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On  Slightly Delta Generalized Pre-Continuous Functions 

 

J.B.Toranagatti 

Department of  Mathematics, Karnatak University’s Karnatak Science College, Dharwad-580 001, Karnataka 

State,  India. e-mail: jagadeeshbt2000@gmail.com 

 

Article History: 

Received: 02-08-2024 

Revised: 11-09-2024 

Accepted: 20-09-2024    

Abstract:  

The purpose of this article is to introduce and investigate the properties of slightly  δgp-

continuous functions. Also, the relationships of slightly  δgp-continuous functions and 

graphs are investigated. 

Keywords- slight continuity, slight pre-continuity, slight gpr-continuity, slight δgp-continuity. 

 

1.Introduction and Preliminaries: 

 In 1982,Mashhour et al. [9] introduced preopen sets and pre-continuity in topology. R.C.Jain  [8]    

presented the idea of slight  continuity and studied  its fundamental characteristics. Balasubramanian 

et al. (2011) introduced the notion of slight gpr-continuity[2] as a generalization of slight  pre-

continuity[2].     J.B. Toranagatti [15,16] recently introduced the concepts of δgp-continuity and 

contra δgp-continuity in topological spaces. 

In this paper, a new strong form of slight gpr-continuity,called slight δgp-continuity,is introduced. It 

is shown that slight δgp-continuity is strictly weaker than contra δgp-continuity and δgp-continuity. 

Relationships between slight δgp-continuity and graphs are investigated. Also,additional properties 

of these functions are investigated.  

Throughout this paper, (X, τ) and (Y, σ) (or X and Y) represents a topological space on which no 

separation axioms are assumed, unless otherwise mentioned. The closure and interior of M X will 

be denoted by cl(M) and int(M) respectively. 

Definition 1.1.    A subset K of a topological space X is called pre-closed[10](resp.,regular 

closed[13]) if cl(int(K))⊆K(resp.,cl(int(K)=K . 

Definition 1.2 A subset K of a topological space X is called  δ-closed[18] if  K = clδ(K)  where  

 clδ (K) = {x ∈ X: int(cl(U))∩K= , U ∈ τ and  x ∈ U }. 

Definition 1.3 A subset K of a topological space X is called  δgp-closed[3](resp,gpr-closed[6] and 

gp-closed[9]) if pcl(K) ⊆ U whenever K ⊆ U and U is  δ-open (resp, regular open and open) in X. 

Definition 1.4 A function f:X→Y  from a topological space X into a topological space Y is called, 

(i)  δ-irresolute [7] if f-1 (M) is δ-closed in X for every δ-closed set M of Y. 

(ii) slightly  continuous[8] (resp, slightly gp-continuous and slightly  gpr-continuous[2]) if   

f-1 (M) is closed  (resp.,gp-closed and gpr-closed) in X for every clopen set M of Y. 

mailto:jagadeeshbt2000@gmail.com


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(iii) δgp-continuous[15](resp, contra δgp-continuous[16]) f-1 (M)  is δgp-open](resp, δgp-closed)  

in X for every open set M of Y. 

(iv) δgp-irresolute [15] if f-1 (M) is δgp-closed in X for every δgp-closed set M of Y. 

(v) pre δgp-closed[16] if the image of every δgp-closed set of X is δgp-closed in Y. 

Definition 1.5 A space X is called, 

(i) locally discrete[11] if every open subset is closed .  

(ii) Submaximal[14] if every pre-open set is open in X. 

(iii) δgp-additive[16]if δGPC(X)  is closed under arbitrary  intersections. 

(iv) δgpT1/2-space[15] if every δgp-closed subset of  X  is pre-closed. 

Theorem 1.6[16]  

If M and N are δgp-open subsets of  a submaximal space X, then M N is                      

δgp-open in X.    

 2. Slightly δgp-Continuous Functions 

Definition 2.1.  

A function f: X → Y is called slightly δ-generalized pre-continuous (briefly slightly δgp-continuous) 

if inverse image of every clopen subset of Y is δgp-open in X.The proof of the following Theorem is 

straightforward and hence omitted. 

Theorem 2.2.  For a function f: X → Y, the following statements are equivalent: 

           i).f is slightly δgp-continuous. 

          ii). inverse image of every clopen subset of Y is δgp-closed in X. 

         iii). inverse image of every clopen subset of Y is δgp-clopen in X. 

Remark 2.3. We have the following diagram for a function f : (X,  τ)→(Y,  σ) :                                                      

                         Slight precontinuity        contra δgp-continuity  

                                      ↓                                  ↓ 

slight gp-continuity → slight δgp-continuity → slight gpr-continuity 

↑ 

δgp-continuity 

                             None of the implications in above diagram is reversible. 

Example 2.3. Let X =Y ={a, b, c,d}, τ = {X, , {a},{b},{a,b},{a,b,c}} and  

σ={Y, , {a},{b, c}{a,b,c},{b,c,d}}. 

(i)Define f : (X,  τ)→(Y,  σ)  by f(a)=a=f(b),f(c)=b and f(d)=c,then f is slightly gpr-continuous but 

not slightly δgp-continuous since {a}  is clopen in Y but  f-1({a})={a,b}  is not  δgp-closed in X . 



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(ii) Define g : (X,  τ)→(Y,  σ)  by g(a)=a=g(c),g(b)=b and g(d)=c,then g is slightly δgp-continuous 

but not slightly  gp-continuous since {a}  is clopen in Y but  g−1({a})={a,c}  is not  gp-closed in X  

(iii) Define h : (X,  τ)→(Y,  σ)  by h(a)=a=h(d),h(b)=d and h(c)=b,then h is slightly δgp-continuous 

but not δgp-continuous since for  closed set {d}, h−1({d})={b}  is not  δgp-closed in X . 

Example 2.4. Let R and Q be the real numbers and rational numbers, respectively. 

Let  M={x  R:x is rational and 0<x<1}.We define two topologies on τ ={R, , M,R\M} and 

 σ = {R,, {0}}. Define f : (R, τ)→(R, σ)  by f(x)=1 if xQ and f(x)=0 if x∉Q. 

Then f is slightly δgp-continuous but not contra δgp-continuous since for  closed set  R\{0},  

f−1(R\{0})=Q  is not  δgp-closed in (R, τ) .                             

Theorem 2.5. If  X is  locally discrete, then the following are equivalent: 

(i) f : X → Y is slightly δgp-continuous; 

(ii) f : X → Y is  δgp-continuous; 

(iii) f : X → Y is contra δgp-continuous. 

Theorem 2.6. If  X is δgpT1/2-space, then the following are equivalent: 

(i) f : X → Y is slightly gpr-continuous; 

(ii) f : X → Y is slightly δgp-continuous; 

(iii) f : X → Y is slightly gp-continuous; 

(iv) f : X → Y is  slightly pre-continuous. 

Theorem 2.7 Let f : (X, τ) → (Y, σ) be a slightly δgp-continuous function, then for each  p  X   

 and each clopen set N containing f(p), there exists δgp-open set M containing such that f(M)  N.  

 Proof. Let p  X and N be a clopen set such that f(N)  Y. Since f is slightly δgp-continuous           

 f-1 (N)  is δgp-open  in X. If we put M = f-1 (N), we have p  M and f(M)  N. 

 Let (X, τ) be a topological space. The quasi-topology  on X is the topology having as base all clopen 

subsets of (X, τ). The open (resp. closed) subsets of the quasi- topology are said to be quasi-open 

(resp. quasi-closed). A point  x of a space X is said to be quasi closure of a subset A of X, denoted by 

clqA, if A  U ≠  for every clopen set U containing x. A subset A is said to be quasi closed if and 

only if A = clq [8] 

Theorem 2.8   

The following are equivalent for a function f:X→Y with X is δgp-additive. 

(i)  f is slightly δgp-continuous; 

(ii) for  each p ∈ X and each clopen set N  of Y containing f(p),  there  exists an δgp-open set   

     M  in X containing  p such that f(M)  N; 



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(iii) f(δgpcl(A))  clq (f(A)  for every subset A of X; 

(iv)  δgpcl(f−1(B))  f−1(clq (B)) for every subset B  of Y. 

Proof: (i)→(ii)It follows  from Theorem  2.7 

(ii)→(i) Let M   Y be a clopen set such that f(p) ∈ M,then  p ∈ f-1 (M).  From  (ii), there exists a  

δgp-open set Up containing p such that f(Up)  M, then p∈ Up f-1 (M). 

Hence f-1 (M) =∪{Up : Up⊂  f-1 (M)} is δgp-open in X . 

(i)→(iii)Let A   X. Suppose y clq (f(A)), then there exists a clopen set B  Y containing y  

such that  f(A) ∩ B =.  So,we have,A ∩ f-1 (B)=φ and  

δgp-cl(A) ∩ f-1(B)=φ which implies f(δgpcl(A)) ∩ B= and hence yf(δgpcl(A)).  

Therefore f(δgpcl(A))   clq (f(A)). 

 (iii)→(iv)Let B   Y, then f-1 (B)   X. By (iii), f (δgpcl(f-1(B))   clq (f (f-1 (B)))   clq (B).  

Thus δgpcl(f-1 (B))  f-1 (clq (B)). 

(iv)→(i) Let M be any clopen subset of Y. Then by (iv),  

δgpcl(f-1(M)  f-1 (clq (M)) = f-1 (M) and δgpcl(f-1 (M)) = f-1 (M).  

Therefore , f-1 (M) is δgp-closed set in X.    

Remark 2.9  The composition of two slightly  δgp-continuous functions need not be slightly  δgp-

continuous as seen from the following examples. 

Example 2.10  Let X=Y={a,b,c,d},Z=={a,b,c}, τ = {X, , {a},{b},{a,b},{a,b,c}} and             

σ={Y, , {a},{b, c},{a,b,c},{b,c,d}} and ƞ=={Z, , {a},{b, c}}. Define f : (X,  τ)→(Y,  σ)  by 

f(a)=a=f(b),f(c)=b and f(d)=c  and g : (Y,  σ)→(Z,  ƞ)  by g(a)=a,g(b)=b,g(c)=c=g(d). Then f and g 

are slightly  δgp-continuous but g⋆f:X→Z is not s l i g h t l y  δgp-continuous, since  for the clopen 

set {a}   in Z , (g⋆f)−1{a}={a,b} is not δgp-closed in X. 

Theorem 2.11  

For  any  two functions  f:X→Y  and  g:Y→Z,the  following hold: 

(i)g⋆f  is slightlly  δgp-continuous  if f is   δgp-irresolute  and  g is slightlly  δgp-continuous. 

(ii)g⋆f is slightlly  δgp-continuous if f is δgp-irresolute and g is δgp-continuous. 

(iii)g⋆f is slightlly δgp-continuous f is δgp-irresolute and g is slightlly continuous.  

Proof:(i) Let N be any clopen set in Z. Then g-1(V) is δgp-open in Y since g is slightlly  δgp-

continuous. Therefore , f-1 [g-1(U)]=(g⋆f )-1(U)  is δgp-closed in X because f is contra δgp-

continuous. 

Hence g⋆f is contra  δgp-continuous. 

The proofs of (ii)  and (iii)  are analogous to (i)  with the obvious changes. 



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Theorem 2.12   

Let f:X→Y  be δgp-open surjection  and g:Y→Z be a function  such  that  g⋆f:X→Z is  slightly  

δgp-continuous,then   g is  slightly  δgp- continuous. 

Proof:Let U be any c lopen set in Z.Then (g⋆f )-1(U)= f-1 (g-1(U )) is δgp-open  in X. Since f is 

a δgp-open surjection, f(f-1(g-1(U)))= g-1(U ) is δgp- open set in Y.  Therefore,  g is slightly 

δgp-continuous. 

Theorem 2.13 

 Let f : X → Y be bijective, δ-irresolute and   pre-closed . Then for every  

δgp-closed set M of  X, f(M) is  δgp-closed in Y. 

Proof. Let M be any δgp-closed set of  X and N a δ-open set of Y containing f(M). Since  

f-1(N) is a δ-open set of X containing M, pcl(M)  f-1 (N) and hence f(pcl(M))  N. Since f is  

pre-closed  and pcl(M) is pre-closed in X, f (pcl(M)) is pre-closed in Y. Since  

pcl (f(M))  pcl(f(pcl(M)))  N, pcl(f(M))  N. Therefore f(M) is δgp-closed in Y.  

Theorem 2.14  Let g:X→X×Y be the graph function of f:X→Y,defined by g(x)=(x,f(x))  

for each x∈X. Then f is slightly δgp-continuous,if g is slightly δgp-continuous. 

Proof:Let V be any c l open   set in Y, then X×V is a clopen set in X×Y.  

It  follows that f-1 (U)= g-1(X×U)  is δgp-closed in X since  g is slightly δgp-continuous.  

Hence f is slightly  δgp-continuous. 

Definition 2.15  A space X is called, 

(i) co-T2 [5] (resp, δgp-Hausdorff[16])if for any pair of distinct points x and y, there exist 

disjoint clopen( δgp-open)sets G and H such that x ∈ G and y ∈ H. 

(ii) co –normal[5](resp, strongly δgp-normal)   if each pair of disjoint clopen(resp, δgp –

closed) sets can be separated  by disjoint open  sets. 

(iii) co-regular[5](resp,strongly  δgp-regular)  if for each clopen(resp, δgp-closed)  set B and each  

x ∉ B,there exist disjoint open sets M and N such that B ⊂ M and x∈ M.  

(iv) mildly compact[12]( δgp-compact[17]) if every clopen(resp, δgp- cover of X has a finite 

sucover 

(v) δgp-connected[16] if  X is not the union of two disjoint nonempty δgp-open sets. 

Theorem 2.16  

If a surjective  function  f:X→Y  is slightly  δgp-continuous and X is δgp-connected space, then 

Y is connected. 

Proof:  Suppose that  Y is not  a connected  space.   Then  there  exist disjoint open sets U and 

V in Y such that Y=U∪V.  Therefore  U and V are clopen  sets in Y. Since f is slightly  δgp-

continuous,    f-1(U)  and  f-1(V)  are  δgp-open sets in X . Also f is surjective , f-1(U)  and  f-1(V)  



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are  non  empty disjoint  and X = f-1(U)∪ f-1(V) which contradicts  the fact that X is δgp-

connected space.             Hence Y is connected. 

Theorem 2.17  If f : X → Y is slightly δgp-continuous injection and Y is co-T2, then X is δgp-

Hausdorff 

Proof. Let x1, x2,  X and x1 ≠ x2. Then since f is injective and Y is co-T2, f(x1) ≠ f(x2) and there 

exist clopen subsets V1, V2  of Y such that f(x1)  V1 and f(x2)  V2 and V1  V2 = . Since f is  

slightly δgp-continuous, xi  f-1 (Vi)  δGPO (X) for  i = 1, 2 and f-1 (V1)  f-1(V2) = . Thus X is 

δgp- Hausdorff. 

Theorem 2.18  

Assume that X is δgp-additive. If f:X→Y and g:X→Y are slightly  δgp-continuous, X is 

submaximal and Y is co-T2 .  Then F={p ∈ X:f(p)=g(p)} is δgp-closed in X. 

Proof:  Let p  F, then  f(p) ≠ g(p). Since Y is co-T2, there exist clopen subsets M1 and M2 of Y  

such that  f(p)∈ M1, g(p)∈ M2  and  M1∩ M2 =.   Since f and g are slightly   δgp-continuous , f-1 

(M1)   and   g -1(M2) are δgp-open sets in X . Let M= f-1 (M1)  and N = g -1(M2), then M and N 

are δgp-open sets containing  p. Set O=M∩N,  then O i s  δgp-open set containing p. Hence  

f(O)∩g(O)=f(M∩N)∩g(M∩N)⊂f(M)∩g(N)= M1∩ M2 = and so O∩F=. 

Then    p ∈   δgpcl(F) and hence,  F is δgp-closed in X.   

Theorem 2.19  

If f:X→Y be slightly  δgp-continuous injective closed function from a strongly δgp-regular space X 

onto a space Y,then Y is co-regular. 

Proof: Let M be a clopen set in Y  and y ∉ M.Take y=f(x).Since f is slightly  δgp-continuous , f-

1(M) is  

δgp-closed. Take N= f-1(M) .We have x ∉ N. Since X is strongly δgp-regular , disjoint o p e n sets U 

and V in Y such that N⊂ U  and x   V. Then we obtain that M=f(N)⊂ f(U) and y=f(x)   f(V) 

such that  f(U) and f(V) are disjoint open sets in Y. This shows Y is co-regular-normal. 

Theorem 2.20  

If f:X→Y be slightly  δgp-continuous injective open function from a strongly δgp-normal space X 

onto a space Y,then Y is co-normal. 

Proof: Let E and F be disjoint clopen subsets of  Y. Since f is slightly  δgp-continuous, f-1 (E) and 

f-1(F) are disjoint δgp-closed sets in X. Since X is strongly δgp-normal, there exist disjoint open sets 

U and V such that f-1(E)⊂U  and f-1 (F)⊂V.  This implies E⊂ f(U) and F⊂ f(V) and  injectivity and 

openness  of f implies f(U) and f(V) are disjoint open sets in Y. This shows Y  is co-normal.  

Theorem 2.21  

If f : X → Y is slightly δgp-continuous surjection, and X is δgp-compact, then Y is mildly compact. 

Proof. Let { V : V  CO (Y),   I } be a cover of Y by clopen sets. Since f is slightly  



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δgp-continuous, {f-1(V) :   I} be δgp-open cover of X so there is a finite subset I0 of I such that  

X =  {f-1 (V ) :   I0 }. Therefore, Y = U { V :   I0} as f is surjective.  

Thus Y is mildly compact.    

Theorem 2.22 

 If f : X → Y is slightly δgp-continuous surjection, and X is δgp-connected, then Y is connected. 

Proof. Assume that Y is disconnected. Then exist disjoint non-empty clopen sets U and V for which            

Y = U  V. Therefore, X = f-1 (U)  f-1 (V) is the union of two disjoint δgp-open nonempty sets and    

hence is not δgp-connected. 

Definition 2.23  The graph G(f ) of a function f:(X,τ )→(Y,σ) is said to be is   δgp-co-closed if 

for each (x,y)∈(X×Y)-G(f ) there exist δgp-open set U in X containing  x and clopen set V in Y 

containing  y such that (U×V)∩G(f )= . 

Theorem 2.24   

The  graph  G(f )  of a  function f:(X,τ )→(Y,σ) is   δgp-co-closed in  X×Y  if and  only for each  

(x,y)∈(X×Y)-G(f )  there  exist δgp-open set U in X containing  x and  clopen set V in Y 

containing  y such that f(U)∩V=. 

Theorem 2.25   

If f:(X,τ )→(Y,σ) is slightly δgp-continuous and Y is co-Hausdorf, then G(f ) is is   δgp-co-closed 

in the product space X×Y. 

Proof:Let (x,y)∈(X×Y)-G(f ),then  f(x) ≠y and there  exists  c l o pen sets U and V such that  

f(x)∈U,y∈V and U∩V= .  Since f is slightly δgp-continuous, then there  exists a δgp-open set 

G such that  x∈G and f(G)⊂ U  and hence we obtain f(G)∩V = .This shows that G(f ) is is   

δgp-co-closed. 

Theorem 2.26   

If f:(X,τ )→(Y,σ) is  slightly δgp-continuous and Y is co-T1, then G(f ) is is   δgp-co-closed in the 

product space X×Y. 

Proof:Let (x,y)∈(X×Y)-G(f ),then  f(x) ≠y and there  exists  a  clopen sets M o f  Y  such that  

f(x)∈ M and y∉ M and U∩V= .  Since f is slightly δgp-continuous, then there  exists a δgp-

open set G such that  x∈G and f(G)⊂ M  and hence we obtain f(G)∩(Y\M) =  and Y\M is 

clopen set containing y.This shows that G(f ) is δgp-co-closed. 

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