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Theory of g-T_g-Maps

Khodabocus m. i. and Sookia n. u. h.

Abstract. Several specific types of generalized maps of a generalized topolog-ical space have been defined and investigated for various purposes from time to time in the literature of topological spaces. Our recent research in the field of a new class of generalized maps of a generalized topological space is reported herein as a starting point for more generalized classes.

Key words and phrases. Generalized topological space, generalized sets, generalized maps, generalized continuous maps, generalized irresolute maps, generalized homeomorphism maps

1. Introduction

The concepts1 _{of (}_T_Ω,_T

Σ)-map π : TΩ → TΣ [1, 2], g- (TΩ,TΣ)-map πg :

T_Ω → T_Σ [9, 18], (Tg,Ω,Tg,Σ)-map π : Tg,Ω → Tg,Σ [3], and g- (Tg,Ω,Tg,Σ)-map πg :Tg,Ω→Tg,Σ [21], called, respectively, ordinary and generalized maps (brieﬂy,

T-mapandg-T-map, respectively) betweenT-spacesTΩandTΣ, and ordinary and

generalized maps (brieﬂy, Tg-map and g-Tg-map, respectively) between Tg-spaces Tg,Ωand Tg,Σare all fundamental concepts that have been introduced and

inves-tigated by several mathematicians [12, 16, 17, 20, 21, 23, 29, 31, 33].

Other concepts called (TΩ,TΣ)-continuous and (TΩ,TΣ)-irresolute maps and

(TΩ,TΣ)-homeomorphism (brieﬂy, T-continuous and T-irresolute maps, and T -homeomorphism, respectively) [6, 24, 32], g- (TΩ,T_Σ)-continuous and g- (TΩ,T_Σ )-irresolute maps andg- (TΩ,T_Σ)-homeomorphism (brieﬂy,g-T-continuous andg-T -irresolute maps, andg-T-homeomorphism, respectively) [6, 13, 14, 26], (Tg,Ω,Tg,Σ

)-continuous and (Tg,Ω,Tg,Σ)-irresolute maps, (Tg,Ω,Tg,Σ)-homeomorphism (brieﬂy,

Tg-continuous, Tg-irresolute maps, andTg-homeomorphism, respectively) [14, 28] and g- (Tg,Ω,Tg,Σ)-continuous and g- (Tg,Ω,Tg,Σ)-irresolute maps, g- (Tg,Ω,Tg,Σ

)-homeomorphism (brieﬂy, g-Tg-continuous and g-Tg-irresolute maps, and g-Tg

-homeomorphism, respectively) [11, 22, 34] are all derived concepts based on the properties ofT-map, g-T-map, Tg-map, andg-Tg-map. Having received extensive studies, all these ordinary and generalized mappings are at this date well-known important notions in ordinary and generalized topologies and their applications.

In this paper, we will show how further contributions can be added to the ﬁeld in a uniﬁed way.

1_{Notes to the reader: The structures}TΩ = (Ω,TΩ) and TΣ = (Σ,TΣ) are called ordinary topological spaces (brieﬂy,T-spaces), and the structuresTg,Ω=Ω,Tg,ΩandTg,Σ=Σ,Tg,Σ are called generalized topological spaces (brieﬂy, Tg-spaces). The mapsπ, πg :TΩ →TΣ and

π,πg :Tg,Ω →Tg,Σ, respectively, stand for ordinary and generalized maps between T-spaces andTg-spaces; the notations (TΩ,TΣ)-map (brieﬂy,T-map),g- (TΩ,TΣ)-map (brieﬂy,g-T-map),

_T

g,Ω,Tg,Σ-map (brieﬂy,Tg-map), andg-Tg,Ω,Tg,Σ-map (brieﬂy,g-Tg-map) emphasize their characters.

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2. Theory

2.1. preliminaries. Our discussion starts by recalling some basic deﬁnitions and

notations of most essential concepts presented in the theory ofg-Tg-sets in a T g-space.

The setUstands for the universe of discourse, ﬁxed within the framework of the theory of g-Tg-maps and containing as elements all sets (Λ-sets: Λ ∈ Ω,Σ,Υ;

TΛ, g-TΛ, TΛ, g-TΛ-sets; Tg,Λ, g-Tg,Λ, Tg,Λ, g-Tg,Λ-sets) considered in this theory,

andI0

ndef=

ν ∈N0_: _ν_≤_n_{; index sets}_I0

∞, In∗, I∞∗ are deﬁned similarly. Let Λ∈

Ω,Σ,Υ⊂Ube a given set and letP(Λ)def= Og,ν⊆Λ : ν ∈I∞∗

be the family of all subsetsOg,1,Og,2, . . ., of Λ. Then every one-valued map of the type Tg,Λ : P(Λ) → P(Λ) satisfying Tg,Λ(∅) = ∅, Tg,Λ(Og)⊆ Og, and Tg,Λ

ν∈I∗

∞Og,ν

=

ν∈I∗

∞Tg,Λ(Og,ν) is called a g-topology on Λ, and the structure Tg,Λ

def

= (Λ,Tg,Λ)

is called aTg,Λ-space, on which no separation axioms are assumed unless otherwise

mentioned [8, 7, 27]. The operator clg_,Λ : P(Λ) → P(Λ) carrying each Tg,Λ-set Sg ⊂Tg,Λ into its closure clg,Λ(Sg) =Tg,Λ−intg,Λ(Tg,Λ\ Sg)⊂Tg,Λ is called a

g-closure operator and the operator intg_,Λ :P(Λ)→ P(Λ) carrying each Tg,Λ-set Sg ⊂Tg,Λ into its interior intg,Λ(Sg) = Tg,Λ−clg,Λ(Tg,Λ\ Sg)⊂Tg,Λ is called a

g-interior operator; for clarity, we will use clg(·), intg(·), respectively, instead of clg_,_Λ(·), intg_,_Λ(·).

LetTg,Λ be a Tg,Λ-space, let Λ :P(Λ) → P(Λ) denotes the absolute

comple-ment with respect to the underlying set Λ⊂U, and letSg ⊂Tg,Λ be anyTg,Λ-set.

The classes

Tg,Λ def=

Og⊂Tg,Λ: Og∈ Tg,Λ

,

¬Tg,Λ def=

Kg ⊂Tg,Λ: Λ(Kg)∈ Tg,Λ

,

(2.1)

respectively, denote the classes of allTg,Λ-open andTg,Λ-closed sets relative to the

g-topology Tg,Λ, and the classes

Csub_T_g_,_Λ[Sg] def= Og∈ Tg,Λ : Og⊆ Sg,

Csup_¬T_g

,Λ[Sg]

def

= Kg∈ ¬Tg,Λ: Kg⊇ Sg

,

(2.2)

respectively, denote the classes ofTg,Λ-open subsets andTg,Λ-closed supersets

(com-plements of the Tg,Λ-open subsets) of the Tg,Λ-set Sg ⊂ Tg,Λ relative to the g

-topologyTg,Λ. To this end, theg-closure and theg-interior of aTg-set Sg⊂Tg in

aTg,Λ-space [3] deﬁne themselves as

intg_,_Λ(Sg)def= Og∈Csub_T_g_,_Λ[Sg]

Og, clg,Λ(Sg)def=

Kg∈Csup_¬T_g ,Λ[Sg]

Kg. (2.3)

Throughout this work, by clg◦intg(·), intg◦clg(·), and clg◦intg◦clg(·), respec-tively, are meant clg(intg(·)), intg(clg(·)), and clg(intg(clg(·))); other composition operators are deﬁned similarly. Also, the backslashTg\Sgrefers to the set-theoretic diﬀerence Tg− Sg. The mapping opg :P(Λ) → P(Λ) is called a g-operation on

P(Λ) if the following statements hold:

∀Sg∈ P(Λ)\∅, ∃(Og,Kg)∈ Tg,Λ\

∅× ¬Tg,Λ\

∅:

opg(∅) =∅

∨¬opg(∅) =∅

, Sg⊆opg(Og)∨Sg⊇ ¬opg(Kg)

,

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where ¬op_g :P(Λ)→ P(Λ) is called the ”complementary g-operation” onP(Λ) and, for allTg-setsSg,Sg,ν,Sg,μ∈ P(Λ)\∅, the following axioms are satisﬁed:

• Ax. i. S_g⊆_op_g₍O_g)∨S_g ⊇ ¬_op_g₍K_g)_,

• Ax. ii. _op_g₍S_g)⊆_op_g◦_op_g₍O_g)∨¬_op_g₍S_g)⊇ ¬_op_g◦¬_op_g₍K_g)_, • Ax. iii. S_g_,ν ⊆ S_g_,μ → op_g(O_g_,ν) ⊆ op_g(O_g_,μ)∨S_g_,μ ⊆ S_g_,ν ←

¬op_g(Kg,μ)⊇ ¬opg(Kg,ν),

• Ax. iv. op_g

σ=ν,μSg,σ

⊆_σ₌_ν,μop_g(Og,σ)∨¬opgσ=ν,μSg,σ

⊇

σ=ν,μ¬opg(Kg,σ)

,

for someTg,Λ-open sets Og, Og,ν,Og,μ∈ Tg,Λ\

∅andTg,Λ-closed setsKg,Kg,ν,

Kg,μ∈ ¬Tg,Λ [4, 19]. The class Lg Ω

=Lω_g Λ× Lκ_g Ω, where

Lg Λdef= opg,νμ(·) =

opg,ν(·),¬opg,μ(·)

: (ν, μ)∈I₃0×I₃0 (2.5)

in the Tg,Λ-space Tg,Λ, stands for the class of all possible g-operators and their

complementaryg-operators in theTg,Λ-spaceTg,Λ. Its elements are deﬁned as:

op_g(·) ∈ Lω_g Λdef= op_g_,₀(·), op_g_,₁(·), op_g_,₂(·), op_g_,₃(·) = intg(·), clg◦intg(·), intg◦clg(·), clg◦intg◦clg(·);

¬op_g(·) ∈ Lκ_g Λdef= ¬op_g_,₀(·), ¬op_g_,₁(·), ¬op_g_,₂(·), ¬op_g_,₃(·) = clg(·), intg◦clg(·), clg◦intg(·), intg◦clg◦intg(·). (2.6)

ATg,Λ-setSg,Λ⊂Tg in aTg,Λ-space is called ag-Tg,Λ-set if and only if there exist

a pair (Og,Kg)∈ Tg,Λ× ¬Tg,Λ ofTg,Λ-open andTg,Λ-closed sets, and ag-operator opg(·)∈ Lg Λsuch that the following statement holds:

(∃ξ) (ξ∈ Sg)∧Sg ⊆opg(Og)

∨Sg ⊇ ¬opg(Kg)

.

(2.7)

Theg-Tg,Λ-setSg⊂Tg,Λis said to be of categoryν if and only if it belongs to the

following class ofg-ν-Tg,Λ-sets:

g-ν-S Tg,Λ

def

= Sg⊂Tg,Λ:

∃Og,Kg,opg,ν(·)

Sg⊆opg,ν(Og)

∨Sg ⊇ ¬opg,ν(Kg). (2.8)

It is called ag-ν-Tg,Λ-open set if it satisﬁes the ﬁrst property ing-ν-S Tg,Λ

and a g-ν-Tg,Λ-closed set if it satisﬁes the second property ing-ν-S Tg,Λ

. The classes of g-ν-Tg,Λ-open andg-ν-Tg,Λ-closed sets, respectively, are deﬁned by

g-ν-O Tg,Λ

def

= Sg⊂Tg,Λ:

∃Og,opg,ν(·)

Sg ⊆opg,ν(Og)

,

g-ν-K Tg,Λ

def

= Sg⊂Tg,Λ:

∃Kg,opg,ν(·)

Sg⊇ ¬opg,ν(Kg)

.

(2.9)

From these classes, the following relation holds: g-S Tg,Λ

= _ν∈I0

3g-ν-S Tg,Λ

= _ν∈I0

3

g-ν-O Tg∪g-ν-KTg

= _ν∈I0

3g-ν-OTg,Λ

∪_ν∈I0

3g-ν-K Tg,Λ

= g-O Tg,Λ

∪g-K Tg,Λ

.

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By omitting the subscriptgin almost all symbols of the above deﬁnitions, we obtain very similar deﬁnitions but in aTΛ-space.

AT_Λ-setS ⊂T_Λin aTΛ-space is called ag-T_Λ-set if and only if there exists a pair (O,K)∈ TΛ× ¬TΛ of TΛ-open andTΛ-closed sets, and an operatorop(·)∈ L Λ such that the following statement holds:

(∃ξ) (ξ∈ S)∧(S ⊆op (O))∨(S ⊇ ¬op (K)). (2.11)

The g-T_Λ-set S ⊂ T_Λ is said to be of category ν if and only if it belongs to the following class ofg-ν-TΛ-sets:

g-ν-S T_Λdef= S ⊂T_Λ: (∃O,K,op_ν(·))

(S ⊆op_ν(O))∨(S ⊇ ¬op_ν(K)). (2.12)

It is called a g-ν-TΛ-open set if it satisﬁes the ﬁrst property in g-ν-S TΛ

and a g-ν-T_Λ-closed set if it satisﬁes the second property in g-ν-S T_Λ. The classes of g-ν-T_Λ-open andg-ν-T_Λ-closed sets, respectively, are deﬁned by

g-ν-O T_Λ def= S ⊂T_Λ: (∃O,op_ν(·)) S ⊆op_ν(O),

g-ν-K T_Λ def= S ⊂T_Λ: (∃K,op_ν(·)) S ⊇ ¬op_ν(K). (2.13)

As in the previous deﬁnitions, from these classes, the following relation holds: g-S T_Λ = _ν∈I0

3g-ν-S TΛ

= _ν∈I0

3

g-ν-O T_Λ∪g-ν-K T_Λ

= _ν∈I0

3g-ν-OTΛ

∪_ν∈I0

3g-ν-K TΛ

= g-O TΛ

∪g-K TΛ

.

(2.14)

The classes O [Tg,Λ] and K [Tg,Λ] denote the families ofTg,Λ-open andTg,Λ-closed

sets, respectively, in Tg,Λ, with S [Tg,Λ] = O [Tg,Λ]∪K [Tg,Λ]; the classes O [TΛ]

and K [T_Λ] denote the families of T-open and T_Λ-closed sets, respectively, in T_Λ, with S [T_Λ] = O [T_Λ]∪K [T_Λ]. (Whenever we feel that the subscript Λ∈Ω,Σ,Υ is understood from the context, it will be omitted for clarity.) We are now in a position to present a carefully chosen set of terms used in the theory ofg-Tg-maps betweenTg-spaces.

A (TΩ,TΣ)-map and a (Tg,Ω,Tg,Σ)-map, respectively, are mappings in the usual

sense betweenT-spaces andTg-spaces.

Definition2.1 ((TΩ,T_Σ), (T_g_,Ω,T_g_,_Σ)-Maps). LetT_Ω= (Ω,TΩ) andT_Σ= (Σ,TΣ)

be T-spaces and, letTg,Ω= (Ω,Tg,Ω) and Tg,Σ= (Σ,Tg,Σ) beTg-spaces. Then, a

map:

• i. π:T_Ω→T_Σis called a (TΩ,T_Σ)-map fromT_Ωinto T_Σ.

• ii. π:T_g_,_Ω→T_g_,_Σ is called a (T_g_,Ω,T_g_,_Σ)-map fromT_g_,_Ωinto T_g_,_Σ.

Ag- (TΩ,TΣ)-map is a generalization of a (TΩ,TΣ)-map and, hence, is a

distin-guished mapping betweenT-spaces which does not exhibit mapping properties in the usual sense but does exhibit mapping properties in the generalized sense.

Definition 2.2 (g-ν- (TΩ,T_Σ)-Map). LetT_Ω = (Ω,TΩ) and T_Σ= (Σ,TΣ) be T

-spaces, and let op(·)∈ L Σ. Then, a map πg :TΩ→TΣ is called a g- (TΩ,TΣ

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sets inT_Ωthere corresponds a pair (O_σ,K_σ)∈ TΣ× ¬TΣofTΣ-open andTΣ-closed sets inT_Σsuch that the following statement holds:

πg(Oω)⊆op (Oσ)∨ πg(Kω)⊇ ¬op (Kσ). (2.15)

Ag- (TΩ,T_Σ)-map is said to be of categoryνif and only if it belongs to the following class ofg-ν- (TΩ,TΣ)-maps:

g-ν-M [TΩ;TΣ]def=

πg: ∀Oω,Kω∃Oσ,Kσ,opν(·)

πg(Oω)⊆opν(Oσ)∨πg(Kω)⊇ ¬opν(Kσ). (2.16)

It is called ag-ν- (TΩ,TΣ)-open map if it satisﬁes the ﬁrst property ing-ν-M [TΩ;TΣ]

and ag-ν- (TΩ,TΣ)-closed map if it satisﬁes the second property ing-ν-M [TΩ;TΣ].

The classes ofg-ν- (TΩ,TΣ)-open and g-ν- (TΩ,TΣ)-closed maps, respectively, are

deﬁned by

g-ν-M_O[TΩ;TΣ]def=

πg:∀Oω∃Oσ,opν(·) πg(Oω)⊆opν(Oσ),

g-ν-M_K[T_Ω;T_Σ]def= πg:∀Kω∃Kσ,opν(·) πg(Kω)⊇opν(Kσ). (2.17)

From the class g-ν-M [TΩ;TΣ], consisting of the classes g-ν-MO[TΩ;TΣ] and

g-ν-M_K[TΩ;TΣ], respectively, ofg-ν- (TΩ,TΣ)-open andg-ν- (TΩ,TΣ)-closed maps,

whereν ∈I₃0, there results in the following deﬁnition.

Definition _2.3. _Let T_Ω _{= (Ω},TΩ_{) and} T_Σ _{= (Σ},TΣ_{) be} T_{-spaces. If, for each} ν ∈ I0

3, g-ν-MO[TΩ;TΣ] and g-ν-MK[TΩ;TΣ], respectively, denote the classes of

g-ν- (TΩ,T_Σ)-open andg-ν- (TΩ,T_Σ)-closed maps, then g-M [TΩ;TΣ] =

ν∈I0

3g-ν-M [TΩ;TΣ]

= _ν∈I0

3

g-ν-M_O[TΩ;TΣ]∪g-ν-MK[TΩ;TΣ]

= _ν∈I0

3g-ν-MO[TΩ;TΣ]

∪_ν∈I0

3g-ν-MK[TΩ;TΣ]

= g-M_O[TΩ;TΣ]∪g-MK[TΩ;TΣ].

(2.18)

As above, theg- (Tg,Ω,Tg,Σ)-map is a generalization of the (Tg,Ω,Tg,Σ)-map and,

thus, is a distinguished mapping betweenTg-spaces which does not exhibit mapping properties in the usual sense but does exhibit mapping properties in the generalized sense.

Definition2.4 (g-ν- (T_g_,Ω,T_g_,_Σ)-Map). LetT_g_,_Ω= (Ω,T_g_,_Ω) andT_g_,_Σ= (Σ,T_g_,_Σ)

be Tg-spaces, and let op_g(·)∈ Lg Σ. Then, a map πg :Tg,Ω →Tg,Σ is called a

g- (Tg,Ω,Tg,Σ)-map if and only if, for every pair (Og,ω,Kg,ω)∈ Tg,Ω×¬Tg,ΩofTg,Ω

-open and Tg,Ω-closed sets in Tg,Ω there corresponds a pair (Og,σ,Kg,σ) ∈ Tg,Σ× ¬Tg,Σ of Tg,Σ-open and TΣ-closed sets in Tg,Σ such that the following statement

holds:

πg(Og,ω)⊆opg(Og,σ)∨ πg(Kg,ω)⊇ ¬opg(Kg,σ). (2.19)

A g- (Tg,Ω,Tg,Σ)-map is said to be of category ν if and only if it belongs to the

following class ofg-ν- (Tg,Ω,Tg,Σ)-maps:

g-ν-M [Tg,Ω;Tg,Σ]def=

πg: ∀Og,ω,Kg,ω∃Og,σ,Kg,σ,opg,ν(·)

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It is called a g-ν- (Tg,Ω,Tg,Σ)-open map if it satisﬁes the ﬁrst property in the

class g-ν-M [Tg,Ω;Tg,Σ] and a g-ν- (Tg,Ω,Tg,Σ)-closed map if it satisﬁes the

sec-ond property ing-ν-M [Tg,Ω;Tg,Σ]. The classes of g-ν- (Tg,Ω,Tg,Σ)-open maps and

g-ν- (Tg,Ω,Tg,Σ)-closed maps, respectively, are deﬁned by

g-ν-M_O[Tg,Ω;Tg,Σ]def=

πg :∀Og,ω∃Og,σ,opg,ν(·)

πg(Og,ω)⊆opg,ν(Og,σ),

g-ν-M_K[Tg,Ω;Tg,Σ]def=

πg:∀Kω∃Kg,σ,opg,ν(·)

πg(Kg,ω)⊇opg,ν(Kg,σ). (2.21)

From the class g-ν-M [Tg,Ω;Tg,Σ], consisting of the classes g-ν-MO[Tg,Ω;Tg,Σ]

andg-ν-M_K[Tg,Ω;Tg,Σ] ofg-ν- (Tg,Ω,Tg,Σ)-open andg-ν- (Tg,Ω,Tg,Σ)-closed maps,

whereν ∈I₃0, respectively, there results in the following deﬁnition.

Definition 2.5. Let T_g_,_Ω = (Ω,T_g_,_Ω) andT_g_,_Σ = (Σ,T_g_,_Σ) be Tg-spaces. If, for

eachν ∈I0

3, g-ν-MO[Tg,Ω;Tg,Σ] and g-ν-MK[Tg,Ω;Tg,Σ], respectively, denote the

classes ofg-ν- (Tg,Ω,Tg,Σ)-open andg-ν- (Tg,Ω,Tg,Σ)-closed maps, then

g-M [Tg,Ω;Tg,Σ] =

ν∈I0

3g-ν-M [Tg,Ω;Tg,Σ] =_ν∈I0

3

g-ν-M_O[Tg,Ω;Tg,Σ]∪g-ν-MK[Tg,Ω;Tg,Σ]

=_ν∈I0

3g-ν-MO[Tg,Ω;Tg,Σ]

∪_ν∈I0

3g-ν-MK[Tg,Ω;Tg,Σ]

=g-M_O[T_Ω;Tg,Σ]∪g-MK[Tg,Ω;Tg,Σ].

(2.22)

Definition 2.6 (g-ν- (T_g_,Ω,T_g_,_Σ)-Continuous). Let T_g_,_Ω = (Ω,T_g_,_Ω) andT_g_,_Σ =

(Σ,Tg,Σ) be Tg-spaces, and let opg(·) ∈ Lg Ω. Then, a map πg : Tg,Ω → Tg,Σ

is said to be g- (Tg,Ω,Tg,Σ)-continuous if and only if, for every pair (Og,σ,Kg,σ)∈

Tg,Σ× ¬Tg,Σ of Tg,Σ-open and TΣ-closed sets in Tg,Σ there corresponds a pair

(Og,ω,Kg,ω)∈ Tg,Ω× ¬Tg,ΩofTg,Ω-open andTg,Ω-closed sets inTg,Ωsuch that the

following statement holds:

π−1

g (Og,σ)⊆opg(Og,ω)∨ πg−1(Kg,σ)⊇ ¬opg(Kg,ω). (2.23)

Ag- (Tg,Ω,Tg,Σ)-continuous map is said to be of categoryνif and only if it belongs

to the following class ofg-ν- (Tg,Ω,Tg,Σ)-continuous maps:

g-ν-C [Tg,Ω;Tg,Σ]def=

πg: ∀Og,σ,Kg,σ∃Og,ω,Kg,ω,opg,ν(·)

π−1

g (Og,σ)⊆opg,ν(Og,ω)∨πg−1(Kg,σ)⊇ ¬opg,ν(Kg,ω). (2.24)

Definition 2.7. Let T_g_,_Ω = (Ω,T_g_,_Ω) and T_g_,_Σ = (Σ,T_g_,_Σ) be Tg-spaces. If,

for eachν ∈I0

3,g-ν-C [Tg,Ω;Tg,Σ] denotes the class ofg-ν- (Tg,Ω,Tg,Σ)-continuous

maps, then

g-C [Tg,Ω;Tg,Σ] =

ν∈I0

3g-ν-C [Tg,Ω;Tg,Σ]. (2.25)

Definition 2.8 (g-ν- (T_g_,Ω,T_g_,_Σ)-Irresolute). Let T_g_,_Ω = (Ω,T_g_,_Ω) and T_g_,_Σ =

(Σ,Tg,Σ) be Tg-spaces, and let opg(·) ∈ Lg Ω. Then, a map πg : Tg,Ω → Tg,Σ

is said to be g- (Tg,Ω,Tg,Σ)-irresolute if and only if, for every pair (Og,σ,Kg,σ) ∈

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(Og,ω,Kg,ω)∈ Tg,Ω× ¬Tg,ΩofTg,Ω-open andTg,Ω-closed sets inTg,Ωsuch that the

following statement holds:

π−1

g

op_g(Og,σ)⊆opg(Og,ω)∨ π−g1

¬op_g(Kg,σ)⊇ ¬opg(Kg,ω). (2.26)

Ag- (Tg,Ω,Tg,Σ)-irresolute map is said to be of categoryν if and only if it belongs

to the following class ofg-ν- (Tg,Ω,Tg,Σ)-irresolute maps:

g-ν-I [Tg,Ω;Tg,Σ]def=

πg : ∀Og,σ,Kg,σ∃Og,ω,Kg,ω,opg,ν(·)

π−1

g

op_g_,ν(Og,σ)⊆opg,ν(Og,ω)∨πg−1

¬op_g_,ν(Kg,σ)⊇

¬op_g_,ν(Kg,ω). (2.27)

Definition 2.9. Let T_g_,_Ω = (Ω,T_g_,_Ω) andT_g_,_Σ = (Σ,T_g_,_Σ) be Tg-spaces. If, for

eachν ∈I0

3,g-ν-I [Tg,Ω;Tg,Σ] denotes the class ofg-ν- (Tg,Ω,Tg,Σ)-irresolute maps,

then

g-I [Tg,Ω;Tg,Σ] =

ν∈I0

3g-ν-I [Tg,Ω;Tg,Σ]. (2.28)

Definition _2.10. _LetT_g_,_Ω_{= (Ω},T_g_,_Ω_{) and}T_g_,_Σ_{= (Σ},T_g_,_Σ_{) be} T_{g-spaces and, let}

T_Ω= (Ω,TΩ) andTΣ= (Σ,TΣ) be T-spaces.

• i. The classes M_O[T_Ω;T_Σ] and M_K[T_Ω;T_Σ] denote the families of T

-open andT-closed maps, respectively, fromTΩintoTΣ, with M [TΩ;TΣ] =

MO[TΩ;TΣ]∪MK[TΩ;TΣ].

• ii. The classes M_O[T_g_,_Ω;T_g_,_Σ] and M_K[T_g_,_Ω;T_g_,_Σ] denote the families

of Tg-open and Tg-closed maps, respectively, from Tg,Ω into Tg,Σ, with

M [Tg,Ω;Tg,Σ] = MO[Tg,Ω;Tg,Σ]∪MK[Tg,Ω;Tg,Σ].

The following sections present the main results of the theory ofg-Tg-maps.

2.2. Main Results. The purpose of the following lines is to explore properties

and characterizations of g- (Tg,Ω;Tg,Σ)-maps πg : Tg,Ω → Tg,Σ belonging to the

classg-M [Tg,Ω;Tg,Σ].

Theorem2.11. Ifπ_g:T_g_,_Ω→T_g_,_Σis a(T_g_,Ω,T_g_,_Σ)-open or a(T_g_,Ω,T_g_,_Σ)-closed map, then

πg∈g-M_O[TΩ;Tg,Σ]∪g-MK[Tg,Ω;Tg,Σ].

(2.29)

Proof. _Letπ_g_:T_g_,_Ω→T_g_,_Σ_{be a (}T_g_,Ω,T_g_,_Σ_{)-map. Then, for every (}O_g_,ω,K_g_,ω₎∈ Tg,Ω× ¬Tg,Ω there exists (Og,,Kg,σ)∈ Tg,Σ× ¬Tg,Σsuch that

πg(Og,ω)⊆ Og,σ∨ πg(Kg,ω)⊇ Kg,σ. But,Og,σ ⊆opg(Og,σ) andKg,σ ⊇ ¬opg(Kg,σ). Consequently,

πg(Og,ω)⊆opg(Og,σ)∨ πg(Kg,ω)⊇ ¬opg(Kg,σ).

Hence,πg∈g-M_O[TΩ;Tg,Σ]∪g-MK[Tg,Ω;Tg,Σ]. Q.e.d.

The converse of Thm. 2.11 is clearly false, because the statement ”π_g :T_g_,_Ω→

Tg,Σ is a (Tg,Ω,Tg,Σ)-map and πg : Tg,Ω →Tg,Σ is not a g- (Tg,Ω,Tg,Σ)-map” is

untrue. The following theorem states that, the image of ag-Tg-set in a Tg-space Tg,Ωis a g-Tg-set in a Tg-spaceTg,Σ if and only if the mapπg :Tg,Ω →Tg,Σ is a

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Theorem 2.12. A necessary and suﬃcient condition for π_g :T_g_,_Ω→T_g_,_Σ to be a

g- (Tg,Ω,Tg,Σ)-map is that, for every (Og,ω,Kg,ω)∈ Tg,Ω× ¬Tg,Ω, πgopg(Og,ω)⊆opg(πg(Og,ω))∨ πg¬opg(Kg,ω)⊇

¬op_g(πg(Kg,ω)). (2.30)

Proof. Necessity. Letπ_g∈g-M [T_g_,_Ω;T_g_,_Σ]. Then for (O_g_,ω,K_g_,ω)∈ T_g_,_Ω×¬T_g_,_Ω

there corresponds (Og,,Kg,σ)∈ Tg,Σ× ¬Tg,Σsuch that

Because Og,ω ⊆opg(Og,ω)∨ Kg,ω ⊇ ¬opg(Kg,ω), it consequently follows that,

πgopg(Og,ω)⊆opg◦opg(Og,σ)∨ πg¬opg(Kg,ω)

⊇ ¬opg◦¬opg(Kg,σ). But, since

opg◦opg(Og,σ)⊆opg(πg(Og,ω)), ¬opg◦¬opg(Kg,σ)⊇ ¬opg(πg(Kg,ω)), the proof at once follows.

Suﬃciency. For every (Og,ω,Kg,ω)∈ Tg,Ω× ¬Tg,Ω, let

πgopg(Og,ω)⊆opg(πg(Og,ω))∨ πg¬opg(Kg,ω)⊇

¬opg(πg(Kg,ω)). Then,

πg(Og,ω)⊆πgopg(Og,ω)⊆opg(πg(Og,ω))⊆opg(Og,σ)

∨ πg(Kg,ω)⊇πg¬opg(Kg,ω)⊇ ¬opg(πg(Kg,ω))⊇ ¬opg(Og,σ), because, Og,ω ⊆ opg(Og,ω), Kg,ω ⊇ ¬opg(Kg,ω), πg(Og,ω) ⊆ opg(Og,σ), and

πg(Kg,ω)⊇ ¬opg(Kg,σ). Therefore,

πg(Og,ω)⊆opg(Og,σ)∨ πg(Kg,ω)⊇ ¬opg(Og,σ).

Thus,πg ∈g-M [Tg,Ω;Tg,Σ], which completes the proof. Q.e.d. Theorem 2.13. If π_g_,α ∈g-M [T_g_,_Ω;T_g_,_Σ]andπ_g_,β ∈g-M [T_g_,_Σ;T_g_,_Υ], thenπ_g_,β◦ πg,α∈g-M [Tg,Ω;Tg,Υ].

Proof. Letπ_g_,α ∈g-M [T_g_,_Ω;T_g_,_Σ] and π_g_,β ∈g-M [T_g_,_Σ;T_g_,_Υ]. Then, for every

(Og,ω,Kg,ω)∈ Tg,Ω× ¬Tg,Ωthere exists (Og,σ,Kg,σ)∈ Tg,Σ× ¬Tg,Σand, for every

(Og,σ,Kg,σ)∈ Tg,Σ× ¬Tg,Σthere exists (Og,υ,Kg,υ)∈ Tg,Υ× ¬Tg,Υ such that πg,α(Og,ω)⊆opg(Og,σ)∨ πg,α(Kg,ω)⊇ ¬opg(Kg,σ),

πg,β(Og,σ)⊆opg(Og,υ)∨ πg,β(Kg,σ)⊇ ¬opg(Kg,υ).

From the ﬁrst line, aided with the second, the logical statement preceding∨becomes

πg,α(Og,ω) ⊆ opg(Og,σ)

πg,β◦πg,α(Og,ω) ⊆ πg,βopg(Og,σ)⊆opg

πg,β(Og,σ)⊆opg

Og,υ, and, that following∨becomes

πg,α(Kg,ω) ⊇ ¬opg(Kg,σ)

πg,β◦πg,α(Kg,ω) ⊇ πg,β¬opg(Kg,σ)⊇ ¬opg

πg,βKg,σ⊇ ¬opg

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Thus,πg,β◦πg,α∈g-M [Tg,Ω;Tg,Υ], which proves the theorem. Q.e.d. Theorem2.14. Letπ_g_,α:T_g_,_Ω→T_g_,_Σbe a(T_g_,Ω,T_g_,_Σ)-map and letπ_g_,β :T_g_,_Σ→

Tg,Υ be a(Tg,Σ,Tg,Υ)-map. Then:

• i. π_g_,α∈g_{-M [}T_g_,_Ω_;T_g_,_Σ_]_implies π_g_,β◦π_g_,α∈g_{-M [}T_g_,_Ω_;T_g_,_Υ_]_. • ii. π_g_,β∈g_{-M [}T_g_,_Σ_;T_g_,_Υ_]_implies π_g_,β◦π_g_,α∈g_{-M [}T_g_,_Ω_;T_g_,_Υ_]_.

Proof. i. Letπ_g_,α be a (T_g_,Ω,T_g_,_Σ)-map andπ_g_,β a (T_g_,Σ,T_g_,_Υ)-map. Then, for

every (Og,ω,Kg,ω) ∈ Tg,Ω× ¬Tg,Ω, there exists (Og,σ,Kg,σ) ∈ Tg,Σ× ¬Tg,Σ and,

for every (Og,σ,Kg,σ)∈ Tg,Σ× ¬Tg,Σ, there exists (Og,υ,Kg,υ)∈ Tg,Υ× ¬Tg,Υ such

that

πg,α(Og,ω)⊆ Og,σ∨ πg,α(Kg,ω)⊇ Kg,σ,

πg,β(Og,σ)⊆ Og,υ∨ πg,β(Kg,σ)⊇ Kg,υ.

The logical statements expressing the relationsπg,α ∈g-M [Tg,Ω;Tg,Σ] andπg,α ∈

g-M [Tg,Σ;Tg,Υ] are, respectively, πg,α(Og,ω)⊆opg

Og,σ ∨ πg,α(Kg,ω)⊇ ¬opg

Kg,σ,

πg,β(Og,σ)⊆opg

Og,υ ∨ πg,β(Kg,σ)⊇ ¬opg

Kg,υ. Therefore, if only the relationπg,α∈g-M [Tg,Ω;Tg,Σ] holds, then

πg,β◦πg,α(Og,ω)⊆πg,βopg

Og,σ∨ πg,β◦πg,α(Kg,ω)

⊇πg,β¬opg

Kg,σ

⇒ πg,β◦πg,α(Og,ω)⊆opg

πg,βOg,σ∨ πg,β◦πg,α(Kg,ω)

⊇ ¬opg

πg,βKg,σ

Og,υ∨ πg,β◦πg,α(Kg,ω)⊇ ¬opg

Kg,υ, and, hence,πg,β◦πg,α∈g-M [Tg,Ω;Tg,Υ].

ii. _{If only the relation}π_g_,β∈g_{-M [}T_g_,_Σ_;T_g_,_Υ_{] holds, then}

πg,β◦πg,α(Og,ω)⊆πg,βOg,σ∨ πg,β◦πg,α(Kg,ω)⊇πg,βKg,σ

Og,υ∨ πg,β◦πg,α(Kg,ω)⊇ ¬opg

Kg,υ,

and, hence,πg,β◦πg,α∈g-M [Tg,Ω;Tg,Υ]. Q.e.d.

Proposition 2.15. Let π_g_,α ∈g-M [T_g_,_Ω;T_g_,_Σ]and π_g_,β ∈g-M [T_g_,_Σ;T_g_,_Ω], satis-fying

πg,β◦πg,α(Og,ω)⊆opg(Og,ω)∨ πg,β◦πg,α(Kg,ω)⊇ ¬opg(Kg,ω),

πg,α◦πg,β(Og,σ)⊆opg(Og,σ)∨ πg,α◦πg,β(Kg,σ)⊇ ¬opg(Kg,σ), (2.31)

respectively. Then, there exist inverse maps π−1

g,α ∈ g-M [Tg,Ω;Tg,Σ] and π−g,β1 ∈ g-M [Tg,Σ;Tg,Ω]such that πg,β =π−g,α1 andπg,α=π−g,β1.

Proof. _{It is clear that,} O_g_,μ ⊆ _op_g₍O_g_,μ_{) or} K_g_,μ ⊇ _op_g₍K_g_,μ_{) for every} μ ∈

ω, σ. But,πg,α ∈g-M [Tg,Ω;Tg,Σ] or πg,β∈g-M [Tg,Σ;Tg,Ω] satisfy

πg,β◦πg,α(Og,ω)⊆opg(Og,ω)∨ πg,β◦πg,α(Kg,ω)⊇ ¬opg(Kg,ω),

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Hence, there exist π−1

g,α ∈ g-M [Tg,Ω;Tg,Σ] and πg−,β1 ∈ g-M [Tg,Σ;Tg,Ω] such that πg,β =π−g,α1 andπg,α=πg−,β1. This proves the proposition. Q.e.d.

Theorem 2.16. Letπ_g∈g-M [T_g_,_Ω;T_g_,_Σ]. Given any T_g_,_Σ-set S_g⊂T_g_,_Σand any pair (Og,ω,Kg,ω)∈ Tg,Ω× ¬Tg,Ω ofTg,Ω-open and Tg,Ω-closed sets satisfying

π−1

g (Sg)⊆opg(Og,ω)∨ πg−1(Sg)⊇ ¬opg(Kg,ω), (2.32)

then:

• i. π_g∈g_-M

K[TΩ;Tg,Σ] implies the existence of aTg,Σ-open setOg,σ ⊇ Sg

such thatπ−1

g (Og,σ)⊆opg(Og,ω).

• ii. π_g∈g-M

O[TΩ;Tg,Σ]implies the existence of aTg,Σ-closed setKg,σ ⊇ Sg such thatπ−1

g (Kg,σ)⊇ ¬opg(Kg,ω).

Proof. i. LetO_g_,σ= Σ−π_gΩ−op_g(O_g_,ω). Then, sinceπ−_g1(Sg)⊆op_g(O_g_,ω)

and πg ∈ g-MK[Tg,Ω;Tg,Σ], there exists a Tg,Σ-open set Og,σ ∈ Tg,Σ such that Og,σ ⊇ Sg. But, since

π−1

g (Og,σ) = Ω−πg−1◦πgΩ−opg(Og,ω)

⊆ Ω−Ω−opg(Og,ω)= opg(Og,ω), the proof ofi.follows.

ii. LetK_g_,σ = Σ−π_gΩ− ¬op_g(K_g_,ω). Then, becauseπ−_g1(Sg)⊇ ¬op_g(K_g_,ω)

and πg ∈ g-MO[Tg,Ω;Tg,Σ], there exists a Tg,Σ-closed Kg,σ ∈ ¬Tg,Σ such that Kg,σ⊇ Sg. But, since

π−1

g (Kg,σ) = Ω−π−g1◦πgΩ− ¬opg(Kg,ω)

⊇ Ω−Ω− ¬opg(Kg,ω)=¬opg(Kg,ω),

the proof ofii.follows. Q.e.d.

We next investigate further properties and give characterizations of those ele-ments which belong to the classg-C [Tg,Ω;Tg,Σ].

Theorem 2.17. If π_g : T_g_,_Ω →T_g_,_Σ is a (T_g_,Ω,T_g_,_Σ)-continuous map, then π_g ∈

g-C [Tg,Ω;Tg,Σ].

Proof. If π_g : T_g_,_Ω → T_g_,_Σ is a (T_g_,Ω,T_g_,_Σ)-continuous map, then, for every

(Og,σ,Kg,σ)∈ Tg,Σ× ¬Tg,Σ, there exists (Og,ω,Kg,ω)∈ Tg,Ω× ¬Tg,Ωsuch that π−1

g (Og,σ)⊆ Og,ω∨ πg−1(Kg,σ)⊇ Kg,ω. But, for every (Og,ω,Kg,ω)∈ Tg,Ω× ¬Tg,Ω,

Og,ω⊆opg(Og,ω)∨ Kg,ω ⊇ ¬opg(Kg,ω), and, consequently,

π−1

g (Og,σ)⊆opg(Og,ω)∨ πg−1(Kg,σ)⊇ ¬opg(Kg,ω).

Hence,πg∈g-C [Tg,Ω;Tg,Σ]. Q.e.d.

Theorem 2.18. If π_g : T_g_,_Ω → T_g_,_Σ is a (T_g_,Ω,T_g_,_Σ)-map satisfying, for every

(Og,σ,Kg,σ)∈ Tg,Σ× ¬Tg,Σ, π−1

g (Og,σ)⊆opg

π−1

g (Og,σ)∨ πg−1(Kg,σ)⊇ ¬opg

π−1

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Proof. For every (O_g_,σ,K_g_,σ)∈ T_g_,_Σ× ¬T_g_,_Σ, it is evident that Og,σ ⊆opg(Og,σ)∨ Kg,σ ⊇ ¬opg(Kg,σ)

⇒ π−1

g (Og,σ)⊆πg−1

op_g(Og,σ)∨ πg−1(Kg,σ)⊇π−g1

¬op_g(Kg,σ)

⇒ π−1

g (Og,σ)⊆opg

π−1

g (Og,σ)∨ πg−1(Kg,σ)⊇ ¬opg

π−1

g (Kg,σ). Hence, there exists (Og,ω,Kg,ω)∈ Tg,Ω× ¬Tg,Ω such thatπ−g1(Og,σ) ⊆ Og,ω and

π−1

g (Kg,σ)⊇ Kg,ω. Consequently,πg∈g-C [Tg,Ω;Tg,Σ], which completes the proof. Q.e.d. Definition 2.19 ((T_g_,_Ω;T_g_,_Σ)-Bijective Map). A (T_g_,Ω,T_g_,_Σ)-map π_g : T_g_,_Ω →

Tg,Σis said to be bijective if and only if it belongs the following class:

g-B [Tg,Ω;Tg,Σ]def=

πg: (∀ζ∈Tg,Σ) (∃!ξ∈Tg,Ω) πg(ξ) =ζ. (2.33)

Theorem 2.20. Ifπ_g∈g-B [T_g_,_Ω;T_g_,_Σ], then

πg∈g-M [Tg,Ω;Tg,Σ] ⇔ πg−1∈g-C [Tg,Σ;Tg,Ω].

(2.34)

Proof. _Necessity. _Letπ−_g1∈g_{-C [}T_g_,_Σ_;T_g_,_Ω_{]. Then} π−1

g −1

(Og,ω)⊆opg(Og,σ)∨ π−g1 −1

(Kg,ω)⊇ ¬opg(Kg,σ).

But πg ∈ g-B [Tg,Ω;Tg,Σ] implies

π−1

g −1

(Sg) = πg(Sg) for every Sg ∈ Tg,Ω∪ ¬Tg,Ω. Consequently,

Hence,πg∈g-M [Tg,Ω;Tg,Σ].

Suﬃciency. Letπg∈g-M [Tg,Ω;Tg,Σ]. Then,

Butπg(Sg) =π−g1 −1

(Sg) for everySg ∈ Tg,Ω∪¬Tg,Ω, sinceπg ∈g-B [Tg,Ω;Tg,Σ].

Consequently,

π−1

g −1

(Og,ω)⊆opg(Og,σ)∨ π−g1 −1

(Kg,ω)⊇ ¬opg(Kg,σ).

Thus,π−1

g ∈g-C [Tg,Σ;Tg,Ω]. Q.e.d.

Theorem 2.21. If π_g_,α ∈g-C [T_g_,_Ω;T_g_,_Σ] and π_g_,β ∈g-C [T_g_,_Σ;T_g_,_Υ], then π_g_,β◦ πg,α∈g-C [Tg,Ω;Tg,Υ].

Proof. Let π_g_,α ∈ g-C [T_g_,_Ω;T_g_,_Σ] and π_g_,β ∈ g-C [T_g_,_Σ;T_g_,_Υ]. Then π_g−1

,α ∈ g-M [Tg,Σ;Tg,Ω] andπg−,β1 ∈g-C [Tg,Υ;Tg,Σ], implying

π−1

g,α(Og,σ)⊆opg(Og,ω) ∨ πg−,α1(Kg,σ)⊇ ¬opg(Kg,ω),

π−1

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respectively. Combining both logical statements, there follows that

π−1

g,β(Og,υ)⊆opg(Og,σ)∨ πg−,β1(Kg,υ)⊇ ¬opg(Kg,σ)

⇒ π−1

g,α◦πg−,β1(Og,υ)⊆πg−,α1

op_g(Og,σ)∨ πg−,α1 ◦πg−,β1(Kg,υ)

⊇π−1

g,α

¬op_g(Kg,σ)

⇒ π−1

g,α◦πg−,β1(Og,υ)⊆opg

π−1

g,α(Og,σ)∨ πg−,α1 ◦πg−,β1(Kg,υ)

⊇ ¬op_gπ−_g_,α1(Kg,σ)

⇒ π−1

g,α◦πg−,β1(Og,υ)⊆opg(Og,ω)∨ πg−,α1 ◦πg−,β1(Kg,υ)⊇ ¬opg(Kg,ω). Since π−1

g,α ◦πg−,β1(Sg) = (πg,β◦πg,α)−1(Sg) for every Sg ∈ Tg,Υ∪ ¬Tg,Υ, there

follows thatπg,β◦πg,α∈g-C [Tg,Ω;Tg,Υ], which was to be proved. Q.e.d. Theorem 2.22. Let π_g_,α : T_g_,_Ω → T_g_,_Σ be a (T_g_,_Ω;T_g_,_Σ)-map and, let the col-lection Og,α_α∈I∗

n : Ω ⊆ opg

α∈I∗

nOg,α

and the collection Kg,α_α∈I∗

n :

Ω ⊆ ¬op_g_α∈I∗

nKg,α

, respectively, be g-Tg-open and g-Tg-closed coverings of Ω, where Og,α_α∈I∗

n and

Kg,α_α∈I∗

n, respectively, denote sequences of Tg-open sets and Tg-closed sets. If, for every α ∈ I_n∗, πg◦ιg,α ∈ g-C [Tg,Ω;Tg,Σ], where ιg,α:Og,α→Tg,Ω orιg,α:Og,α→Tg,Ω, thenπg∈g-C [Tg,Ω;Tg,Σ].

Proof. For every α ∈ I_n∗, let π_g◦ι_g_,α ∈ g-C [T_g_,_Ω;T_g_,_Σ]. Then, for every pair

Og,σ(α),Kg,σ(α)

∈ Tg,Σ× ¬Tg,Σ, there exists

Og,ω(α),Kg,ω(α)

∈ Tg,Ω× ¬Tg,Ω

such that

(πg◦ιg,α)−1Og,σ(α)

⊆op_gOg,ω(α)

∨ (πg◦ιg,α)−1Kg,σ(α)

⊇ ¬opg

Kg,ω(α)

⇒ _α∈I∗

n(πg◦ιg,α)

−1_O

g,σ(α)

⊆_α∈I∗

nopg

Og,ω(α)

∨ _α∈I∗

n(πg◦ιg,α)

−1_K

g,σ(α)

⊇_α∈I∗

n¬opg

Kg,ω(α)

⇒ _α∈I∗

n(πg◦ιg,α)

−1_O

g,σ(α)

⊆opg_α∈I∗

nOg,ω(α)

∨ _α∈I∗

n(πg◦ιg,α)

−1_K

g,σ(α)

⊇ ¬opg_α∈I∗

nKg,ω(α)

.

Since the following relations hold

π−1

g

Og,σ(α)

= _α∈I∗

n(πg◦ιg,α)

−1_O

g,σ(α)

, π−1

g

Kg,σ(α)

= _α∈I∗

n(πg◦ιg,α)

−1_K

g,σ(α)

,

the proof of the theorem follows. Q.e.d.

Henceforth, we investigate some properties and give some characterizations of g- (Tg,Ω,Tg,Σ)-irresolute maps.

Theorem_2.23. _A₍T_g_,Ω,T_g_,_Σ₎_-mapπ_g_:T_g_,_Ω→T_g_,_Σ_{is a}g_{- (}T_g_,Ω,T_g_,_Σ₎_-irresolute map if and only if, for every(Og,σ,Kg,σ)∈ Tg,Σ× ¬Tg,Σ,

π−1

g

op_g(Og,σ)⊆opg

π−1

g (Og,σ)∨ π−g1

¬op_g(Kg,σ)⊇

(13)

Proof. Necessity. Let π_g ∈ g-I [T_g_,_Ω;T_g_,_Σ]. Then, there exists (O_g_,ω,K_g_,ω) ∈ Tg,Ω× ¬Tg,Ω such that, for every (Og,σ,Kg,σ)∈ Tg,Σ× ¬Tg,Σ,

π−1

g

¬op_g(Kg,σ)⊇ ¬opg(Kg,ω). But since π−g1

opg(Og,σ) ⊆ πg−1(Og,σ) and πg−1

¬opg(Kg,σ) ⊇ π−g1(Kg,σ), it follows that

π−1

g

opg(Og,σ)⊆opg

π−1

¬opg(Kg,σ)⊇

¬opg

π−1

g (Kg,σ).

Suﬃciency. Let πg : Tg,Ω → Tg,Σ be a (Tg,Ω,Tg,Σ)-map satisfying, for every

(Og,σ,Kg,σ)∈ Tg,Σ× ¬Tg,Σ, π−1

g

op_g(Og,σ)⊆opg

π−1

¬op_g(Kg,σ)⊇

¬op_gπ−1

g (Kg,σ). But, π−1

g

opg(Og,σ) ⊆ πg−1(Og,σ) and π−g1

¬opg(Kg,σ) ⊇ πg−1(Kg,σ). There-fore, there exists (Og,ω,Kg,ω) ∈ Tg,Ω× ¬Tg,Ω such that π−g1(Og,σ) ⊆opg(Og,ω) andπ−1

g (Kg,σ)⊇ ¬opg(Kg,ω). Consequently,

π−1

g

opg(Og,σ)⊆opg(Og,ω)∨ π−g1

¬opg(Kg,σ)⊇ ¬opg(Kg,ω).

Thus,πg ∈g-I [Tg,Ω;Tg,Σ], which completes the proof. Q.e.d. Theorem 2.24. A g- (T_g_,Ω,T_g_,_Σ)-map π_g : T_g_,_Ω → T_g_,_Σ is a g- (T_g_,Ω,T_g_,_Σ) -irresolute map if and only if, for every(Og,ω,Kg,)∈ Tg,Ω× ¬Tg,Ω,

πgopg(Og,ω)⊇opg(πg(Og,ω))∨ πg¬opg(Kg,ω)⊆

¬opg(πg(Kg,ω)). (2.36)

Proof. Necessity. Let π_g ∈ g-I [T_g_,_Ω;T_g_,_Σ]. Then, there exists (O_g_,ω,K_g_,ω) ∈ Tg,Ω× ¬Tg,Ω such that, for every (Og,σ,Kg,σ)∈ Tg,Σ× ¬Tg,Σ,

π−1

g

op_g(Og,σ)⊆opg(Og,ω)∨ πg−1

¬op_g(Kg,σ)⊇ ¬opg(Kg,ω)

⇒ πgopg(Og,ω)⊇opg(Og,σ)∨ πg¬opg(Kg,ω)⊆ ¬opg(Kg,σ). But since op_g(Og,σ) ⊇ πgopg(Og,ω) and ¬opg(Kg,σ) ⊆ πg¬opg(Kg,ω), it follows that

¬op_g(πg(Kg,ω)).

Suﬃciency. Letπg :Tg,Ω→Tg,Σbe a g- (Tg,Ω,Tg,Σ)-map satisfying, for every

(Og,ω,Kg,ω)∈ Tg,Ω× ¬Tg,Ω,

¬op_g(πg(Kg,ω)). Then,

π−1

g

opg(πg(Og,ω))⊆opg(Og,ω)∨ π−g1

¬opg(πg(Kg,ω))⊇

(14)

But,πg∈g-M [Tg,Ω;Tg,Σ] equivalently implies the existence of (Og,σ,Kg,σ)∈ Tg,Σ×

¬Tg,Σsuch that, for every (Og,ω,Kg,ω)∈ Tg,Ω×¬Tg,Ω, opg(πg(Og,ω))⊆opg(Og,σ) or¬opg(πg(Kg,ω))⊇ ¬opg(Kg,σ). Consequently,

π−1

g

opg(Og,σ)⊆opg(Og,ω)∨ π−g1

¬opg(Kg,σ)⊇ ¬opg(Kg,ω).

Thus,πg ∈g-I [Tg,Ω;Tg,Σ], which completes the proof. Q.e.d. Theorem 2.25. Letπ_g:T_g_,_Ω→T_g_,_Σbe a(T_g_,Ω,T_g_,_Σ)-map. Then,

πg∈g-M [Tg,Ω;Tg,Σ]∩g-C [Tg,Ω;Tg,Σ] ⇒ πg∈g-I [Tg,Ω;Tg,Σ].

(2.37)

Proof. Let π_g ∈ g-M [T_g_,_Ω;T_g_,_Σ]∩g-C [T_g_,_Ω;T_g_,_Σ]. Then, there exists a pair

(Og,ω,Kg,ω)∈ Tg,Ω× ¬Tg,Ωsuch that, for every (Og,σ,Kg,σ)∈ Tg,Σ× ¬Tg,Σ, πg(Og,ω)⊆opg(Og,σ)∨ πg(Kg,ω)⊇ ¬opg(Og,σ),

and there exists (Og,σ,Kg,σ) ∈ Tg,Σ× ¬Tg,Σ such that, for every (Og,ω,Kg,ω) ∈

Tg,Ω× ¬Tg,Ω, π−1

g (Og,σ)⊆opg(Og,ω)∨ πg−1(Kg,σ)⊇ ¬opg(Kg,ω). From the ﬁrst statement, there follows that

Og,ω⊆π−g1

op_g(Og,σ)∨ Kg,ω ⊇π−g1

¬op_g(Kg,σ). But,

π−1

g

op_g(Og,σ)⊆opg

π−1

¬op_g(Kg,σ)⊇

¬op_gπ−_g1(Kg,σ); and, from the second statement, there follows that

op_gπ_g−1(Og,σ)⊆opg(Og,ω)∨ ¬opg

π−1

g (Kg,σ)⊇ ¬opg(Og,ω). From these last two logical statements, it consequently follows that

π−1

g

¬op_g(Kg,σ)⊇ ¬opg(Kg,ω),

and, hence,πg∈g-I [Tg,Ω;Tg,Σ], which completes the proof. Q.e.d. Theorem 2.26. If g-C [T_g_,_Ω;T_g_,_Σ] and g-I [T_g_,_Ω;T_g_,_Σ], respectively, denote the classes ofg- (Tg,Ω,Tg,Σ)-continuous andg- (Tg,Ω,Tg,Σ)-irresolute maps, then