈2)(μ,λ).

This shows ζ〈〈1o〈〈2 � ζ〈〈1 oT ζ〈〈2 and the result follows. �Now, consider the operator ω from the family of all topologies on a set X to the family of all I -topologies on X.

Proposition 6.1. (See [12].) If T is a topology on a set X, then ω(T) is an I -topology on X has a basis{α ∧ 1H: α ∈ I, H ∈ T}, called topologically generated.

Proposition 6.2. If Ω is a syntopogenous structure on a set X and ℘Ω = {ζ〈〈: 〈〈 ∈ Ω}, then ℘Ω is a T -syntopogenousstructure on X with τ(℘Ω) = ω(T(Ω)).

Proof. The first assertion follows immediately from the preceding lemma.Now, let λ ∈ τ(℘Ω) and θ ∈ I1. Then for every x ∈ λθ , we have

θ < λ(x) = λo(x), because λ is τ(℘Ω)-open

= supζ〈〈∈℘Ω

ζ〈〈(1x, λ), by Proposition 1.1

= supε,γ∈I0

{ε T γ : (1x)

1−ε 〈〈 λγ ∗ , for some 〈〈 ∈ Ω}

= supγ∈I0

{γ : {x} 〈〈 λγ ∗, for some 〈〈 ∈ Ω

}.

This means that, for some γ ∈ I0 with γ � θ and some 〈〈 ∈ Ω , we get {x} 〈〈 λγ ∗ , that is x ∈ intT(Ω)(λγ ∗).Since γ � θ , we have, λγ ∗ ⊆ λθ , therefore, x ∈ intT(Ω)λ

θ .Consequently λθ = intT(Ω)λ

θ is T(Ω)-open and so λ is T(Ω)-lower semicontinuous. Hence λ ∈ ω(T(Ω)).Which implies τ(℘Ω) ⊆ ω(T(Ω)).On the other hand, since ω(T(Ω)) has a basis {α ∧ 1H: α ∈ I, H ∈ T(Ω)} and α ∧ 1H ∈ τ(℘Ω), then ω(T(Ω)) ⊆

τ(℘Ω), and this completes the proof. �7. Fuzzy T -neighbourhood syntopogenous spaces

In this section, we deduce the concept of fuzzy T -neighbourhood syntopogenous structures. Also, we show thereis a one-to-one correspondence between the T -neighbourhood systems on a set and the fuzzy T -neighbourhood syn-topogenous structures on the same set.

The phrase “fuzzy T -neighbourhood” is abbreviated to fTn-.

Definition 7.1. A fTn-topogenous order on a set X is a relation Θ between the crisp subsets and the fuzzy subsetsof X, which satisfies: for all H,M ⊆ X and μ,λ ∈ IX;

(TNT1) X Θ 1 and Ø Θ 0;(TNT2) if H Θ μ, then 1H � μ;(TNT3) if H ⊆ M Θ μ� λ, then H Θ λ;(TNT4) if H Θ μ and M Θ λ, then H ∩ M Θ μ ∧ λ and H ∪ M Θ μ ∨ λ.

The pair (X, {Θ}) is said to be a fTn-topogenous space.

Definition 7.2. A fTn-syntopogenous structure on a set X is a family H of fTn-topogenous orders on X satisfying thefollowing conditions:



(TNS1) H is directed in the sense that for every Θ1,Θ2∈H there is Θ∈H which is finer than Θ1 and Θ2.(TNS2) if Θ∈ H, then for every ε ∈ I0, there is Θε∈ H such that whenever H Θ 1M ∨ α, for all H,M ⊆ X and α ∈ I ,

there is C ⊆ X, such that H Θε 1C ∨ α + ε and C Θε 1M ∨ α + ε.

The pair (X,H) is said to be a fTn-syntopogenous space.

Now, we supply a one-to-one corresponding between the family of T -topogenous orders on a set X and the familyof fTn-topogenous orders on X, considered as fuzzy relations on the ordinary power set of X.

Lemma 7.1. Let ζ be a T -topogenous order on a set X,μ,λ ∈ IX and α ∈ I1 such that α < ζ(μ,λ). Then μθ∗ ⊆λ(1−β)∗ , for every θ,β ∈ I0, with θ T β � 1 − α.

Proof. Let μ,λ ∈ IX and α ∈ I1 such that α < ζ(μ,λ). Then, by (TT4) we have α < 1 − hgt[μ T (1 − λ)], that is

Ø = [μ T (1 − λ)](1−α)∗

=⋃

εT γ�1−α

[με∗ ∩ (1 − λ)γ ∗

], by Lemma 1.1(ii)

⊇ μθ∗ ∩ (1 − λ)β∗ , for every θ,β ∈ I0 with θ T β � 1 − α.

Therefore,

μθ∗ ⊆ X\(1 − λ)β∗ = λ(1−β) ⊆ λ(1−β)∗ , for every θ,β ∈ I0 with θ T β � 1 − α.

Which proves our assertion. �Proposition 7.1. Let ζ : IX × IX −→ I be a T -topogenous order on X. Define the relation Θζ between the crispsubsets and the fuzzy subsets of X, by: 1H Θζ μ iff ζ(1H,μα) > 1 − α, ∀α ∈ I0,1, H ⊆ X, μ ∈ IX.

Then Θζ is a fTn-topogenous order on X.

Proof. Let H,M ∈ 2X and μ,λ ∈ IX. Then

(TNT1) It is clearly that X Θζ 1 and Ø Θζ 0.(TNT2) Let 1H Θζ μ. Then for every α ∈ I0,1, ζ(1H,μα) > 1−α, hence by Lemma 7.1, H = Hθ∗ ⊆ (μα)(1−β)∗ = μα .

Consequently H � μ.(TNT3) Let 1H � 1M Θζ μ� λ. Then for every α ∈ I1, we get

1 − α < ζ(1M,μα

)� ζ(1H, λα

), by (TT2).

Which implies 1H Θζ λ.(TNT4) Let 1H Θζ μ and 1M Θζ λ. Then by (TT2) we have for every α ∈ I1,

1 − α < ζ(1H,μα

)� ζ(1H∩M,μα

)and

1 − α < ζ(1M, λα

)� ζ(1H∩M, λα

). Hence,

1 − α < ζ(1H∩M,μα

)∧ ζ(1H∩M, λα

)= ζ(1H∩M,μα ∧ λα

)= ζ(1H∩M, (μ ∧ λ)α

).

This implies 1H∩M Θζ μ ∧ λ.Moreover, by (TT2) we have

1 − α < ζ(1H,μα

)� ζ(1H, (μ ∨ λ)α

)and

1 − α < ζ(1M, λα

)� ζ(1M, (μ ∨ λ)α

). Hence,

1 − α < ζ(1H, (μ ∨ λ)α

)∧ ζ(1M, (μ ∨ λ)α

)= ζ(1H∪M, (μ ∨ λ)α

).



Which implies 1H∪M Θζ μ ∨ λ.

This proves Θζ is a fTn-topogenous order on X. �Proposition 7.2. Let Θ be a fTn-topogenous order on X. Define the map ζΘ : 2X × 2X −→ I , by

ζΘ(1H,1M) = 1 − inf{α ∈ I : 1H Θ 1M ∨ α}, H,M ⊆ X.

Then ζΘ is a T -topogenous order on X.

Proof. It is immediately that ζΘ satisfies the axioms in Theorem 5.1. �Example 3. Let ζ : IX × IX −→ I , be a T -topogenous order on X defined by

ζ(λ,μ) = 1 − hgt[λ T (1 − μ)

], μ,λ ∈ IX.

Then, the fTn-topogenous order Θζ on X, induced by ζ , has the formula:

1H Θζ μ iff H ⊆ μ1∗, H ⊆ X,μ ∈ IX.

Because,

1H Θζ μ

iff ζ(1H,μα

)> 1 − α, ∀α ∈ I0,1

iff 1 − hgt[1H T(1 − μα

)]> 1 − α, ∀α ∈ I0,1

iff hgt[1H T(1 − μα

)]< α, ∀α ∈ I0,1

iff 1H ∩ (1 − μα)= Ø, ∀α ∈ I0,1

iff H ⊆ μα, ∀α ∈ I0,1

iff H ⊆ μ1∗ .

We end this section, by showing how a fTn-topogenous space may generate a T -neighbourhood space.

Theorem 7.1. Let (X, {Θ}) be a fTn-topogenous space, and define the operator − : 2X −→ IX, by:

(1H)−(x) = inf{α ∈ I : x Θ 1(X\H) ∨ α}, H ⊆ X, x ∈ X.

Then (X,− ) is a T -neighbourhood space.

Proof. We prove that the operator − satisfies the axioms in [4, Theorem 2.3].

(a) For all x ∈ X, 0−(x) = inf{α ∈ I : x Θ 1(X\Ø) ∨ α} = 0. Hence 0− = 0.(b) For all H ⊆ X and x ∈ H, we have

(1H)−(x) = inf{α ∈ I : x Θ 1(X\H) ∨ α} = 1, by (TNT2).

This shows (1H)− � (1H).(c) For all H,M ⊆ X and x ∈ X,

(1H ∨ 1M)−(x) = (1(H∪M))−(x)

= inf{α ∈ I : x Θ 1(X\(H∪M)) ∨ α}= inf{α ∈ I : x Θ (1(X\H) ∧ 1(X\M)) ∨ α

}= inf{α ∈ I : x Θ (1(X\H) ∨ α) ∧ (1(X\M) ∨ α)

}= inf{α ∈ I : x Θ 1(X\H) ∨ α and x Θ 1(X\M) ∨ α}



= max{inf{α ∈ I : x Θ 1(X\H) ∨ α}, inf{α ∈ I : x Θ 1(X\M) ∨ α}}

= (1H)−(x) ∨ (1M)−(x).

That is (1H ∨ 1M)− = (1H)− ∨ (1M)−.(d) Suppose that H ⊆ X, x ∈ X and θ,β ∈ I0, are such that

x /∈ [(1H)−](θT β)∗ ,

this mean that (1H)−(x) < θ T β .So, we can choose γ, ε ∈ I0, for which (1H)−(x) < γ < γ + ε < θ T β , therefore from the definition of theoperator −, x Θ 1(X\H) ∨ γ .

Hence, from axiom (TNS2) in Definition 7.2, there is C ⊆ X such that

x Θ 1C ∨ γ + ε and C Θ 1(X\H) ∨ γ + ε.

Then from the definition of − once again,

(1(X\C))−(x) � γ + ε < β and (1H)− ∧ 1C � γ + ε < θ.

This implies ((1H)−)θ∗ ∩ C = Ø and hence ((1H)−)θ∗ ⊆ X\C, therefore (((1H)−)θ∗)−(x) � (1(X\C))−(x) < β ,

that is, x /∈ [(((1H)−)θ∗)−]β∗ .Consequently, [(((1H)−)θ∗)−]β∗ ⊆ [(1H)−](θT β)∗ .

This proves that − satisfies the axiom (d) of [4, Theorem 2.3], and completes the proof of (X,− ) is a T -neighbourhoodspace. �Theorem 7.2. For a given T -neighbourhood space (X, τ (Σ)), define a binary relation �Σ between the crisp subsetsand the fuzzy subsets of X by:

1HΘΣμ iff μ ∈ Σ(H), for all H ⊆ X, μ ∈ IX.

Then (X, {ΘΣ }) is a fTn-topogenous space.

Proof. We show that ΘΣ satisfies the axioms in Definition 7.1.

(TNT1) Obviously, Ø ΘΣ 0 and X ΘΣ 1;(TNT2) Let 1H ΘΣ μ. Then μ ∈ Σ(H) =⋂x∈H Σ(x), hence by [4, Theorem 1.1], we have μ(x) = 1, for every x ∈ H,

which implies H � μ;(TNT3) Let 1H � 1M ΘΣ μ� λ. Then

μ ∈ Σ(M) =⋂x∈M

Σ(x) ⊆⋂x∈H

Σ(x) = Σ(H),

since Σ(H) is an I -filter, we get λ ∈ Σ(H), that is 1H ΘΣ λ;(TNT4) Let 1H ΘΣ μ and 1M ΘΣ λ. Then

μ ∈ Σ(H) ⊆ Σ(H ∩ M), also λ ∈ Σ(M) ⊆ Σ(H ∩ M),

since Σ(H ∩ M) is an I -filter, we get μ ∧ λ ∈ Σ(H ∩ M), that is

H ∩ M ΘΣ μ ∧ λ.

On the other hand, since both Σ(H) and Σ(M) are I -filters, then μ∨λ ∈ Σ(H) and μ∨λ ∈ Σ(M), therefore

μ ∨ λ ∈ Σ(H) ∩ Σ(M) =[⋂

x∈H

Σ(x)

]∩[ ⋂

y∈M

Σ(y)

]=⋂

x∈H∪M

Σ(x) = Σ(H ∪ M),

which proves 1H∪M ΘΣ μ ∨ λ,

and this completes the proof. �



Proposition 7.3. With the notations in Theorem 7.2, the association (X, τ (Σ)) �−→ (X, {ΘΣ }) is a one-to-one corre-spondence between T -neighbourhood spaces and fTn-topogenous spaces.

Proof. By Theorem 7.2, the mapping defined there, between T -neighbourhood spaces and fTn-topogenous spaces,is well defined. Also, this mapping is evidently the inverse mapping to that given in Theorem 7.1. Hence, each of themis a one-to-one correspondence. �8. Conclusion

This manuscript shows the compatibility between T -syntopogenous spaces and other structures such as fuzzyT -neighbourhood spaces, fuzzy T -uniform spaces and fuzzy T -proximity spaces. It also shows that functional fuzzyT -separatedness and syntopogenous structure induce a T -syntopogenous structure. Moreover, it gives the notion offuzzy T -neighbourhood syntopogenous structures.

Acknowledgements

The authors are grateful to an anonymous referee for his/her generous advice that significantly improved the pre-sentation of this article.

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