On Generalization Of Generalized Star Generalized Continuous Function In Topological Spaces

In this paper our aim is to study a based on (gg)*g-continuous function. The family of continuous function plays an important role in topology. Observing these, Csaszar introduced the concept of generalized open sets [10]. In 1970, Levin [29] introduced the concept of generalized closed sets and discussed about the properties of sets, closed and open maps, normal and separation axioms, compactness. In 1982, Malghan [22] introduced and studied the concept of generalized closed maps. In 2017, Basavaraj M. Ittanagi and H.G Govardhana Reddy [3] introduced gg -closed sets in topological spaces. In 2018, I. Christal Bai and T. Shyla Isac Mary [9] introduced (gg)* - closed sets in topological spaces. In 2026, X. Josphine Selva Rani and T. Shyla Isac Mary [20] introduced (gg)^*g – closed sets in topological spaces. The aim of this paper is to be continue the study of (gg)*g  – continuous function and analyzed the different aspects.

PRELIMINARIES

Throughout this paper always means the topological spaces X,τ  and Y,σ

 on which no separation axioms are assumed unless explicitly stated. For a subset A of a topological space X,τ , the closure of A and the interior of A are denoted by cl(A) and int(A) respectively, A^c denotes the compliment of A in X,τ .

Definition:1

The closure of a subset A of a topological space (X, τ) is the smallest closed set containing A is denoted by clA

The generalized closure (briefly g  - closure) of a subset A of a topological space (X, τ) is the smallest g  - closed set containing A is denoted by gclA

Definition:2

Generalized - closed set (briefly ???? - closed) if ????????(????)⊆ ???? [29] whenever ????⊆???? and ???? is open in X.

Generalization of generalized closed set (briefly ???????? - closed) if ????????????(????)⊆ ???? [3] whenever ???? ⊆ ???? and U is regular semi - open in X.

Generalization of generalized star closed set (briefly (????????)∗ - closed) if ????????????(????)⊆ ???? [9] whenever ???? ⊆ ???? and U is ???????? - open in X.

Generalization of generalized star generalized closed set (briefly (????????)^∗???? - closed) if ????????????(????)⊆???? [20] whenever ????⊆???? and ???? is (????????)^∗- open in (X, τ).

Definition:3

A function f:(X,τ)→(Y,σ) is called 
    Continuous [30] if the inverse image of every closed set V in (Y,σ) is closed in (X,τ).
    regular continuous [31] if the inverse image of every closed set V in (Y,σ) is regular closed in (X,τ).
    π - continuous function [14] if the inverse image of every closed set V in (Y,σ) is π - closed in (X,τ).
    g^*  - continuous function [32] if the inverse image of every closed set V in (Y,σ) is g^*  -closed in (X,τ).
    〖(gαr)〗^(**)  - continuous function [39] if the inverse image of every closed set V in (Y,σ) is 〖(gαr)〗^(**)  - closed in (X,τ).
    〖gr〗^*  - continuous function [17] if the inverse image of every closed set V in (Y,σ) is 〖gr〗^*  - closed in (X,τ).
    〖(gg)〗^*  - continuous function [9] if the inverse image of every closed set V in (Y,σ) is 〖(gg)〗^*- closed in (X,τ).
    rβ- continuous function [23] if the inverse image of every closed set V in (Y,σ) is rβ-closed in (X,τ).
    〖gp〗^*  - continuous function [18] if the inverse image of every closed set V in (Y,σ) is 〖gp〗^*  - closed in (X,τ).
    〖(gsp)〗^*  - continuous function [33] if the inverse image of every closed set V in (Y,σ) is 〖(gsp)〗^*  - closed in (X,τ).
    g^#  - continuous function [4] if the inverse image of every closed set V in (Y,σ) is g^#  - closed in (X,τ).
    〖(gs)〗^*  - continuous function [1] if the inverse image of every closed set V in (Y,σ) is 〖(gs)〗^*- closed in (X,τ).
    g^* sr - continuous function [43] if the inverse image of every closed set V in (Y,σ) is g^* sr - closed in (X,τ).
    r^* g^*  - continuous function [27] if the inverse image of every closed set V in (Y,σ) is r^* g^*  - closed in (X,τ).
    〖(g〗^* 〖p)〗^*  - continuous function [34] if the inverse image of every closed set V in (Y,σ) is 〖(g〗^* 〖p)〗^*  - closed in (X,τ).
    r^∧ g - continuous function [37] if the inverse image of every closed set V in (Y,σ) is r^∧ g - closed in (X,τ).
    rwg – continuous function [26] if the inverse image of every closed set V in (Y,σ) is rwg  - closed in (X,τ).
    R^* - Continuous function [4] if the inverse image of every closed set V in (Y,σ) is R^*   - closed in (X,τ).
    g^* p – Continuous function [38] if the inverse image of every closed set V in (Y,σ) is g^* p  - closed in (X,τ). 
    g^* s^* - Continuous function [1] if the inverse image of every closed set V in (Y,σ) is g^* s^*   - closed in (X,τ).
     strongly g^* - Continuous function [32] if the inverse image of every closed set V in (Y,σ) is strongly g^*  - closed in (X,τ).
    g^* s - Continuous function [43] if the inverse image of every closed set V in (Y,σ) is g^* s  - closed in (X,τ).
    wg - Continuous function [26] if the inverse image of every closed set V in (Y,σ) is wg  - closed in (X,τ).
    pg - Continuous function [18] if the inverse image of every closed set V in (Y,σ) is pg  - closed in (X,τ).
    ws - Continuous function [12] if the inverse image of every closed set V in (Y,σ) is ws  - closed in (X,τ).
    gp - Continuous function [13] if the inverse image of every closed set V in (Y,σ) is gp  - closed in (X,τ).
    αg - Continuous function [24] if the inverse image of every closed set V in (Y,σ) is αg  - closed in (X,τ).
    gs - Continuous function [36] if the inverse image of every closed set V in (Y,σ) is gs - closed in (X,τ).
    gsp - Continuous function [33] if the inverse image of every closed set V in (Y,σ) is gsp  - closed in (X,τ).
    sg - Continuous function [24] if the inverse image of every closed set V in (Y,σ) is sg  - closed in (X,τ).
    gα - Continuous function [21] if the inverse image of every closed set V in (Y,σ) is gα  - closed in (X,τ).
    α - Continuous function [42] if the inverse image of every closed set V in (Y,σ) is α  - closed in (X,τ).
    b - Continuous function [11] if the inverse image of every closed set V in (Y,σ) is b - closed in (X,τ).
    semi - Continuous function [8] if the inverse image of every closed set V in (Y,σ) is semi - closed in (X,τ).
    gb - Continuous function [40] if the inverse image of every closed set V in (Y,σ) is gb  - closed in (X,τ).
    〖gb〗^* - Continuous function [45] if the inverse image of every closed set V in (Y,σ) is 〖gb〗^*  - closed in (X,τ).
     g^(∧*) s - Continuous function [1] if the inverse image of every closed set V in (Y,σ) g^(∧*) s - closed in (X,τ). 
    On Generalization of Generalized star generalized continuous function in Topological Spaces
Definition:1.1
A function f:(X,τ)→(Y,σ) is called 〖(gg)〗^* g- continuous if the inverse image of 
every closed set V in (Y,σ) is 〖(gg)〗^* g - closed set in (X,τ).
Proposition:1.2
Every continuous function is 〖(gg)〗^* g - continuous function.
Proof:
Let f:(X,τ)→(Y,σ) be a continuous function and let V be a closed set in (Y,σ). 
Since f is continuous, then by Definition 3[i], f^(-1) (V) is closed in (X,τ). By Proposition 3.3[20], every closed set is 〖(gg)〗^* g - closed. So f^(-1) (V) is 〖(gg)〗^* g - closed in (X,τ). By Definition 1.1, f is 〖(gg)〗^* g- continuous function.
Remark: The converse part of the above proposition need not be true as shown in the following example.
Example:1.3
Let X=Y={a,b,c,d}, τ={φ,{a},{b},{a,b},X}, σ={φ,{c,d},{a,c},Y} 
and τ_c={φ,{b,c,d},{a,c,d},{c,d},X}. A function f:(X,τ)→(Y,σ) is defined by f(a)=c,f(b)=a,f(c)=b,f(d)=d. f is 〖(gg)〗^* g – continuous function but not continuous function, because for closed set V = {c,d} in (Y,σ), f^(-1) (V)={a,d}, {a,d} is not closed in (X,τ).
Proposition:1.4
Let f:(X,τ)→(Y,σ)  be a function
    Every regular continuous function is 〖(gg)〗^* g- continuous function.
    Every π - continuous function is 〖(gg)〗^* g- continuous function.
    Every g^*  - continuous function is 〖(gg)〗^* g- continuous function.
    Every 〖(gαr)〗^(**)  - continuous function is 〖(gg)〗^* g- continuous function.
    Every 〖(gr)〗^*  - continuous function is 〖(gg)〗^* g- continuous function.
    Every 〖(gg)〗^*  - continuous function is 〖(gg)〗^* g- continuous function.
    Every rβ- continuous function is 〖(gg)〗^* g- continuous function.
    Every 〖gp〗^*  - continuous function is 〖(gg)〗^* g- continuous function.
    Every 〖(gsp)〗^*  - continuous function is 〖(gg)〗^* g- continuous function.
    Every g^#  - continuous function is 〖(gg)〗^* g- continuous function.
    Every 〖(gs)〗^*  - continuous function is 〖(gg)〗^* g- continuous function.
    Every g^* sr - continuous function is 〖(gg)〗^* g- continuous function.
    Every r^* g^*  - continuous function is 〖(gg)〗^* g- continuous function.
Every 〖(g〗^* 〖p)〗^*  - continuous function is 〖(gg)〗^* g- continuous function.
Proof: 
    Let f:(X,τ)→(Y,σ)  be a regular continuous function and let V be a
closed set in (X,τ). Since f is regular continuous function, then by Definition 3[ii], f^(-1) (V) is regular closed in (X,τ). By Proposition 3.3[20], every regular closed set is 〖(gg)〗^* g - closed. Then f^(-1) (V) is 〖(gg)〗^* g – closed in (X,τ). By Definition 1.1, f is 〖(gg)〗^* g- continuous function.
    Let f:(X,τ)→(Y,σ) be a π - continuous function and let V be a
closed set in (X,τ). Since f is  π - continuous function, then by Definition 3[iii], f^(-1) (V) is a π - closed in (X,τ). By Proposition 3.3[20], every  π - closed set is 〖(gg)〗^* g - closed. Then f^(-1) (V) is 〖(gg)〗^* g – closed in (X,τ). By Definition 1.1, f is 〖(gg)〗^* g- continuous function.
    Let f:(X,τ)→(Y,σ) be a g^*- continuous function and let V be a
closed set in (Y,σ). Since f is g^*- continuous, then by Definition 3[iv], f^(-1) (V) is g^*- closed in (X,τ). By Proposition 3.3[20], every g^*- closed set is 〖(gg)〗^* g - closed. So f^(-1) (V) is 〖(gg)〗^* g - closed in (X,τ). By Definition 1.1, f is 〖(gg)〗^* g- continuous function.
    Let f:(X,τ)→(Y,σ) be a 〖(gαr)〗^(**)- continuous function and let V 
be a closed set in (X,σ). Since f is 〖(gαr)〗^(**)  - continuous, then by Definition 3[v], f^(-1) (V) is 〖(gαr)〗^(**)- closed in (X,τ). By Proposition 3.3[20], every 〖(gαr)〗^(**)- closed set is 〖(gg)〗^* g - closed. So f^(-1) (V) is 〖(gg)〗^* g - closed in (X,τ). By Definition 1.1, f is 〖(gg)〗^* g- continuous function.
    Let f:(X,τ)→(Y,σ) be a 〖(gr)〗^*- continuous function and let V be a 
closed set in (Y,σ). Since f is 〖(gr)〗^*- continuous, then by Definition 3[vi], f^(-1) (V) is 〖(gr)〗^*- closed in (X,τ). By Proposition 3.3[20], every 〖(gr)〗^*- closed set is 〖(gg)〗^* g - closed. So f^(-1) (V) is 〖(gg)〗^* g - closed in (X,τ). By Definition 1.1, f is 〖(gg)〗^* g- continuous function.
    Let f:(X,τ)→(Y,σ) be a 〖(gg)〗^*- continuous function and let V be a 
closed set in (Y,σ). Since f is 〖(gg)〗^*- continuous, then by Definition 3[vii], f^(-1) (V) is 〖(gg)〗^*- closed in (X,τ). By Proposition 3.3[20], every 〖(gg)〗^*- closed set is 〖(gg)〗^* g - closed. So f^(-1) (V) is 〖(gg)〗^* g - closed in (X,τ). By Definition 1.1, f is 〖(gg)〗^* g- continuous function.
    Let f:(X,τ)→(Y,σ) be a rβ - continuous function and let V be a 
closed set in (Y,σ). Since f is rβ – continuous, then by Definition 3[viii], f^(-1) (V) is rβ- closed in (X,τ). By Proposition 3.3[20], every rβ - closed set is 〖(gg)〗^* g - closed. So f^(-1) (V) is 〖(gg)〗^* g - closed in (X,τ). By Definition 1.1, f is 〖(gg)〗^* g- continuous function.
    Let f:(X,τ)→(Y,σ) be a 〖gp〗^*- continuous function and let V be a 
closed set in (Y,σ). Since f is 〖gp〗^*- continuous, then by Definition 3[ix], f^(-1) (V) is 〖gp〗^* - closed in (X,τ). By Proposition 3.3[20], every 〖gp〗^*- closed set is 〖(gg)〗^* g - closed. So f^(-1) (V) is 〖(gg)〗^* g - closed in (X,τ). By Definition 1.1, f is 〖(gg)〗^* g- continuous function.
    Let f:(X,τ)→(Y,σ) be a 〖(gsp)〗^*- continuous function and let V be a 
closed set in (Y,σ). Since f is 〖(gsp)〗^*- continuous, then by Definition 3[x], f^(-1) (V) is 〖(gsp)〗^*- closed in (X,τ). By Proposition 3.3[20], every 〖(gsp)〗^*- closed set is 〖(gg)〗^* g - closed. So f^(-1) (V) is 〖(gg)〗^* g - closed in (X,τ). By definition 1.1, f is 〖(gg)〗^* g- continuous function.
    Let f:(X,τ)→(Y,σ) be a g^#- continuous function and let V be a 
closed set in (Y,σ). Since f is g^#- continuous, then by Definition 3[xi],  f^(-1) (V) is g^#- closed in (X,τ). By Proposition 3.3[20], every g^#- closed set is 〖(gg)〗^* g - closed. So f^(-1) (V) is 〖(gg)〗^* g - closed in (X,τ). By Definition 1.1, f is 〖(gg)〗^* g- continuous function.
    Let f:(X,τ)→(Y,σ) be a 〖(gs)〗^*- continuous function and let V be a 
closed set in (Y,σ). Since f is 〖(gs)〗^*  - continuous, then by Definition 3[xii], f^(-1) (V) is 〖(gs)〗^*  - closed in (X,τ). By Proposition 3.3[20], every 〖(gs)〗^*  - closed set is 〖(gg)〗^* g - closed. So f^(-1) (V) is 〖(gg)〗^* g - closed in (X,τ). By Definition 1.1, f is 〖(gg)〗^* g- continuous function.
    Let f:(X,τ)→(Y,σ) be a g^* sr - continuous function and let V be a 
closed set in (Y,σ). Since f is g^* sr – continuous, then by Definition 3[xiii], f^(-1) (V) is g^* sr - closed in (X,τ). By Proposition 3.3[20], every g^* sr - closed set is 〖(gg)〗^* g - closed. So f^(-1) (V) is 〖(gg)〗^* g - closed in (X,τ). By Definition 1.1, f is 〖(gg)〗^* g- continuous function.
    Let f:(X,τ)→(Y,σ) be a r^* g^*- continuous function and let V be a 
closed set in (Y,σ). Since f is r^* g^*- continuous, then by Definition 3[xiv], f^(-1) (V) is r^* g^*- closed in (X,τ). By proposition 3.3[20], every r^* g^*- closed set is 〖(gg)〗^* g - closed. So f^(-1) (V) is 〖(gg)〗^* g - closed in (X,τ). By Definition 1.1, f is 〖(gg)〗^* g- continuous function.
    Let f:(X,τ)→(Y,σ) be a 〖(g〗^* 〖p)〗^*- continuous function and let V be a 
closed set in (Y,σ). Since f is 〖(g〗^* 〖p)〗^*- continuous then by Definition 3[xv], f^(-1) (V) is 〖(g〗^* 〖p)〗^*- closed in (X,τ). By proposition 3.3[20], every 〖(g〗^* 〖p)〗^*- closed set is 〖(gg)〗^* g - closed. So f^(-1) (V) is 〖(gg)〗^* g - closed in (X,τ). By Definition 1.1, f is 〖(gg)〗^* g- continuous function.
Remark: The converse of the above proposition need not be true as shown in the following examples.
Example:1.5
Let X=Y={a,b,c,d}, τ={φ,{c},{d},{c,d},{a,c,d},X}, σ={φ,{d},{a,c},Y}
and τ_c={φ,{a,b,d},{a,b,c},{a,b},{b},X}. A function f:(X,τ)→(Y,σ) is defined by f(a)=a,f(b)=d,f(c)=b,f(d)=c. f is 〖(gg)〗^* g – continuous function but not regular continuous function, because for closed set V = {a,b,c} in (Y,σ), f^(-1) (V)={a,c,d}, {a,c,d} is not regular closed in (X,τ).
Example:1.6
Let X=Y={a,b,c,d}, τ={φ,{a},{b},{a,b},X}, σ={φ,{c,d},{a,c},Y} 
and τ_c={φ,{b,c,d},{a,c,d},{c,d},X}. An identity function f:(X,τ)→(Y,σ) is defined by f(a)=a,f(b)=b,f(c)=c,f(d)=d. By f is 〖(gg)〗^* g – continuous function but not π and g^*- continuous function, because for closed set V = {a,d} in (Y,σ), f^(-1) (V)={a,d}, {a,d} is not π and g^*- closed in (X,τ).
Example:1.7
Let X=Y={a,b,c,d}, τ={φ,{c},{d},{c,d},{a,c,d},X},
σ={φ,{a},{a,b},{b,c,d},Y} and τ_c={φ,{a,b,d},{a,b,c},{a,b},{b},X}. 
A function f:(X,τ)→(Y,σ) is defined by f(a)=b,f(b)=d,f(c)=b,f(d)=c. f is 〖(gg)〗^* g – continuous function but not 〖(gαr)〗^(**)- continuous function, because for closed set V = {a,b,c} in (Y,σ), f^(-1) (V)={b,c,d}, {b,c,d} is not 〖(gαr)〗^(**)- closed in (X,τ).
Example:1.8
Let X=Y={a,b,c}, τ={φ,{b},{a,c},X}, σ={φ,{b},Y}, τ_c={φ,{b},{b,c},X}. 
A function f:(X,τ)→(Y,σ) is defined by f(a)=b,f(b)=a,f(c)=b,f(c)=c. By f is 〖(gg)〗^* g – continuous function but not 〖(gr)〗^* and r^* g^*- continuous function, because for closed set V = {b} in (Y,σ), f^(-1) (V)={a}, {a} is not 〖(gr)〗^* and r^* g^*- closed in (X,τ).
Example:1.9
Let X=Y={a,b,c,d}, τ={φ,{c},{d},{c,d},{a,c,d},X}, σ={φ,{a},{a,b},Y}
and τ_c={φ,{a,b,d},{a,b,c},{a,b},{b},X}. A function f:(X,τ)→(Y,σ) is defined by f(a)=c,f(b)=d,f(c)=a,f(d)=b. f is 〖(gg)〗^* g – continuous function but not 〖(gg)〗^*- continuous function, because for closed set V = {a,b} in (Y,σ), f^(-1) (V)={c,d}, {c,d} is not 〖(gg)〗^*- closed in (X,τ).
Example:1.10
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{a,b},{b,c},{b,c,d},X},
σ={φ,{a},{b},Y} and τ_c={φ,{a,b,d},{a,b,c},{a,b},{b},X}. A function f:(X,τ)→(Y,σ) is defined by f(a)=b,f(b)=d,f(c)=b,f(d)=c. f is 〖(gg)〗^* g – continuous function but not rβ- continuous function, because for closed set V = {b} in (Y,σ), f^(-1) (V)={a}, {a} is not rβ- closed in (X,τ).
Example:1.11
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{a,b},{b,c},{b,c,d},X},
σ={φ,{a},{d},{c,d},Y}, τ_c={φ,{a,b,d},{a,b,c},{a,b},{b},X}. A function f:(X,τ)→(Y,σ) is defined by f(a)=c,f(b)=d,f(c)=b,f(d)=a. f is (gg)^* g - continuous function but not 〖gs〗^*, 〖gp〗^*,g^* sr and (gsp)^*- continuous function, because for closed set V = {c,d} in (Y,σ), f^(-1) (V)={a,b}, {a,b} is not 〖gs〗^*, 〖gp〗^*,g^* sr and 〖(gsp)〗^*- closed in (X,τ).
Example:1.12
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{b,c},{b,c,d},X},
σ={φ,{a},{c,d},{a,c,d},Y}, τ_c={φ,{a,c,d},{a,b,d},{a,d},{a},X}. A function f:(X,τ)→(Y,σ) is defined by f(a)=a,f(b)=d,f(c)=c,f(d)=b. f is (gg)^* g - continuous function but not g^⋕ and (g^* p)^*- continuous function, because for closed set V = {c,d} in (Y,σ), f^(-1) (V)={a,b}, {a,b} is not g^⋕ and 〖(g^* p)〗^*- closed in (X,τ).
Proposition:1.13
Let f:(X,τ)→(Y,σ) be a function
    Every 〖(gg)〗^* g - continuous function is r^∧ g - continuous.
     Every 〖(gg)〗^* g - continuous function is rwg - continuous.
Proof: 
    Let f:(X,τ)→(Y,σ) be a (gg)^* g- continuous function and let V be a 
closed set in (Y,σ). Since f is (gg)^* g – continuous, then by Definition 3[xvi], f^(-1) (V) is 〖(gg)〗^* g - closed in (X,τ). By Proposition 3.3[20], every (〖gg)〗^* g - closed set is r^∧ g - closed. So f^(-1) (V) is r^∧ g - closed in (X,τ). By Definition 1.1, f is r^∧ g -continuous function.
    Let f:(X,τ)→(Y,σ) be a (gg)^* g- continuous function and let V be a 
closed set in (Y,σ). Since f is (gg)^* g – continuous, then by Definition 3[xvii], f^(-1) (V) is 〖(gg)〗^* g - closed in (X,τ). By Proposition 3.3[20], every (〖gg)〗^* g - closed set is rwg - closed. So f^(-1) (V) is rwg - closed in (X,τ). By Definition 1.1, f is rwg -continuous function.
Remark: The converse part of the proposition need not be true as shown in the following examples.
Example:1.14
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{b,c},{b,c,d},X},
σ={φ,{b},{a,b},{a,b,c},{a,c,d},Y}, τ_c={φ,{a,c,d},{a,b,d},{a,d},{a},X}, 
A function f:(X,τ)→(Y,σ) is defined by f(a)=d,f(b)=b,f(c)=a,f(d)=c. f is r^∧ g - continuous function but not (gg)^* g - continuous function, because for closed set V = {a,b} in (Y,σ), f^(-1) (V)={b,c}, {b,c} is not 〖(gg)〗^* g - closed in (X,τ).
Example:1.15
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{b,c},{b,c,d},X},
σ={φ,{a},{a,b},{a,b,c},Y}, τ_c={φ,{a,c,d},{a,b,d},{a,d},{a},X}. A function f:(X,τ)→(Y,σ) is defined by f(a)=b,f(b)=c,f(c)=d,f(d)=a. f is r^∧ g - continuous function but not (gg)^* g - continuous function, because for closed set V = {a} in (Y,σ), f^(-1) (V)={d}, {d} is not 〖(gg)〗^* g - closed in (X,τ).
    Independent set of 〖(gg)〗^* g - continuous functions with other continuous function
The following example shows that the concept of 〖(gg)〗^* g - continuous function is independent from R^* - Continuous, g^* p - Continuous, g^* s^* - Continuous, strongly g^* - Continuous, g^* s - Continuous, wg - Continuous, pg - Continuous, ws - Continuous, gp - Continuous, αg - Continuous, gs - Continuous, gsp - Continuous, sg - Continuous, gα - Continuous, α - Continuous, b - Continuous, semi - Continuous, gb - Continuous, 〖gb〗^* - Continuous, g^(∧*) s - Continuous.
Example:2.1
Let X=Y={a,b,c,d}, τ={φ,{a},{c},{a,c},{c,d},{a,c,d},X}, 
σ={φ,{b},{a,b},{a,b,c},{a,c,d},Y}, τ_c={φ,{b,c,d},{a,b,d},{b,d},{a,b},{b},X}. 
A function f:(X,τ)→(Y,σ) is defined by f(a)=a,f(b)=c,f(c)=b,f(d)=d. 
f is R^* - continuous function but not (gg)^* g - continuous function, because for closed set V = {a,b} in (Y,σ), f^(-1) (V)={a,c}, {a,c} is not 〖(gg)〗^* g - closed in (X,τ).
A function f:(X,τ)→(Y,σ) is defined by f(a)=d,f(b)=b,f(c)=a,f(d)=c. 
f is 〖(gg)〗^* g - continuous function but not R^*  - continuous function, because for closed set V = {a,b} in (Y,σ), f^(-1) (V)={b,c}, {b,c} is not R^*  - closed in (X,τ).
Example:2.2
Let X=Y={a,b,c,d}, τ={φ,{c},{d},{c,d},{a,c,d},X}, 
σ={φ,{d},{a,d},{a,c,d},Y}, τ_c={φ,{a,b,d},{a,b,c},{a,b},{b},X}. 
An identity function f:(X,τ)→(Y,σ) is defined by f(a)=a,f(b)=b,f(c)=
c,f(d)=d. f is 〖(gg)〗^* g - continuous function but not g^* p - continuous 
function, because for closed set V = {a,c,d} in (Y,σ), f^(-1) (V)={a,c,d}. {a,c,d} is not g^* p - closed in (X,τ).
A function f:(X,τ)→(Y,σ) is defined by f(a)=d,f(b)=b,f(c)=c,f(d)=a. 
f is g^* p - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {d} in (Y,σ), f^(-1) (V)={a}, {a} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.3
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{b,c},{b,c,d},X},
σ={φ,{a},{d},{a,b,c},Y}, τ_c={φ,{a,c,d},{a,b,d},{a,d},{a},X}.
A function f:(X,τ)→(Y,σ) is defined by f(a)=d,f(b)=a,f(c)=b,f(d)=c. 
f is 〖(gg)〗^* g - continuous function but not g^* s^*  - continuous function, because for closed set V = {a,b,c} in (Y,σ), f^(-1) (V)={b,c,d}. {b,c,d} is not g^* s^*  - closed in (X,τ).
A function f:(X,τ)→(Y,σ) is defined by f(a)=b,f(b)=d,f(c)=c,f(d)=a. 
f is g^* s^* - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {d} in (Y,σ), f^(-1) (V)={b}, {b} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.4
Let X=Y={a,b,c,d}, τ={φ,{c},{d},{c,d},{a,c,d},X},
σ={φ,{c},{a,b,c},Y}, τ_c={φ,{a,b,d},{a,b,c},{a,b},{b},X}. 
A function f:(X,τ)→(Y,σ) is defined by f(a)=a,f(b)=d,f(c)=c,f(d)=b. 
f is 〖(gg)〗^* g - continuous function but not strongly g^*  - continuous function, because for closed set V = {a,b,c} in (Y,σ), f^(-1) (V)={a,c,d}. {a,c,d} is not strongly g^*  - closed in (X,τ).
A function f:(X,τ)→(Y,σ) is defined by f(a)=c,f(b)=d,f(c)=b,f(d)=a. 
f is strongly g^* - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {c} in (Y,σ), f^(-1) (V)={a}, {a} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.5
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{b,c},{b,c,d},X},
σ={φ,{b},{a,d},{a,b,d},Y}, τ_c={φ,{a,c,d},{a,b,d},{a,d},{a},X}.
A function f:(X,τ)→(Y,σ) is defined by f(a)=a,f(b)=b,f(c)=d,f(d)=c. 
f is 〖(gg)〗^* g - continuous function but not g^* s - continuous function, because for closed set V = {a,b,d} in (Y,σ), f^(-1) (V)={a,b,c}. {a,b,c} is not g^* s - closed in (X,τ).
A function f:(X,τ)→(Y,σ) is defined by f(a)=c,f(b)=d,f(c)=b,f(d)=a. 
f is g^* s - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {b} in (Y,σ), f^(-1) (V)={c}, {c} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.6
Let X=Y={a,b,c,d}, τ={φ,{c},{d},{c,d},{a,c,d},X},
σ={φ,{b}{a,c,d),Y}, τ_c={φ,{a,b,d},{a,b,c},{a,b},{b},X}.
A function f:(X,τ)→(Y,σ) is defined by f(a)=b,f(b)=a,f(c)=c,f(d)=d. 
f is 〖(gg)〗^* g - continuous function but not wg - continuous function, because for closed set V = {a,c,d} in (Y,σ), f^(-1) (V)={b,c,d}. {b,c,d} is not wg - closed in (X,τ).
A function f:(X,τ)→(Y,σ) is defined by f(a)=b,f(b)=d,f(c)=c,f(d)=a. 
f is wg - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {b} in (Y,σ), f^(-1) (V)={a}, {a} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.7
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{a,b},{b,c},{b,c,d},X},
σ={φ,{d},{a,d},Y}, τ_c={φ,{a,c,d},{a,b,d},{c,d}{a,d},{a},X}. 
A function f:(X,τ)→(Y,σ) is defined by f(a)=a,f(b)=d,f(c)=b,f(d)=c. 
f is 〖(gg)〗^* g - continuous function but not pg - continuous function, because for closed set V = {a,d} in (Y,σ), f^(-1) (V)={a,b}. {a,b} is not pg - closed in (X,τ).
Let X=Y={a,b,c,d}, τ={φ,{c},{d},{c,d},{a,c,d},X},
σ={φ,{d},{a,d},Y}, τ_c={φ,{a,c,d},{a,b,c},{a,b},{b},X}.
A function f:(X,τ)→(Y,σ) is defined by f(a)=c,f(b)=b,f(c)=b,f(d)=d. 
f is pg - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {d} in (Y,σ), f^(-1) (V)={d}, {d} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.8
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{b,c},{b,c,d},X},
σ={φ,{a}{b,c,d),Y}, τ_c={φ,{a,b,d},{a,b,c},{a,b},{b},X}. 
An identity function f:(X,τ)→(Y,σ) is defined by f(a)=a,f(b)=b,f(c)=
c,f(d)=d. f is 〖(gg)〗^* g - continuous function but not ws - continuous function, because for closed set V = {b,c,d} in (Y,σ), f^(-1) (V)={b,c,d}. {b,c,d} is not ws - closed in (X,τ).
A function f:(X,τ)→(Y,σ) is defined by f(a)=c,f(b)=b,f(c)=a,f(d)=d. 
f is ws - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {a} in (Y,σ), f^(-1) (V)={c}, {c} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.9
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{a,b},{b,c},{b,c,d},X},
σ={φ,{b},Y}, τ_c={φ,{a,c,d},{a,b,d},{c,d}{a,d},{a},X}. 
A function f:(X,τ)→(Y,σ) is defined by f(a)=d,f(b)=a,f(c)=b,f(d)=c. 
f is 〖(gg)〗^* g - continuous function but not gp - continuous function, because for closed set V = {b} in (Y,σ), f^(-1) (V)={c}. {c} is not gp - closed in (X,τ).
Let X=Y={a,b,c,d}, τ={φ,{c},{d},{c,d},{a,c,d},X},
σ={φ,{b},{a,d},Y}, τ_c={φ,{a,c,d},{a,b,c},{a,b},{b},X}. 
A function f:(X,τ)→(Y,σ) is defined by f(a)=b,f(b)=d,f(c)=a,f(d)=c. 
f is gp - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {b} in (Y,σ), f^(-1) (V)={a}, {a} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.10
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{a,b},{b,c},{b,c,d},X},
σ={φ,{b},{d},{b,d},{b,c,d},Y}, τ_c={φ,{a,c,d},{a,b,d},{c,d}{a,d},{a},X}.
A function f:(X,τ)→(Y,σ) is defined by f(a)=d,f(b)=b,f(c)=c,f(d)=a. 
f is 〖(gg)〗^* g - continuous function but not αg - continuous function, because for closed set V = {b,d} in (Y,σ), f^(-1) (V)={a,b}. {a,b} is not αg - closed in (X,τ).
Let X=Y={a,b,c,d}, τ={φ,{c},{d},{c,d},{a,c,d},X},
σ={φ,{b},{b,d},Y}, τ_c={φ,{a,c,d},{a,b,c},{a,b},{b},X}. 
A function f:(X,τ)→(Y,σ) is defined by f(a)=b,f(b)=a,f(c)=c,f(d)=d. 
f is αg - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {b} in (Y,σ), f^(-1) (V)={a}, {a} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.11
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{a,b},{b,c},{b,c,d},X},
σ={φ,{b},{d},{b,d},Y}, τ_c={φ,{a,c,d},{a,b,d},{c,d}{a,d},{a},X}.
A function f:(X,τ)→(Y,σ) is defined by f(a)=d,f(b)=b,f(c)=a,f(d)=c. 
f is 〖(gg)〗^* g - continuous function but not gs - continuous function, because for closed set V = {b} in (Y,σ), f^(-1) (V)={b}. {b} is not gs - closed in (X,τ).
A function f:(X,τ)→(Y,σ) is defined by f(a)=b,f(b)=d,f(c)=c,f(d)=a. 
f is gs - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {b,d} in (Y,σ), f^(-1) (V)={a,b}, {a,b} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.12
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{a,b},{b,c},{b,c,d},X},
σ={φ,{b},{b,c,d},Y}, τ_c={φ,{a,c,d},{a,b,d},{c,d}{a,d},{a},X}. 
A function f:(X,τ)→(Y,σ) is defined by f(a)=b,f(b)=a,f(c)=c,f(d)=d. 
f is 〖(gg)〗^* g - continuous function but not gsp - continuous function, because for closed set V = {b} in (Y,σ), f^(-1) (V)={a}. {a} is not gsp - closed in (X,τ).
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{b,c},{b,c,d},X},
σ={φ,{b},{b,d},Y}, τ_c={φ,{a,c,d},{a,b,d},{a,d},{a},X}.
A function f:(X,τ)→(Y,σ) is defined by f(a)=c,f(b)=b,f(c)=a,f(d)=d. 
f is αg - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {b} in (Y,σ), f^(-1) (V)={b}, {b} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.13
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{b,c},{b,c,d},X},
σ={φ,{a},{a,b,c},Y}, τ_c={φ,{a,c,d},{a,b,d},{a,d},{a},X}.
An identity function f:(X,τ)→(Y,σ) is defined by f(a)=a,f(b)=b,f(c)=
c,f(d)=d. f is 〖(gg)〗^* g - continuous function but not sg- continuous function, because for closed set V = {a,b,c} in (Y,σ), f^(-1) (V)={a,b,c}. {a,b,c} is not sg - closed in (X,τ).
A function f:(X,τ)→(Y,σ) is defined by f(a)=c,f(b)=d,f(c)=a,f(d)=b. 
f is sg - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {a} in (Y,σ), f^(-1) (V)={c}, {c} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.14
Let X=Y={a,b,c,d}, τ={φ,{a},{c},{a,c},{c,d},{a,c,d},X},
σ={φ,{b},{b,c}{b,c,d),Y}, τ_c={φ,{b,c,d},{a,b,d},{b,d},{a,b},{b},X}.
A function f:(X,τ)→(Y,σ) is defined by f(a)=b,f(b)=c,f(c)=a,f(d)=d.
f is 〖(gg)〗^* g - continuous function but not gα - continuous function, because for closed set V = {b,c} in (Y,σ), f^(-1) (V)={a,b}. {a,b} is not gα - closed in (X,τ).
A function f:(X,τ)→(Y,σ) is defined by f(a)=c,f(b)=d,f(c)=a,f(d)=b. 
f is gα - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {b} in (Y,σ), f^(-1) (V)={d}, {d} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.15
Let X=Y={a,b,c}, τ={φ,{b},{a,c},X},
σ={φ,{a},{a,b},Y}, τ_c={φ,{a,c},{b},X}, 
A function f:(X,τ)→(Y,σ) is defined by f(a)=c,f(b)=b,f(c)=a. 
f is 〖(gg)〗^* g - continuous function but not α - continuous function, because for closed set V = {a,b} in (Y,σ), f^(-1) (V)={b,c}. {b,c} is not α - closed in (X,τ).
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{b,c},{b,c,d},X},
σ={φ,{a},{b,d},Y}, τ_c={φ,{a,c,d},{a,b,d},{a,d},{a},X}.
A function f:(X,τ)→(Y,σ) is defined by f(a)=d,f(b)=c,f(c)=b,f(d)=a. 

f is α - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {a} in (Y,σ), f^(-1) (V)={d}, {d} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.16
Let X=Y={a,b,c,d}, τ={φ,{c},{d},{c,d},{a,c,d},X},
σ={φ,{b},{a,b},{b,c,d},Y}, τ_c={φ,{a,b,d},{a,b,c},{a,b},{b},X}.
A function f:(X,τ)→(Y,σ) is defined by f(a)=b,f(b)=a,f(c)=c,f(d)=d. 
f is 〖(gg)〗^* g - continuous function but not b - continuous function, because for closed set V = {b,c,d} in (Y,σ), f^(-1) (V)={a,c,d}. {b,c,d} is not b - closed in (X,τ).
f is b - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {b} in (Y,σ), f^(-1) (V)={a}, {a} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.17
Let X=Y={a,b,c,d}, τ={φ,{a},{c},{a,c},{c,d},{a,c,d},X},
σ={φ,{a},{b,c},Y}, τ_c={φ,{b,c,d},{a,b,d},{b,d},{a,b},{b},X}.
A function f:(X,τ)→(Y,σ) is defined by f(a)=a,f(b)=c,f(c)=b,f(d)=d. 
f is 〖(gg)〗^* g - continuous function but not semi continuous function, because for closed set V = {b,c} in (Y,σ), f^(-1) (V)={b,c}. {b,c} is not semi closed in (X,τ).
f is semi continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {a} in (Y,σ), f^(-1) (V)={a}, {a} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.18
Let X=Y={a,b,c,d}, τ={φ,{c},{d},{c,d},{a,c,d},X},
σ={φ,{b}{b,c,d),Y}, τ_c={φ,{a,b,d},{a,b,c},{a,b},{b},X}.
A function f:(X,τ)→(Y,σ) is defined by f(a)=c,f(b)=a,f(c)=b,f(d)=d. 
f is 〖(gg)〗^* g - continuous function but not 〖gb〗^*  and gb continuous function, because for closed set V = {b,c,d} in (Y,σ), f^(-1) (V)={a,c,d}. {a,c,d} is not 〖gb〗^*  and gb - closed in (X,τ).
A function f:(X,τ)→(Y,σ) is defined by f(a)=b,f(b)=d,f(c)=c,f(d)=a. 
f is 〖gb〗^*  and gb - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {b} in (Y,σ), f^(-1) (V)={c}, {c} is not 〖(gg)〗^* g - closed in (X,τ).
Example:2.19
Let X=Y={a,b,c,d}, τ={φ,{b},{c},{b,c},{b,c,d},X},
σ={φ,{a},{a,b,d},Y}, τ_c={φ,{a,c,d},{a,b,d},{a,d},{a},X}.
A function f:(X,τ)→(Y,σ) is defined by f(a)=c,f(b)=a,f(c)=b,f(d)=d. 
f is 〖(gg)〗^* g - continuous function but not g^(∧*) s - continuous function, because for closed set V = {a,b,d} in (Y,σ), f^(-1) (V)={b,c,d}. {b,c,d} is not g^(∧*) s  - closed in (X,τ).
A function f:(X,τ)→(Y,σ) is defined by f(a)=c,f(b)=a,f(c)=d,f(d)=b. 
f is g^(∧*) s - continuous function but not 〖(gg)〗^* g - continuous function, because for closed set V = {a} in (Y,σ), f^(-1) (V)={b}, {b} is not 〖(gg)〗^* g - closed in (X,τ).
Remark: From the above discussion and known results for the relationship between 〖(gg)〗^* g - continuous function and other existing continuous function are established in figure 1.

 
Figure 1
In the Figure 1, A→B means the set A implies B, but not conversely. 
In the Figure 2, A↮B means the set A and B are independent of each other.

Figure 2
 
    Aspects of 〖(gg)〗^* g - continuous function
Theorem:3.1 If  f:(X,τ)→(Y,σ) is 〖(gg)〗^* g - continuous function and 
g:(Y,σ)→(Z,η) is continuous function then (gof): (X,σ)⟶(Z,η) is 〖(gg)〗^* g – continuous function.
Proof:
Let us take U be any closed set in (Z,η). Since g is continuous function then by
Definition 3[i], g^(-1) (U) is closed in (Y,σ). Since f is 〖(gg)〗^* g - continuous function then by Definition 1.1, f^(-1) (g^(-1) (U)) is 〖(gg)〗^* g - closed set in (X,τ). Therefore (gof)^(-1) (U) is 〖(gg)〗^* g - closed set in (X,τ). Hence (gof) is 〖(gg)〗^* g -continuous function.
Remark:  Composition of two 〖(gg)〗^* g - continuous function need not be 〖(gg)〗^* g - continuous function.
Example:3.2
Let X=Y=Z={a,b,c,d}, τ={φ,{b},{c},{b,c},{b,c,d},X}
σ={φ,{a},{c},{a,c},{c,d},{a,c,d},Y}. A functions f:(X,τ)→(Y,σ) and g:(Y,σ)→(Z,η) are defined by f(a)=a,f(b)=b,f(c)=c,f(d)=d and g(a)=a,g(b)=b,g(c)=d,g(d)=c. f and g are 〖(gg)〗^* g - continuous function but not 〖(gof)〗^(-1)  is 〖(gg)〗^* g - continuous function, because for closed set V = {c,d} in (Y,σ),  〖(gof)〗^(-1) (V)={c,d}. {c,d} is not  〖(gg)〗^* g - closed in (X,τ).
Theorem:3.3
A function f:(X,τ)→(Y,σ). Then the following statements are equivalent.
    f is 〖(gg)〗^* g – continuous function.
     f^(-1) (U) is 〖(gg)〗^* g  - open set in (X,τ) for every open set U in (Y,σ).
     〖 f〗^(-1) (V) is 〖(gg)〗^* g  - closed set in (X,τ) for every closed set V in (Y,σ).
Proof:
(i)⇒(ii) Let f be a 〖(gg)〗^* g – continuous function and U be an open set in (Y,σ).Then U^c is closed set in (Y,σ). Since f is 〖(gg)〗^* g – continuous function, by Definition 1.1, f^(-1) 〖(U〗^c) is 〖(gg)〗^* g  - closed set in (X,τ). Therefore f^(-1) 〖(U〗^c)=X-f^(-1) (U) is 〖(gg)〗^* g  - closed set in (X,τ) and hence f^(-1) (U) is 〖(gg)〗^* g  - open set in (X,τ).
(ii)⇒(iii) Let us assume that f^(-1) (V) is 〖(gg)〗^* g  - open set in (X,τ) for every open set U in (Y,σ). Let V be a closed set in (Y,σ). Then V^c is open set in (Y,σ).
By assumption f^(-1) (V^c) is 〖(gg)〗^* g  - open set in (X,τ). Therefore f^(-1) (V^c )=X-f^(-1) (V) is 〖(gg)〗^* g  - open set in (X,τ) and hence f^(-1) (V) is 〖(gg)〗^* g  - closed set in (X,τ).
(ii)⇒(iii) Let f be a 〖(gg)〗^* g - continuous function and let V be a closed set in (Y,σ). Then V^c is open set in (Y,σ). Since f is 〖(gg)〗^* g – continuous function, by Definition 1.1, f^(-1) (V^c )=X-f^(-1) (V) is 〖(gg)〗^* g  - open set in (X,τ). Hence f^(-1) (V) is 〖(gg)〗^* g  - closed set in (X,τ).
(iii)⇒(i) Let us assume that f^(-1) (V) is 〖(gg)〗^* g  - closed set in (X,τ) for every closed set V in (Y,σ). Let W be an open set in (Y,σ), then W^c is closed set in (Y,σ). By assumption f^(-1) (W^c ) is 〖(gg)〗^* g  - closed set in (X,τ). Therefore f^(-1) (W^c )=X-f^(-1) (W) is 〖(gg)〗^* g  - closed set in (X,τ) and hence f is 〖(gg)〗^* g – continuous function.
Theorem:3.4 
Let (X,τ) be a topological space in which every singleton set is 〖(gg)〗^* - closed.
Then the function f:(X,τ)→(Y,σ) is 〖(gg)〗^* g - continuous if x∈gint〖(f〗^(-1) (V)) for every open sub set V of (Y,σ) containing f(x).
Proof:
           Assume that a function f:(X,τ)→(Y,σ) is 〖(gg)〗^* g – continuous function. 
To prove x∈gint〖(f〗^(-1) (V)). Let x∈X and V be an open set in (Y,σ) containing f(x). That is f(x)∈U. Since f is 〖(gg)〗^* g – continuous function, by Theorem 3.3 f^(-1) (V) is 〖(gg)〗^* g  - open set in (X,τ) and since {x} is 〖(gg)〗^* - closed, then x∈gint〖(f〗^(-1) (V)).
Conversely assume that x∈gint〖(f〗^(-1) (V))  for every open subset V of  (Y,σ) containing f(x). To prove f is 〖(gg)〗^* g – continuous function.
    Let V be an open set in (Y,σ). Suppose that G⊆f^(-1) (V) and G is 〖(gg)〗^* g - closed. Let x∈G⊆f^(-1) (V)
⇒x∈f^(-1) (V)
⇒f(x)∈V
By hypothesis, x∈gint〖(f〗^(-1) (V)) and from the assumption f^(-1) (V) is an 〖(gg)〗^* g  - open set in (X,τ). By Theorem 3.3, f is 〖(gg)〗^* g – continuous function.
Theorem:3.5
           If f is 〖(gg)〗^* g - continuous function then f〖((gg)〗^* g cl(A))⊆cl(f(A)) for every subset A of (X,τ).
Proof:
Let f:(X,τ)→(Y,σ) is 〖(gg)〗^* g - continuous function and let A⊆X then cl(f(A)) 
is closed in (Y,σ).
Since f(A)⊆cl(f(A)), A⊆f^(-1) (cl(f(A)))
We know that cl(f(A)) is closed in (Y,σ) and also f is 〖(gg)〗^* g – continuous function. 
By Definition 1.1, f^(-1) (cl(f(A))) is 〖(gg)〗^* g  - closed in (X,τ).
Let us assume that y∈f^(-1) 〖((gg)〗^* g-cl(A)). Then y=f(x), where x∈(gg)^* g-cl(A). Let G be an open set containing y=f(x). Since f is (gg)^* g – continuous function, by Theorem 3.3, f^(-1) (G) is (gg)^* g  - open set in (X,τ). By Theorem 5.8[20], f^(-1) (G)∩A≠φ. 
⇒〖f(f〗^(-1) (G)∩A)≠φ
⇒〖f(f〗^(-1) (G))∩f(A)≠φ.
〖f(f〗^(-1) (G))∈G
⇒G∩f(A)≠φ.
Therefore y∈cl(f(A)) because for any open set G containing y, G∩f(A)≠φ is a characterization of y being in the closure of f(A). Hence f((gg)^* g-cl(A))⊆cl(f(A)).
Theorem:3.6 
Let a function f:(X,τ)→(Y,σ) then the following statements are equivalent.
    For every point x∈(X,τ) and each open set V containing f(x) in (Y,σ), there is an 
(gg)^* g  - open set U in (X,τ) such that x∈U and f(U)⊆V.
    For each A⊆(X,τ),f((gg)^* g-cl(A))⊆cl f(A) .
    For each B⊆(X,τ),(gg)^* gcl(f^(-1) (B))⊆f^(-1) (cl(B)).

Proof:
 (i)⇒(ii) Suppose (i) is hold. Let us assume that y∈f((gg)^* g-cl(A)). Then there exists an element x∈(gg)^* g-cl(A) such that y=f(x). Let V be an open set containing y that is y∈V,f(x)∈V. Since x∈(gg)^* gcl(A), by Theorem 5.8[20], there exists an (gg)^* g  - open set U containing a point x such that U∩A≠φ.
⇒f(U∩A)≠φ
⇒f(U)∩f(A)≠φ
By hypothesis f(U)⊆V 
⇒f(U)∩f(A)⊆V∩f(A)≠φ
⇒ V∩f(A)≠φ
Thus y∈cl(f(A)), because for any open set V containing y, V∩f(A)≠φ is a characterization of y being in the closure of f(A).
Therefore f((gg)^* g-cl(A))⊆cl(f(A)).
(ii)⇒(i) Suppose (ii) is hold. Let V be an open set containing f(x) and let x∈(X,τ). And let A=f^(-1) (V^c ). Since f((gg)^* g-cl(A))⊆cl(f(A))
                                                   =cl((V^c ))                                           
                       =V^c
Therefore f((gg)^* gcl(A))⊆V^c
⇒(gg)^* gcl(A)⊆f^(-1) (V^c )=A
⇒(gg)^* gcl(A)⊆A
Also we know that A⊆(gg)^* gcl(A) that implies (gg)^* gcl(A)=A.
Since f(x)∈V,x∈f^(-1) (V)
This x∉A then x∉(gg)^* gcl(A) 
By Theorem 5.8[20], there exists (gg)^* g open set U containing x such that U∩A≠φ
That implies V⊆A^c⇒f(U)⊆f(A^c)⊆V. 
Hence f(U)⊆V.
(ii)⇒(iii) Suppose (ii) is hold. Let B⊆(Y,σ) and replacing A by f^(-1) (B) in (ii).
We get f((gg)^* gcl(f^(-1) (B)))⊆cl(f(f^(-1) (B)))=cl(B)
⇒f((gg)^* gcl (f^(-1) (B))⊆cl(B)
⇒(gg)^* gcl (f^(-1) (B))⊆f^(-1) (cl(B))
(iii)⇒(ii) Suppose (iii) is hold. Let A⊆(X,σ) and take f(A)=B in (iii)
We get (gg)^* gcl(f^(-1) (f(A)))⊆f^(-1) (cl(f(A)))
⇒(gg)^* gcl(A)⊆f^(-1) (cl(f(A)))
⇒f((gg)^* gcl(A))⊆cl(f(A)).

Theorem:3.7

If f:(X,τ)→(Y,σ) is closed and (gg)*g - continuous function and B is (gg)*g–

closed set of (y,σ) then f-1(B)  is (gg)*g- closed set in x,τ.

Proof:

Let B be (gg)*g- closed set of (Y,σ)and let f-1(B)⊆U, where U is an open set of

X,τ.  Since f is closed, there is an open set V such that B⊆V and f-1B⊆U. Since B is (gg)*g  - closed set gcl(B)⊆U , that implies f-1(gclB)⊆U . Since gclB⊆cl(B) ,

by assumption f is closed and gg*g  - continuous, f-1clB  is gg*g  - closed set in X,τ.

⇒gcl(f-1clB)⊆U

⇒gcl(f-1B)⊆U .

Therefore f-1B  is gg*g  - closed set in X,τ.

 Theorem:3.8

If f:(X,τ)→(Y,σ)  is  gg*g  - continuous function, closed and

g :Y,σ→(Z,η)  is (gg)*g  - continuous function then gof :(X,τ)→(Z,η)  is (gg)*g  – continuous function.

Proof:

           Let us take U be any closed set in (Z,η) . Then f-1(U)  is closed set in Y,σ.

Since g  is  (gg)*g  - continuous function and closed then g-1(U)  is closed in Y,σ  and also f is (gg)*g  – continuous function f-1(g-1U)  is gg*g  - closed set in X,τ.  Hence gof  is (gg)*g  – continuous function.

Theorem:3.9

In extremely disconnected space (X,τ) , If f:(X,τ)→(Y,σ)  is g  - continuous function and open then (gg)*g  - continuous function.

Proof:

Let f:(X,τ)→(Y,σ)   be a g-continuous function and let V be a

closed set in (X,τ) . Since f is g-continuous function and open, then f-1(V)  is g-closed in (X,τ) . By Theorem 5.4[20], f-1(V)  is (gg)*g  – closed, then f-1(V)  is (gg)*g  – closed in (X,τ) . By Definition 1.1, f is (gg)*g - continuous function.

CONCLUSION

The study of gg*g  – continuous function in topological spaces literary investigated. The concept of gg*g  – continuous function will be extended to strongly gg*g  – continuous function, perfectly gg*g  – continuous function, contra gg*g  – continuous function.  Also this study can be elaborated to bitopological spaces and fuzzy topological spaces.

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