Strong 


GP



-continuity and weakly 


GP



-closed functions on 


GPT



 spaces

Hanan Al-Saadi; Huda Al-Malki

doi:10.1016/j.jksus.2024.103259

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08 2024

:36;

103259

doi:

10.1016/j.jksus.2024.103259

Strong $GP$ -continuity and weakly $GP$ -closed functions on $GPT$ spaces

Hanan Al-Saadi^a,⁎, Huda Al-Malki^b

a Department of Mathematics, Faculty of Sciences, Umm Al-Qura University, Makkah 21955, Saudi Arabia

b Department of Mathematics, Adham University College, Umm Al-Qura University, Saudi Arabia

⁎Corresponding author. hsssaadi@uqu.edu.sa (Hanan Al-Saadi),

Received: 2024-2-13, Accepted: 2024-5-13,

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Abstract

In this research article, we present the procedure of generating $GPT$ spaces in two different ways: using the generalized neighbourhood system and the monotonic operator. Then, we introduce several types of generalized primal continuous functions. Some characteristics have been dissected, and the relationships among them have been studied. We use the technique of Császár, which changes the “generalized topology” to other “generalized topologies” weaker than it, to show some important results. Furthermore, we show that the notion of “strong $GP$ -continuity” coincides with the notion “ $GP$ -continuity” under some conditions. We present these results on a simple graph to make it easier for the reader. Finally, study the preservation of the notions of “ $GP$ -connected” and “ $GP$ -hyperconnected” by different types of generalized primal continuous functions.

Keywords

54A05

54C10

54A10

GP-continuous

GPN-continuous

Strong GP-continuous

Weakly GP-closed

GP-connected

GP-hyperconnected

1 Introduction

Continuity has been a core concept in the branch of pure mathematics in general and in the theory of topology. A function between two topological spaces is continuous if the pre-image of every open set is open. To judge whether a function is continuous, we need to examine the structure of the spaces. This nice kind of function is quite important for the role that it plays in preserving the topological structure of the domain space in the co-domain space.

In Császár (1997) the concept of generalization was established when Császár introduced the concept of generalized open set, which led to the definition of a new mathematical structure weaker than the topology, that is, having relaxing conditions. Following the scientific approach, any “generalization” to the basic concepts in the “theory of topology” must be studied. This happened when many articles drew attention to this generalized space and examined every topological characteristic. Moreover, the notion of “generalized continuous” functions between two generalized topological spaces is defined.

In Al-Saadi and Al-Malki (2023), the same methodology was followed. The author gave a new mathematical structure that connected the generalized topological properties with the primal set $P$ given on the same set. The basic definitions were given, and some operators with nice behaviour are present in Al-Saadi and Al-Malki (2024).

In this paper, we continue our study of this new space. The study focuses on “generalized continuity” under the influence of the primal set. The topic of continuity is one of the major topics in topology theory, and for that, we investigate in depth many types of continuous functions. These types are already studied in the theory of “generalization”. Here we will study them in the sense of primal collection. Also, we will study the relationship between them and state the necessary conditions to develop some weak kinds of them into strong types. The importance of continuous functions in topology comes from their effective role in preserving topological properties.

This article contains five sections. Section 1 is divided in two parts: the first one summarizes the previous studies in this research; the second part is recalling the basic definitions and the fundamental theorems. In Section 2, we present three types of continuous functions on $GPT$ spaces. First, we generate $GPT$ spaces in two different ways. One by using the generalized neighbourhood system, and the other by using the monotonic operator. Then, we give the definitions of $GPN$ -continuous, $GP$ - $θ$ -continuous, and almost $GP$ -continuous; and we discuss their characteristics. The relationship among them and counterexamples have been studied. In Section 3, we introduce the notions of “strong $GP$ -continuous function”, “strongly $GP$ - $θ$ -continuous”, and “super $GP$ -continuous”. Then, study the relationship among them. Also, we present the concepts of “weakly $GP$ -closed function” and “ $GP$ -regular space”; and we discuss their characteristics. In Section 4, we present the notions of “ $GP$ -connected” and “ $GP$ -hyperconnected”. Then, we study the preservation of these notions by different types of generalized primal continuous functions. Section 5 is a discussion of all theimportant results that we present.

1.1

1.1 Literature review

In Császár (2002) the notion of “generalized topology” was presented. The collection $g$ of the power set $2^{T}$ of a non-empty set $T$ is named a generalized topology $GT$ if the empty set belongs to $g$ , and the countable union of every set that belongs to $g$ also belongs to $g$ . Thus, the pair $(T, g)$ is assumed to be a generalized topological space, denoted by $GT$ space.

According to Császár (2008b), this space member is named $g$ -open sets; however, $g$ -closed means its complements. Also, $C_{g} (T)$ refers to the whole set of $g$ -closed sets, whereas $c_{g} (M)$ and $i_{g} (M)$ mean the closure and interior of $M \subset T$ , respectively, which are defined as in the general case. Studies continued in this generalized space. Császár (2004a) presented the concept of topological disconnection under the influence of generalization, while Császár (2005) presented the concept of generalized continuous functions. Császár (2008a) introduced the notion of “strong generalized topological space”, which is given as follows: A generalized topology is named strong if $T \in g$ .

Moreover, this field has received a lot of attention since the literature has studied and examined its topological characteristics in detail. For example, Császár (2004b) gave the separation axioms by replacing the notion of open sets with a more general expression, while a lot of research gave the definition of generalized continuity with different types of more general kinds of continuity, for example, Min (2009b) presented the notion of almost continuity, Min (2009a) introduced the concept of $θ (g, g^{‵})$ -continuity, and Jayanthi (2012) gave the definition of contra continuous function.

In another direction, literature has spread that presents new mathematical structures as useful tools that lead to a deeper understanding and broader application of the concepts of topological spaces in various fields. For example, the mathematical structure “ideal” in Janković and Hamlett (1990) and its dual structure “filter” in Kuratowski (2014). These two structures have applications in science and society. The notion of ideal $m$ -space is given in Al-Omari and Noiri (2012), including some type of operator.

On the same approach, the notion of “grill” appeared in Choquet (1947). The associated topology for the grill is given in Roy and Mukherjee (2007). New types of mapping were presented and compactness in the light of a grill in Roy et al. (2008). The generalized type of continuity and its decomposition via grill set are defined and studied in Hatir and Jafari (2010).

In 2022, the dual structure of the “grill” is defined and named “primal” in Acharjee et al. (2022). A family $P \subseteq 2^{T}$ is called a primal over $T$ , if the next are satisfied for $M, N \subseteq T$ : (i) $T \notin P$ . (ii) Whenever $M \subseteq N$ and $N \in P$ , thus $M \in P$ . (iii) Whenever $M \cap N \in P$ , thus $M \in P$ or $N \in P$ . Some operators are given via a “primal set” and presented in AL-Omari et al. (2022). Also, Modak (2013) follows the same approach and definition of new space via the notion of filter and grill. In Al-shami et al. (2023) a new space is defined by joining the nice properties of primal and soft sets.

1.2

1.2 $GPT$ spaces

Through this part, we will review the basic definitions of $GPT$ spaces, that are presented in Al-Saadi and Al-Malki (2023) and Al-Saadi and Al-Malki (2024)

Definition 1

The triple $(T, g, P)$ refers to the generalized primal topological space $GPT$ space for short. Moreover, $(g, P)$ -open sets is the symbol for the element of this space, and $(g, P)$ -closed sets denote their complement. The entire set of $(g, P)$ -closed set is symbolized by $C_{(g, P)} (T)$ and $c l_{(g, P)} (M)$ denotes the closure of $M \subseteq T$ , that is qualified as in the general situation.

Definition 2

Let $(T, g, P)$ be a $GPT$ space. Consider an operator $ψ : T \to 2^{2^{T}}$ given as $t \in O$ , for all $O \in ψ (t)$ . Hence, $O \in ψ (t)$ is a generalized primal neighbourhood or $GPN$ for $t \in T$ , and $ψ$ is a generalized primal neighbourhood system over $T$ or $GPN$ system. The set of all $GPN$ systems on $T$ is denoted by $Ψ (T)$ .

Definition 3

Let $(T, g, P)$ be a $GPT$ space with $M \subseteq T$ . The operator ${(.)}^{♢} : 2^{T} \to 2^{T}$ is defined by $M^{♢} (T, g, P) = {t \in T : M^{c} \cup O^{c} \in P, \forall O \in ψ (t)} .$

Theorem 4

Let $(T, g, P)$ be a $GPT$ space. The next holds for $M, N \subseteq T$ .

$ϕ^{♢} = ϕ$ .
$M^{♢}$ is $(g, P)$ -closed i.e., $c l_{(g, P)} (M^{♢}) = M^{♢}$ .
${(M^{♢})}^{♢} \subseteq M^{♢}$ .
$M^{♢} \subseteq N^{♢}$ , whenever, $M \subseteq N$ .
$^{♢} M \cup N^{♢} = {(M \cup N)}^{♢}$ .
${(M \cap N)}^{♢} \subseteq M^{♢} \cap N^{♢}$ .

Definition 5

Let $(T, g, P)$ be a $GPT$ space with $M \subseteq T$ . We define the operator $c l^{♢} : 2^{T} \to 2^{T}$ by $c l^{♢} (M) = M \cup M^{♢}$ .

Theorem 6

Let $(T, g, P)$ be a $GPT$ space. The next holds for $M, N \subseteq T$ .

$c l^{♢} (ϕ) = ϕ$ ,
$M \subseteq c l^{♢} (M)$ ,
$c l^{♢} (c l^{♢} (M)) = c l^{♢} (M)$ ,
$c l^{♢} (M) \subseteq c l^{♢} (N)$ , whenever $M \subseteq N$ ,
$c l^{♢} (M) \cup c l^{♢} (N) = c l^{♢} (M \cup N)$ .

Theorem 7

Let $(T, g, P)$ be a $GPT$ space with $M, N \subseteq T$ . We have $M \cap N^{♢} \subseteq {(M \cap N)}^{♢}$ , whenever $M$ is $(g, P)$ -open.

Definition 8

Let $(T, g, P)$ be a $GPT$ space and $M \subseteq T$ .

Whenever $M \subseteq c l^{♢} (i_{g} M)$ , $M$ is called $(g, P)$ -semi-open set.
Whenever $M \subseteq i_{g} (c l^{♢} (M))$ , $M$ is called $(g, P)$ -pre-open set.
Whenever $M = i_{g} (c l^{♢} (M))$ , $M$ is called $(g, P)$ -regular open set.
Whenever $M \subseteq c_{g} (i_{g} (c l^{♢} (M)))$ , $M$ is called $(g, P)$ - $β$ -open set.
Whenever $M \subseteq i_{g} (c l^{♢} (i_{g} (M)))$ , $M$ is called $(g, P)$ - $α$ -openset.

The entire set of $(g, P)$ -semi-opensets is denoted via $S g_{P}$ , and the entire set of $(g, P)$ -pre-open setsis denoted via $P g_{P}$ . Moreover, the entire set of $(g, P)$ - $α$ -open sets is denoted via $α g_{P}$ , while $β g_{P}$ represents the whole set of $(g, P)$ - $β$ -open sets. Finally, $R g_{P}$ represents the whole set of $(g, P)$ -regularopen sets.

Definition 9

Let $(T, g, P)$ be a $GPT$ space and $M \subseteq T$ . If $(T ∖ M)$ is a $(g, P)$ -semi-open (resp. $(g, P)$ -pre-open, $(g, P)$ -regular open, $(g, P)$ - $α$ -open, $(g, P)$ - $β$ -open), hence $M$ is called a $(g, P)$ -semi-closed (resp. $(g, P)$ -pre-closed, $(g, P)$ -regular closed, $(g, P)$ - $α$ -closed, $(g, P)$ - $β$ -closed).

Proposition 10

Let $(T, g, P)$ be a $GPT$ space. We have:

$⋃_{δ \in Δ} M_{δ}$ such that each $M_{δ} \in S g_{P}$ also belongs to $S g_{P}$ .
$⋃_{δ \in Δ} M_{δ}$ such that each $M_{δ} \in P g_{P}$ also belongs to $P g_{P}$ .
$⋃_{δ \in Δ} M_{δ}$ such that each $M_{δ} \in α g_{P}$ also belongs to $α g_{P}$ .
$⋃_{δ \in Δ} M_{δ}$ such that each $M_{δ} \in β g_{P}$ also belongs to $β g_{P}$ .
$⋃_{δ \in Δ} M_{δ}$ such that each $M_{δ} \in R g_{P}$ also belongs to $R g_{P}$ .

Corollary 11

Let $(T, g, P)$ be a $GPT$ space. All of the families $S g_{P}, P g_{P}, α g_{P}, β g_{P}$ and $R g_{P}$ constitute a $GPT$ space with a primal set $P$ over $T$ .

Definition 12

Let $(T, g, P)$ be a $GPT$ space and $(T^{‵}, g^{‵})$ is $GT$ space. The mapping $p : T \to T^{‵}$ is called $(g, P)$ -semi-continuous (resp. $(g, P)$ -pre-continuous, $(g, P)$ - $α$ -continuous, $(g, P)$ - $β$ -continuous)if $p^{- 1} (M)$ , for all $M$ is $g$ -open is $(g, P)$ -semi-open (resp. $(g, P)$ -pre-open, $(g, P)$ - $α$ -open, $(g, P)$ - $β$ -open).

Theorem 13

Let $(T, g, P)$ be a $GPT$ space and $M \subseteq T$ . We find:

$M \in α g_{P}$ if and only if $M \in S g_{P}$ and $M \in P g_{P}$ .
If $M \in S g_{P}$ , then $M \in β g_{P}$ .
If $M \in P g_{P}$ , then $M \in β g_{P}$ .

Corollary 14

Let $(T, g, P)$ be a $GPT$ space. We have:

$P g_{P} \cap S g_{P} = α g_{P}$ .
$g − open \subset α g_{P} \subset S g_{P} \subset β g_{P}$ .
$α g_{P} \subset P g_{P} \subset β g_{P}$ .

Definition 15

Let $(T, g, P)$ be a $GPT$ space. Consider $M \subseteq T$ . If $c l^{♢} (M) = T$ , hence $M$ is called $(g, P)$ -dense set.

2 Main results

This part is about demonstrating the possibility of determining new types of continuity on a $GPT$ space. First, let us make a supplement to the results from Al-Saadi and Al-Malki (2023) about a $GPT$ space characteristic.

Definition 16

Consider $η : 2^{T} \to 2^{T}$ as a map that has the monotonic property, that is, if $M \subset N$ of $T$ , then $η (M) \subset η (N)$ . The whole set of monotonic maps on a set $T$ is denoted by $Γ (T)$ .

Remark 17

Let $M \subset T$ , then the next is true:

$M$ is $η$ -open iff $M \subset η (M)$ . Thus, $ϕ$ is $η$ -open.
The countableunion of $η$ -open setsis $η$ -open; see Császár (2002). Hence, the set of all $η$ -open sets forms a $GT$ and is denoted by $g_{η}$ . Consequentially, $(T, g_{η}, P)$ is a $GPT$ space, where $P$ is a primal set over $T$ .

By the above consideration, it is obvious that we can produce all $GPT$ space over $T$ for some $η \in Γ$ .

Corollary 18

Let $(T, g, P)$ be a $GPT$ space. Then, there exists $η \in Γ$ satisfying the following properties:

$η (ϕ) = ϕ$ and $η (M) \subset M$ ,
$η (η (M)) = η (M)$ .

Proof

Consider $g_{η} = g$ and $η (M) = {\cup G : G \subset M, \forall G \in g}$ . Hence, $η (M) \in g$ . So, $η (M) \subset M$ and $η (ϕ) = ϕ$ , hence we prove (i).

Moreover, from the definition $M \subset η (M)$ , but from (i) $η (M) \subset M$ implies $η (M) = M$ . Therefore, $η (M) \in g$ and $η (η (M)) = η (M)$ . □

Another method to obtain a $GPT$ space is by using the map $ψ$ , that is mentioned in Definition 2 (for deep details, see Császár, 2002).

Definition 19

If $ψ \in Ψ (T)$ and $G \in g$ satisfy the condition: if $t \in G$ , there exists $O \in ψ (t)$ satisfying $O \subset G$ . Then, $g$ is a $GT$ denoted by $g_{ψ}$ . Consequentially, $(T, g_{ψ}, P)$ is a $GPT$ space, where $P$ is a primal set over $T$ . Conversely, let $g$ be a $GT$ , then $\exists ψ \in Ψ (T)$ such that $g = g_{ψ}$ , which means $ψ$ satisfies the condition: $\forall t \in T, \exists O \in ψ (t), O \in g$ .

In the first case, consider $ψ = ψ_{g}$ and for the second case consider $ψ \in Ψ_{g} (T)$ . For $g = g_{η}$ , where $η \in Γ$ , considering $i_{g_{η}}$ as $i_{η}$ and $c_{g_{η}}$ as $c_{η}$ . Also, for $g = g_{ψ}$ , where $ψ \in Ψ (T)$ , considering $i_{g_{ψ}}$ as $i_{ψ}$ and $c_{g_{ψ}}$ as $c_{ψ}$ .

Definition 20

Let $(T, g, P)$ be a $GPT$ space, $ψ \in Ψ$ , and $M \subseteq T$ . Then,

$ι_{ψ} (M) = {t \in T : \exists O \in ψ (t) satisfies O \subset M}$ .
$η_{ψ} (M) = {t \in T : O^{c} \cup M^{c} \in P, forall O \in ψ (t)}$ .

Note that $ι_{ψ}, η_{ψ} \in Γ$ . Moreover, $i_{ψ} (M) \subset ι_{ψ} (M)$ as well as $c l_{ψ}^{♢} (M) \subset η_{ψ} (M)$ . However, when $g_{ψ} = g$ , we have $ι_{ψ} = i_{ψ}$ and $η_{ψ} = c l_{ψ}^{♢}$ .

2.1

2.1 $G P$ -continuous function

This part of our article is a discussion of three different types of generalized primal continuity. We depend on two types of $GPT$ spaces: those generated by a generalized neighbourhood system and monotonic maps.

Definition 21

Let $(T, g, P)$ and $(T^{‵}, g^{‵}, P^{‵})$ be two $GPT$ spaces. Consider $O$ as $(g^{‵}, P^{‵})$ -open. Define a function $p : T \to T^{‵}$ . Then, $p$ is named $GP$ -Continuous if and only if $p^{- 1} (O)$ is $(g, P)$ -open.

The more general type of $GP$ -continuity based on the concept of generalized neighbourhood systems as follows:

Definition 22

Consider $ψ \in Ψ (T)$ and $ψ^{‵} \in Ψ (T^{‵})$ . Hence, $p$ is named $GPN$ -continuous iff for every $t \in T$ and $O^{‵} \in ψ^{‵} (p (t))$ , there is $O \in ψ (t)$ satisfying $p (O) \subset O^{‵}$ .

Since $c l^{♢} (M) \subset c l^{♢} (N)$ if $M \subseteq N$ , then $c l^{♢} \in Γ$ . The generalized primal neighbourhood system is defined as following:

Definition 23

Let $(T, g, P)$ be $GPT$ space, $M \subset T$ , $G$ be $(g, P)$ -open set, and $t \in G$ . If $O = c l^{♢} (G)$ , then $O \in ψ (c l^{♢}, g) (t)$ . Therefore, $ψ (c l^{♢}, g) \in Ψ (T)$ .

The connection between $GPN$ -continuity and $GP$ -continuity is described in the following.

Theorem 24

Every $GPN$ -continuous function is $GP$ -continuous, where $g$ produce by the neighbourhood system.

Proof

Consider $p$ as a function from $(T, g_{ψ}, P)$ to $(T^{‵}, g_{ψ}^{‵}, P^{‵})$ . Let $G^{‵} \in g_{ψ^{‵}}^{‵}$ , $t \in T$ and $p (t) \in G^{‵}$ . Thus, there is $O^{‵} \in ψ^{‵} (p (t))$ satisfies $O^{‵} \subset G^{‵}$ . Hence, $\exists O \in ψ (t)$ such that $p (O) \subset O^{‵}$ . Therefore, $p (O) \subset G^{‵}$ and $O \subset p^{- 1} (G^{‵})$ implies $p^{- 1} (G^{‵}) \in g_{ψ}$ . □

There is a $GP$ -continuous map produced by the neighbourhood system, which is not $GPN$ -continuous; the following example makes that clear.

Example 25

Let $(T, g, P)$ be a $GPT$ space, where $T = {t_{1}, t_{2}, t_{3}}$ , $g = {ϕ, {t_{1}}, {t_{2}}, {t_{1}, t_{2}}, T}$ , and $P = 2^{T} ∖ T$ . Suppose that $(T^{‵}, g^{‵}, P^{‵})$ is a $GPT$ space, where $T^{‵} = T$ , $g^{‵} = {ϕ, {t_{1}}, {t_{3}}, {t_{1}, t_{3}}, T^{‵}}$ , and $P^{‵} = 2^{T^{‵}} ∖ T^{‵}$ . Consider $ψ = ψ (c l^{♢}, g)$ and $ψ^{‵} = ψ (c l^{♢}, g^{‵})$ . Since if $O \in g$ and $t_{1} \in O$ , then ${t_{1}} \in g$ and $c l^{♢} {t_{1}} = {t_{1}, t_{3}} \notin g$ . If $O \in g$ and $t_{2} \in O$ , then ${t_{2}} \in g$ and $c l^{♢} {t_{2}} = {t_{2}, t_{3}} \notin g$ . If $O \in g$ and $t_{3} \in O$ , then $T \in g$ and $c l^{♢} (T) = T \in g$ . Hence, $g_{ψ} = {ϕ, T}$ . By the same way, we have $g_{ψ^{‵}} = {ϕ, T}$ . Let $p : T \to T^{‵}$ be the identity map. Thus, $p$ is a $GP$ -continuous. However, $p$ is not $GPN$ -continuous since $p (t_{1}) \in {t_{1}} \in g^{‵}$ and $c l^{♢} ({t_{1}}) = {t_{1}, t_{2}}$ implies ${t_{1}, t_{2}} \in ψ^{‵} {t_{1}}$ if $O \in ψ ({t_{1}})$ and $c l^{♢} ({t_{1}}) = {t_{1}, t_{3}}$ and ${t_{1}, t_{3}} ⊈ {t_{1}, t_{2}}$ .

The extra condition that makes the inverse of Theorem 24 true is described in the following.

Theorem 26

Every $GP$ -continuous function is $GPN$ -continuous, whenever $ψ^{‵} = ψ_{g^{‵}}^{‵}$ , where $g$ produced by the neighbourhood system.

Proof

Consider $p$ as a function from $(T, g_{ψ}, P)$ to $(T^{‵}, g_{ψ}^{‵}, P^{‵})$ . Put $t \in T$ and $O^{‵} \in ψ^{‵} (p (t))$ . Thus, $p (t) \in O^{‵}$ and $O^{‵} \in g_{ψ^{‵}}^{‵}$ . Hence, $t \in p^{- 1} (O^{‵}) \in g_{ψ}$ , then there exists $O \in ψ (t)$ satisfies $O \subset p^{- 1} (O^{‵})$ and implies $p (O) \subset O^{‵}$ . □

The map $η_{ψ}$ in Definition 20 is a useful tool to define the $GPN$ -continuous in a different way as follows:

Theorem 27

Consider $p : T \to T^{‵}, ψ \in Ψ (T)$ , $ψ^{‵} \in Ψ (T^{‵}), M \subset T, N \subset T^{‵}$ . Then, the next are equivalent:

$p$ is $GPN$ -continuous;
$p (η_{ψ} (M)) \subset η_{ψ^{‵}} (p (M));$
$η_{ψ} (p^{- 1} (N)) \subset p^{- 1} (η_{ψ^{‵}} (N))$ .

Proof

$(i) \Rightarrow (i i)$ . Let $t \in η_{ψ} (M)$ . Thus $p (t) \in η_{ψ^{‵}} (p (M))$ . Because if that does not hold, then $\exists O \in ψ^{‵} (p (t))$ satisfies $O^{c} \cup {(p (M))}^{c} \notin P$ . We get, $\exists S \in ψ (t)$ such that $p (S) \subset O$ and ${(p (S))}^{c} \cup {(p (M))}^{c} \notin P$ . Hence, $S^{c} \cup M^{c} \notin P$ , which is a contradiction.

$(i i) \Rightarrow (i i i)$ . Consider $M = p^{- 1} (N)$ . Thus, from (ii), we get $p (η_{ψ} (M)) \subset η_{ψ^{‵}} (p (M)) = η_{ψ^{‵}} (p (p^{- 1} (N))) \subset η_{ψ^{‵}} (N) .$ Therefore, $η_{ψ} (p^{- 1} (N)) \subset p^{- 1} (η_{ψ^{‵}} (N))$ .

$(i i i) \Rightarrow (i)$ . Consider $O \in ψ^{‵} (p (t))$ . Put $N = (T^{‵} ∖ O)$ and $p (t) \notin η_{ψ^{‵}} (N)$ . Thus, $t \notin p^{- 1} (η_{ψ^{‵}} (N))$ . From (iii) $t \notin η_{ψ} p^{- 1} (N)$ we get $\exists S \in ψ (t)$ satisfies $S^{c} \cup {(p^{- 1} (N))}^{c} \notin P$ . Thus, ${(p (S))}^{c} \cup N^{c} \notin P$ . Hence, $p (S) \subset O$ . □

2.2

2.2 $GP$ - $θ$ -continuous function

By using a generalized primal neighbourhood system $ψ (c l^{♢}, g) \in Ψ (T)$ that is mentioned in Definition 23, we can present a new kind of continuous function as follows:

Definition 28

Let $(T, g, P)$ and $(T^{‵}, g^{‵}, P^{‵})$ be two $GPT$ spaces. Let $ψ = ψ (c l^{♢}, g)$ and $ψ^{‵} = ψ^{‵} (c l^{♢}, g^{‵})$ . Hence $p : T \to T^{‵}$ is named $GP$ - $θ$ -Continuous if it is $GPN$ -Continuous.

In anther word if $t \in T$ , $O^{‵} \in g^{‵}$ , and $p (t) \in O^{‵}$ , then $\exists O \in g : t \in O$ satisfies $p (c l^{♢} (O)) \subset c l^{♢} (O^{‵})$ , where $c l^{♢} (O), c l^{♢} (O^{‵})$ restricted on $(T, g, P)$ and $(T^{‵}, g^{‵}, P^{‵})$ respectively.

Theorem 29

Let $(T, g, P)$ and $(T^{‵}, g^{‵}, P^{‵})$ be two $GPT$ spaces $M \subset T, N \subset T^{‵}$ , and $p : T \to T^{‵}$ . If $O \in ψ (c l^{♢}, g)$ and $O^{‵} \in ψ^{‵} (c l^{♢}, g^{‵})$ satisfy the next two conditions:

$(1) t \in η_{ψ} (M)$ iff $t \in G \in g$ implies ${(c l^{♢} (G))}^{c} \cup M^{c} \in P$ .

$(2) t^{‵} \in η_{ψ}^{‵} (N)$ iff $t^{‵} \in G^{‵} \in g^{‵}$ implies ${(c l^{♢} (G^{‵}))}^{c} \cup N \in P$ , then the next are equivalent:

$p$ is $GP$ - $θ$ -continuous;
$p (η_{ψ} (M)) \subset η_{ψ^{‵}} (p (M));$
$η_{ψ} (p^{- 1} (N)) \subset p^{- 1} (η_{ψ^{‵}} (N))$ .

2.3

2.3 Almost $GP$ -continuity

This part is a discussion of the notion of “almost continuous” in the generalized field. Min (2009b) presented this notion between two generalized topological spaces. Here, we will give it with consideration for the primal set.

Definition 30

Let $(T, g, P)$ and $(T^{‵}, g^{‵}, P^{‵})$ be two $GPT$ spaces. Consider $p : T \to T^{‵}$ . Then, $p$ is named almost $G P$ -continuous at $t \in T$ if for every $(g^{‵}, P^{‵})$ -open set $G^{‵} : p (t) \in G^{‵}$ , there existsa $(g, P)$ -open set $G : t \in G$ satisfies $p (G) \subset i_{g^{‵}} (c l^{♢} (G^{‵}))$ . Moreover, $p$ is named almost $GP$ -continuous if it is almost $GP$ -continuous at every $t \in T$ .

The more general concept between $GP$ -continuity and almost $GP$ -continuity is qualified in the following.

Theorem 31

Every $GP$ -continuous function is almost $GP$ -continuous.

Proof

For $t \in T$ let $G^{‵}$ be a $(g^{‵}, P^{‵})$ -open set such that $p (t) \in G^{‵}$ . Since $p$ is $GP$ -continuous, $p^{- 1} (G^{‵})$ is $(g, P)$ -open set. Put $p^{- 1} (G^{‵}) = G$ . Thus $t \in G$ . Now, $p (G) = G^{‵} \subset i_{g^{‵}} (c l^{♢} (G^{‵}))$ . Hence, $p$ is almost $GP$ -continuous at $t \in T$ . Since $t$ is arbitrary, $p$ is almost $GP$ -continuous. □

To say these two notions coincide, we need to discuss the technique that Császár presented in Császár (2008a). This technique changes the generalized topology of other generalized topologies, which is smaller than it. Here we will do the same, but this time with consideration of a primal set as follows:

Definition 32

Let $(T, g, P)$ be a $GPT$ space. Then, $ϑ (g)$ and $σ (g)$ are two $GPT$ spaces, given as follows for each $M \subseteq T$ :

(i) $ϑ (g) = {M \subseteq T : \forall t \in M, \exists (g, P) − closedset N satisfies t \in i_{g} (N) \subset M}$ .

(ii) $σ (g) = {M \subseteq T : \forall t \in M, \exists (g, P) − openset N satisfies t \in c l^{♢} (N) \subset M}$ .

Note that the member of $ϑ (g)$ coincides with the union of all $(g, P)$ -regular open set. By another way the family of all $(g, P)$ - regular open sets $R g_{P}$ is a based for the $GPT$ space $ϑ (g)$ .

So, the concept of almost $GP$ -continuous coincides with the notion of $GP$ -continuity if we make a suitable change to the $GPT$ space as follows:

Proposition 33

$p : (T, g, P) \to (T^{‵}, g^{‵}, P^{‵})$ is almost $GP$ -continuous iff $p : (T, g, P) \to (T^{‵}, ϑ (g^{‵}), P^{‵})$ is $GP$ -continuity.

Proof

Since $p : (T, g, P) \to (T^{‵}, g^{‵}, P^{‵})$ is almost $GP$ -continuous, then for every $t \in T$ and for every $(g^{‵}, P^{‵})$ -open set $G^{‵} : p (t) \in G^{‵}$ , there existsa $(g, P)$ -openset $G : t \in G$ satisfies $p (G) \subset i_{g^{‵}} (c l^{♢} (G^{‵}))$ . Hence, $p (G) \subset i_{ϑ (g^{‵})} (c l^{♢} (G^{‵})) \subseteq p (G),$ from the definition of $ϑ (g^{‵})$ . By regularity of $G^{‵}$ we have $i_{ϑ (g^{‵})} (c l^{♢} (G^{‵})) = G^{‵}$ . Hence $p^{- 1} (G^{‵}) = G$ . Therefore, $p : (T, g, P) \to (T^{‵}, ϑ (g^{‵}), P^{‵})$ is $GP$ -continuity. □

3 Strong $GP$ -continuity

In this part, special types of $GP$ -continuity will be introduced. In fact, the concepts of super $(g, g^{‵})$ -continuous functions and strongly $(g, g^{‵})$ - $θ$ -continuous functions are given in Min and Kim (2011). Also, the notion of strong $(g, g^{‵})$ -continuous functions is presented by Kim and Min (2012). All these notions are described according to $GT$ space features. Here, we will study these types of continuity in light of the primal set. First, let us define the set $ω_{g} = ⋃ {M \subset T : M isa (g, P) − open}$ .

Definition 34

Let $(T, g, P)$ and $(T^{‵}, g^{‵}, P^{‵})$ be two $GPT$ spaces. Consider $p : T \to T^{‵}$ . Then, $p$ is named a strong $GP$ -continuity if for every $t \in T$ and $(g^{‵}, P^{‵})$ -open set $G^{‵} : p (t) \in G^{‵}$ , there is a $(g, P)$ -open set $G : t \in G$ satisfies $c l^{♢} (p (G)) \cap ω_{g^{‵}} \subset G^{‵}$ .

Theorem 35

Consider $p : T \to T^{‵}$ as strong $GP$ -continuous function. If $p (ω_{g}) \subseteq ω_{g^{‵}}$ , then $p (c l^{♢} (H)) \subseteq c l^{♢} (p (H))$ , for all $(g, P)$ -open set $H \subseteq T$ .

Proof

Assume $H \subseteq T$ is $(g, P)$ -open. Let $t \in c_{g} (H)$ . Let $G^{‵} \subseteq T^{‵}$ be $(g^{‵}, P^{‵})$ -open such that $p (t) \in G^{‵}$ . As $p$ is a strong $GP$ -continuity, there existsa $(g, P)$ -openset $G \subseteq T : t \in G$ and $c l^{♢} (p (G)) \cap ω_{g^{‵}} \subset G^{‵}$ . Moreover, as $t \in c l^{♢} (H)$ and $G \subseteq T : t \in G$ , $H \cap G \neq ϕ$ . But $p (ω_{g}) \subseteq ω_{g^{‵}}$ . Hence, $ϕ \neq p (G \cap H) \subseteq p (G) \cap p (H), \subseteq p (H) \cap c l^{♢} (p (G)), = (p (H) \cap ω_{g^{‵}}) \cap c l^{♢} (p (G)), \subseteq p (H) \cap G^{‵} .$ Thus, $p (H) \cap G^{‵} \neq ϕ$ and $p (t) \in c l^{♢} (p (H))$ . Hence, $p (c l^{♢} (H)) \subseteq c l^{♢} (p (H))$ . □

Theorem 36

Consider $p : T \to T^{‵}$ . If $p (ω_{g}) \subseteq ω_{g^{‵}}$ , then the next are identical:

$p$ is a strong $GP$ -continuity;
For every $t \in T$ and a $(g^{‵}, P^{‵})$ -open set $G^{‵} : p (t) \in G^{‵}$ , there is a $(g, P)$ -open set $G : t \in G$ satisfies $c l^{♢} (p (c l^{♢} (G))) \cap ω_{g^{‵}} \subseteq G^{‵};$
For every $t \in T$ and a $(g^{‵}, P^{‵})$ -closed set $H^{‵} : p (t) \notin H^{‵}$ , there is a $(g, P)$ -open set $G : t \in G$ and a $(g^{‵}, P^{‵})$ -open set $G^{‵}$ satisfies $H^{‵} \cap ω_{g^{‵}} \subseteq G^{‵}$ , and $p (c l^{♢} (G)) \cap G^{‵} = ϕ;$
For every $t \in T$ and a $(g^{‵}, P^{‵})$ -closed set $H^{‵} : p (t) \notin H^{‵}$ , there is a $(g, P)$ -open set $G : t \in G$ and a $(g^{‵}, P^{‵})$ -open set $G^{‵}$ satisfies $H^{‵} \cap ω_{g^{‵}} \subseteq G^{‵}$ , and $p (G) \cap G^{‵} = ϕ$ .

Proof

$(i) \Rightarrow (i i)$ . For $t \in T$ , let $G^{‵}$ be a $(g^{‵}, P^{‵})$ -open set such that $p (t) \in G^{‵}$ . So, there is a $(g, P)$ -open set $G : t \in G$ satisfies $c l^{♢} (p (G)) \cap ω_{g^{‵}} \subseteq G^{‵}$ . From Theorem 35, we get $p (c l^{♢} (G)) \subseteq c l^{♢} (p (G))$ . Hence, $p (c l^{♢} (G)) \cap ω_{g^{‵}} \subseteq G^{‵}$ .

$(i i) \Rightarrow (i i i)$ . For $t \in T$ , let $H^{‵}$ be a $(g^{‵}, P^{‵})$ -closed set such that $p (t) \notin H^{‵}$ . Since $p (t) \in (X^{‵} ∖ H^{‵})$ and $(X^{‵} ∖ H^{‵})$ is a $(g^{‵}, P^{‵})$ -open set, there is a $(g, P)$ -open set $G : t \in G$ satisfies $c l^{♢} (p (c l^{♢} (G))) \cap ω_{g^{‵}} \subseteq (X^{‵} ∖ H^{‵})$ . Put $G^{‵} = (X^{‵} ∖ c l^{♢} (p (c l^{♢} (G))))$ . Hence, $G^{‵}$ is a $(g^{‵}, P^{‵})$ -open set satisfies $H^{‵} \cap ω_{g^{‵}} \subseteq G^{‵}$ , and $p (c l^{♢} (G)) \cap G^{‵} = ϕ$ .

$(i i i) \Rightarrow (i v)$ . It is clear.

$(i v) \Rightarrow (i)$ . Let $t \in T$ and $G^{‵}$ be a $(g^{‵}, P^{‵})$ -open set with $p (t) \in G^{‵}$ . Hence, $(X^{‵} ∖ G^{‵})$ is a $(g^{‵}, P^{‵})$ -closed set with $p (t) \notin (X^{‵} ∖ G^{‵})$ . So, there is a $(g, P)$ -open set $G : t \in G$ and $(g^{‵}, P^{‵})$ -open set $W$ satisfy $(X^{‵} ∖ G^{‵}) \cap ω_{g^{‵}} \subseteq W$ and $p (G) \cap W = ϕ$ . Hence, $c l^{♢} (p (G)) \cap ω_{g^{‵}} \subseteq c l^{♢} ((X^{‵} ∖ G^{‵})) \cap ω_{g^{‵}} = (X^{‵} ∖ G^{‵}) \cap ω_{g^{‵}} \subseteq G^{‵} .$ Therefore, $p$ is a strong $GP$ -continuous. □

Corollary 37

Consider $p : T \to T^{‵}$ . If $T^{‵}$ is strong, then the next are identical:

$p$ is a strong $GP$ -continuity;
For every $t \in T$ and a $(g^{‵}, P^{‵})$ -open set $G^{‵} : p (t) \in G^{‵}$ , there is a $(g, P)$ -open set $G : t \in G$ satisfies $c l^{♢} (p (c l^{♢} (G))) \subseteq G^{‵};$
For every $t \in T$ and a $(g^{‵}, P^{‵})$ -closed set $H^{‵} : p (t) \notin H^{‵}$ , there is a $(g, P)$ -open set $G : t \in G$ and a $(g^{‵}, P^{‵})$ -open set $G^{‵}$ satisfies $H^{‵} \subseteq G^{‵}$ and $p (c l^{♢} (G)) \cap G^{‵} = ϕ;$
For every $t \in T$ and a $(g^{‵}, P^{‵})$ -closed set $H^{‵} : p (t) \notin H^{‵}$ , there is a $(g, P)$ -open set $G : t \in G$ and a $(g^{‵}, P^{‵})$ -open set $G^{‵}$ satisfies $H^{‵} \subseteq G^{‵}$ and $p (G) \cap G^{‵} = ϕ$ . Hence, $G^{‵}$ is satisfies $H^{‵} \cap ω_{g^{‵}} \subseteq G^{‵}$ , and $p (c l^{♢} (G)) \cap G^{‵} = ϕ$ .

Definition 38

Let $(T, g, P)$ and $(T^{‵}, g^{‵}, P^{‵})$ be two $GPT$ spaces. Then, $p : T \to T^{‵}$ is named strongly $GP$ - $θ$ -continuous, if for every $t \in T$ and every $(g^{‵}, P^{‵})$ -open set $G^{‵} : p (t) \in G^{‵}$ , there existsa $(g, P)$ -openset $G : t \in G$ satisfies $p (c l^{♢} (G)) \subseteq G^{‵}$ .

The next result studies the relationship between the notions of a strong $GP$ -continuous function and a strongly $GP$ - $θ$ -continuous.

Theorem 39

If $p : T \to T^{‵}$ is a strong $GP$ -continuous and $T^{‵}$ is strong, then $p$ is strongly $GP$ - $θ$ -continuous.

Proof

It automatically from Corollary 37. □

There is a strongly $GP$ - $θ$ -continuous function, which is not strong $GP$ -continuous; the following example makes that clear.

Example 40

Let $(T, g, P)$ be a $GPT$ space, where $T = {t_{1}, t_{2}, t_{3}}$ , $g = {ϕ, {t_{1}}}$ , and $P = {ϕ, {t_{1}}, {t_{2}}, {t_{1}, t_{2}}}$ . Suppose that $(T^{‵}, g^{‵}, P^{‵})$ is a $GPT$ space, where $T^{‵} = {a, b, c}$ , $g^{‵} = {ϕ, {a}, T^{‵}}$ , and $P^{‵} = 2^{T^{‵}} ∖ T^{‵}$ . Define $p : T \to T^{‵}$ by $p (t_{1}) = p (t_{2}) = p (t_{3}) = a .$ Thus, $p$ is strongly $GP$ - $θ$ -continuous function.

However, it is not strong $GP$ -continuous since $c l^{♢} (p ({t_{1}})) = T^{‵} ⊈ {a}$ .

Definition 41

Let $(T, g, P)$ and $(T^{‵}, g^{‵}, P^{‵})$ be two $GPT$ spaces. Then, $p : T \to T^{‵}$ is named super $GP$ -continuous, if for every $t \in T$ and every $(g^{‵}, P^{‵})$ -open set $G^{‵} : p (t) \in G^{‵}$ , there existsa $(g, P)$ -open set $G : t \in G$ satisfies $p (i_{g} (c l^{♢} (G))) \subseteq G^{‵}$ .

The connection between the previous types of continuous functions and the $GP$ -continuous function is qualified in the following.

Theorem 42

Every strongly $GP$ - $θ$ -continuous function is super $GP$ -continuous.

Proof

Let $p : (T, g, P) \to (T^{‵}, g^{‵}, P^{‵})$ be strongly $GP$ - $θ$ -continuous. Then, for every $t \in T$ and every $(g^{‵}, P^{‵})$ -open set $G^{‵} : p (t) \in G^{‵}$ , there existsa $(g, P)$ -open set $G : t \in G$ satisfies $p (c l^{♢} (G)) \subseteq G^{‵}$ . But $i_{g} (c l^{♢} (G)) \subseteq c l^{♢} (G)$ . Thus, $p (i_{g} (c l^{♢} (G))) \subseteq G^{‵}$ . Therefore, $p$ is super $GP$ -continuous. □

Theorem 43

Every super $GP$ -continuous function is $GP$ -continuous.

Proof

Let $p : (T, g, P) \to (T^{‵}, g^{‵}, P^{‵})$ be super $GP$ -continuous. Then, for every $t \in T$ and every $(g^{‵}, P^{‵})$ -open set $G^{‵} : p (t) \in G^{‵}$ , there existsa $(g, P)$ -open set $G : t \in G$ satisfies $p (i_{g} (c l^{♢} (G))) \subseteq G^{‵}$ . Hence, $p (i_{g} (G)) \subseteq G^{‵}$ . Since $t \in p^{- 1} (G^{‵})$ , $p^{- 1} (G^{‵}) \subseteq G$ . Thus, $i_{g} (G) \subseteq p^{- 1} (G^{‵}) \subseteq G$ . Therefore, $p^{- 1} (G^{‵})$ is $(g, P)$ -open. Hence, $p$ is $GP$ -continuous. □

Example 44

Let $(T, g, P)$ be a $GPT$ space, where $T = {t_{1}, t_{2}, t_{3}}$ , $g = {ϕ, {t_{1}, t_{3}}, T}$ , and $P = 2^{T} ∖ T$ . Suppose that $(T^{‵}, g^{‵}, P^{‵})$ is a $GPT$ space, where $T^{‵} = {a, b, c}$ , $g^{‵} = {ϕ, {a}}$ , and $P^{‵} = 2^{T^{‵}} ∖ T^{‵}$ . Define $p : T \to T^{‵}$ by $p (t_{1}) = p (t_{3}) = a, and p (t_{3}) = b .$ Thus, $p$ is $GP$ -continuous function.

However, it is not super $GP$ -continuous since $p (i_{g} (c l^{♢} ({t_{1}, t_{3}}))) = p (T) ⊈ {a}$ .

3.1

3.1 Weakly $GP$ -closed and $GP$ -open functions

In this part, we introduce the notions of “weakly $GP$ -closed function” and “ $GP$ -regular space”. These two concepts are useful tools to show the converse of the relationship that we gave in Theorem 39. This result studies the connection between the notions of “strong $GP$ -continuous function” and “strongly $GP$ - $θ$ -continuous”.

Definition 45

Let $(T, g, P)$ and $(T^{‵}, g^{‵}, P^{‵})$ be two $GPT$ spaces. Then, $p : T \to T^{‵}$ is named weakly $GP$ -closed, if for every $(g, P)$ -closed set $F$ , we have $c l^{♢} (p (i_{g} (F))) \subseteq p (F)$ .

Proposition 46

If $p$ is a weakly $GP$ -closed, then for each $(g, P)$ -open set $G$ , we have $c l^{♢} (p (i_{g} (G))) \subseteq p (c l^{♢} (G))$ .

Proof

Since $c l^{♢} (G)$ is $(g, P)$ -closed set, $G \subseteq c l^{♢} (G)$ , and $p$ is a weakly $GP$ -closed. Then, $c l^{♢} (p (i_{g} (G))) \subseteq p (c l^{♢} (G))$ . □

Theorem 47

If $p$ is a weakly $GP$ -closed and strongly $GP$ - $θ$ -continuous, then $p$ is strong $GP$ -continuous.

Proof

For $t \in T$ , let $G^{‵}$ be a $(g^{‵}, P^{‵})$ -open : $p (t) \in G^{‵}$ . Thus, there existsa $(g, P)$ -open : $t \in G$ satisfies $p (c l^{♢} (G)) \subseteq G^{‵}$ . By Proposition 46, we get $c l^{♢} (p (G)) \cap ω_{g^{‵}} \subseteq p (c l^{♢} (G)) \cap ω_{g^{‵}} \subseteq G^{‵} .$ Hence, by Theorem 36 (ii), $p$ is strong $GP$ -continuous. □

Definition 48

Let $(T, g, P)$ be a $GPT$ space. Then, $T$ is called $GP$ -regular on $ω_{g}$ if for $t \in ω_{g}$ and a $(g, P)$ -closed set $F : t \notin F$ , there exist $G_{1}, G_{2} \in g$ satisfies $t \in G_{1}, F \cap ω_{g} \subseteq G_{2}$ and $G_{1} \cap G_{2} = ϕ$ .

Proposition 49

Let $(T, g, P)$ be a $GPT$ space. Then, $T$ is $GP$ -regular iff for $t \in ω_{g}$ and a $(g, P)$ -open set $G : t \in G$ , there exists a $(g, P)$ -open set $H : t \in H$ satisfies $t \in H \subseteq c l^{♢} (H) \cap ω_{g} \subseteq G$ .

Proof

It is direct from Definition 48 and considering $H = (T ∖ F)$ . □

Theorem 50

If $p$ is a strongly $GP$ - $θ$ -continuous and $T^{‵}$ is $GP$ -regular, then $p$ is strong $GP$ -continuous.

Proof

For $t \in T$ , let $G^{‵}$ be a $(g^{‵}, P^{‵})$ -open set $: p (t) \in G_{1}^{‵}$ . Since $T^{‵}$ is $GP$ -regular, there is a $(g^{‵}, P^{‵})$ -open set $G_{2}^{‵} : p (t) \in G_{2}^{‵}$ satisfies $p (t) \in G_{2}^{‵} \subseteq c l^{♢} (G_{2}^{‵}) \cap ω_{g^{‵}} \subseteq G_{1}^{‵} .$ Since $p$ is a strongly $GP$ - $θ$ -continuous, there existsa $(g, P)$ -open $G : t \in G$ satisfy $p (c l^{♢} (G)) \subseteq G_{2}^{‵}$ . Hence, $c l^{♢} (p (c l^{♢} (G))) \cap ω_{g^{‵}} \subseteq c l^{♢} (G_{2}) \cap ω_{g^{‵}} \subseteq G_{1}^{‵} .$ By using Theorem 36 (ii), $p$ is strong $GP$ -continuous. □

Corollary 51

Let $(T, g, P)$ and $(T^{‵}, g^{‵}, P^{‵})$ be two $GPT$ spaces. Consider $p : T \to T^{‵}$ . If $T^{‵}$ is $GP$ -regular and strong, then the next are identical:

$p$ is a strong $GP$ -continuous;
$p$ is a strongly $GP$ - $θ$ -continuous;
$p$ is a $GP$ -continuous.

Proof

$(i) \Rightarrow (i i)$ . It is directly from Theorem 39.

$(i i) \Rightarrow (i i i)$ . It comes directly from Theorems 42 and 43.

$(i i i) \Rightarrow (i)$ . It comes from the $GP$ -regularity of the $T^{‵}$ Theorem 36 (iv). □

Definition 52

Let $(T, g, P)$ and $(T^{‵}, g^{‵}, P^{‵})$ be two $GPT$ spaces. Consider $p : T \to T^{‵}$ . Then, $p$ is named $GP$ -open function, if for all $(g, P)$ -open set $G$ in $T$ , we have $p (G)$ is $(g^{‵}, P^{‵})$ -open.

Theorem 53

If $p : T \to T^{‵}$ is $GP$ -open and strong $GP$ -continuous with $p (ω_{g}) = ω_{g^{‵}}$ , then $T^{‵}$ is $GP$ -regular.

Proof

Suppose $s \in ω_{g^{‵}}$ and $G^{‵}$ is any $(g^{‵}, P^{‵})$ -open such that $s \in G^{‵}$ . Put $p (t) = s$ for $t \in T$ . But $p$ is strong $GP$ -continuous. Thus, there existsa $(g, P)$ -open set $G : t \in G$ satisfies $c l^{♢} (p (G)) \cap ω_{g^{‵}} \subseteq G^{‵}$ . Also, $p$ is $GP$ -open, and hence $p (G)$ is a $(g^{‵}, P^{‵})$ -open such that $s \in p (G)$ . Therefore, $p (G) = p (G) \cap ω_{g^{‵}} \subseteq c l^{♢} (p (G)) \cap ω_{g^{‵}} \subseteq G^{‵} .$ Thus, by Proposition 49, we have $T^{‵}$ is $GP$ -regular. □

4 $GP$ -connected and $GP$ -hyperconnected spaces

This part is an application of the notions studied in previous sections. We will study the preservation of the notions of “ $GP$ -connected” and “ $GP$ -hyperconnected” by different types of generalized primal continuous functions.

Definition 54

Let $(T, g, P)$ be a $GPT$ space. Then, $T$ is named $GP$ -connected space if there are no non-empty disjoint $(g, P)$ -open sets $G$ and $H$ satisfies $T = G \cup H$ .

Definition 55

Let $(T, g, P)$ be a $GPT$ space. Then, $T$ is named:

$GP$ - $α$ -connected space if there are no non-empty disjoint $(g, P)$ - $α$ -open sets $G$ and $H$ satisfies $T = G \cup H$ .
$GP$ - $β$ -connected space if there are no non-empty disjoint $(g, P)$ - $β$ -open sets $G$ and $H$ satisfies $T = G \cup H$ .
$GP$ -semi-connected space if there are no non-empty disjoint $(g, P)$ -semi-open sets $G$ and $H$ satisfies $T = G \cup H$ .
$GP$ -pre-connected space if there are no non-empty disjoint $(g, P)$ -pre-open sets $G$ and $H$ satisfies $T = G \cup H$ .

Definition 56

Let $(T, g, P)$ and $(T^{‵}, g^{‵}, P^{‵})$ be two $GPT$ spaces. Then, $p : T \to T^{‵}$ is named:

Contra $GP$ -continuous if for every $(g^{‵}, P^{‵})$ -open set $G^{‵} \subseteq T^{‵}$ , $p^{- 1} (G^{‵})$ is $(g, P)$ -closed in $T$ .
Contra $GP$ - $α$ -continuous if for every $(g^{‵}, P^{‵})$ -open set $G^{‵} \subseteq T^{‵}$ , $p^{- 1} (G^{‵})$ is $(g, P)$ - $α$ -closed in $T$ .
Contra $GP$ -semi-continuous if for every $(g^{‵}, P^{‵})$ -open set $G^{‵} \subseteq T^{‵}$ , $p^{- 1} (G^{‵})$ is $(g, P)$ -semi-closed in $T$ .
Contra $GP$ -pre-continuous if for every $(g^{‵}, P^{‵})$ -open set $G^{‵} \subseteq T^{‵}$ , $p^{- 1} (G^{‵})$ is $(g, P)$ -pre-closed in $T$ .
Contra $GP$ - $β$ -continuous if for every $(g^{‵}, P^{‵})$ -open set $G^{‵} \subseteq T^{‵}$ , $p^{- 1} (G^{‵})$ is $(g, P)$ - $β$ -closed in $T$ .

Theorem 57

Let $p : (T, g, P) \to (T^{‵}, g^{‵}, P^{‵})$ be a contra $GP$ - $α$ -continuous onto function. Consider $T$ as $GP$ - $α$ -connected space. Then, $T^{‵}$ is $GP$ -connected space.

Proof

Consider $p : (T, g, P) \to (T^{‵}, g^{‵}, P^{‵})$ as a contra $GP$ - $α$ -continuous onto function. Let $T$ be $GP$ - $α$ -connected space. Suppose that $T^{‵}$ is not $GP$ -connected space. Thus, there exist non-empty disjoint $(g^{‵}, P^{‵})$ -open sets $G^{‵}$ and $H^{‵}$ satisfies $G^{‵} \cup H^{‵} = T^{‵}$ . Therefore, we can say that $G^{‵}, H^{‵}$ are $(g^{‵}, P^{‵})$ -closed sets. Hence, $p^{- 1} (G^{‵})$ and $p^{- 1} (H^{‵})$ are $(g, P)$ - $α$ -open since $p$ is contra $GP$ - $α$ -continuous. But $p^{- 1} (H^{‵}) \cap p^{- 1} (G^{‵}) = ϕ$ , and $p^{- 1} (H^{‵}) \cup p^{- 1} (G^{‵}) = T$ . Thus, $T$ is not $GP$ - $α$ -connected space, which is a contradiction. Hence, $T^{‵}$ is $GP$ -connected space. □

Corollary 58

Let $p : (T, g, P) \to (T^{‵}, g^{‵}, P^{‵})$ be a contra $GP$ - $β$ -continuous (resp. $GP$ -semi-continuous, $GP$ -pre-continuous) onto function. Consider $T$ as $GP$ - $β$ -connected (resp. $GP$ -semi-connected, $GP$ -pre-connected) space. Then, $T^{‵}$ is $GP$ -connected space.

According to Theorem 13 and Corollary 14, we conclude that:

$g$ -open $\Rightarrow$ $(g, P)$ - $α$ -open $\Rightarrow$ $(g, P)$ -pre-open $\Rightarrow$ $(g, P)$ - $β$ -open.
$(g, P)$ - $α$ -open $\Rightarrow$ $(g, P)$ -semi-open $\Rightarrow$ $(g, P)$ - $β$ -open.

However, the relationship between the types of $GP$ -connected spaces takes the following path:

Theorem 59

Let $(T, g, P)$ be a $GPT$ space. Then, we have:

$T$ is $GP$ - $β$ -connected $\Rightarrow$ $GP$ -semi-connected $\Rightarrow$ $GP$ - $α$ -connected $\Rightarrow$ $GP$ -connected.
$T$ is $GP$ - $β$ -connected $\Rightarrow$ $GP$ -pre-connected $\Rightarrow$ $GP$ - $α$ -connected.

Proof

(i) (1) $GP$ - $β$ -connected $\Rightarrow$ $GP$ -semi-connected: Let $T$ be $GP$ - $β$ -connected, but not $GP$ -semi-connected. Hence, there are non-empty disjoint $(g, P)$ -semi-open sets $G$ and $H$ in $T$ satisfies $G \cup H = T$ which implies that there are non-empty disjoint $(g, P)$ - $β$ -open sets $G$ and $H$ in $T$ satisfies $G \cup H = T$ . Hence, $T$ is not $GP$ - $β$ -connected, which is contradiction. Therefore, $T$ is $GP$ -semi-connected.

(2) $GP$ -semi-connected $\Rightarrow$ $GP$ - $α$ -connected: Let $T$ be $GP$ -semi- connected, but not $GP$ - $α$ -connected. Hence, there are non-empty disjoint $(g, P)$ - $α$ -open sets $G$ and $H$ in $T$ satisfies $G \cup H = T$ which implies that there are non-empty disjoint $(g, P)$ -semi-open sets $G$ and $H$ in $T$ satisfies $G \cup H = T$ . Hence, $T$ is not $GP$ -semi-connected, which is contradiction. Therefore, $T$ is $GP$ - $α$ -connected.

(3) $GP$ - $α$ -connected $\Rightarrow$ $GP$ -connected: Let $T$ be $GP$ - $α$ -connected, but not $GP$ -connected. Hence, there are non-empty disjoint $(g, P)$ -open sets $G$ and $H$ in $T$ satisfies $G \cup H = T$ which implies that there are non-empty disjoint $(g, P)$ - $α$ -open sets $G$ and $H$ in $T$ satisfies $G \cup H = T$ . Hence, $T$ is not $GP$ - $α$ -connected, which is contradiction. Therefore, $T$ is $GP$ -connected.

By using the same technique, we can show the implications in (ii). □

Definition 60

Let $(T, g, P)$ be $GPT$ space. Then, $T$ is named:

$(g, P)$ -hyperconnected if each $(g, P)$ -open set $G$ of $T$ is $(g, P)$ -dense.
$(g, P)$ - $α$ -hyperconnected if each $(g, P)$ - $α$ -open set $G$ of $T$ is $(g, P)$ -dense.
$(g, P)$ -semi-hyperconnected if each $(g, P)$ -semi-open set $G$ of $T$ is $(g, P)$ -dense.
$(g, P)$ -pre-hyperconnected if each $(g, P)$ -pre-open set $G$ of $T$ is $(g, P)$ -dense.
$(g, P)$ - $β$ -hyperconnected if each $(g, P)$ - $β$ -open set $G$ of $T$ is $(g, P)$ -dense.

Theorem 61

Let $(T, g, P)$ be a $GPT$ space. Then, we have:

$T$ is $GP$ - $β$ -hyperconnected $\Rightarrow$ $GP$ -semi-hyperconnected $\Rightarrow$ $GP$ - $α$ -hyperconnected $\Rightarrow$ $GP$ -hyperconnected.
$T$ is $GP$ - $β$ -hyperconnected $\Rightarrow$ $GP$ -pre-hyperconnected $\Rightarrow$ $GP$ - $α$ -hyperconnected.

Proof

(i) (1) $GP$ - $β$ -hyperconnected $\Rightarrow$ $GP$ -semi-hyperconnected: Let $T$ be $GP$ - $β$ -hyperconnected, but not $GP$ -semi-hyperconnected. Hence, there is non-empty $(g, P)$ -semi-open set $M$ in $T$ satisfies $c l^{♢} (M) \neq T$ which implies that there is non-empty $(g, P)$ - $β$ -open set $M$ in $T$ satisfies $c l^{♢} (M) \neq T$ . Hence, $T$ is not $GP$ - $β$ -hyperconnected, which is contradiction. Therefore, $T$ is $GP$ -semi-hyperconnected.

(2) $GP$ -semi-hyperconnected $\Rightarrow$ $GP$ - $α$ -hyperconnected: Let $T$ be $GP$ -semi-hyperconnected, but not $GP$ - $α$ -hyperconnected. Hence, there is non-empty $(g, P)$ - $α$ -open set $M$ in $T$ satisfies $c l^{♢} (M) \neq T$ which implies that there is non-empty $(g, P)$ -semi-open set $M$ in $T$ satisfies $c l^{♢} (M) \neq T$ . Hence, $T$ is not $GP$ -semi-hyperconnected, which is contradiction. Therefore, $T$ is $GP$ - $α$ -hyperconnected.

(3) $GP$ - $α$ -hyperconnected $\Rightarrow$ $GP$ -hyperconnected: Let $T$ be $GP$ - $α$ -hyperconnected, but not $GP$ -hyperconnected. Hence, there is non-empty $(g, P)$ -open set $M$ in $T$ satisfies $c l^{♢} (M) \neq T$ which implies that there is non-empty $(g, P)$ - $α$ -open sets $M$ in $T$ satisfy $c l^{♢} (M) \neq T$ . Hence, $T$ is not $GP$ - $α$ -hyperconnected, which is contradiction. Therefore, $T$ is $GP$ -hyperconnected.

By using the same technique, we can show the implications in (ii). □

5 Conclusion

Recently, Al-Saadi and Al-Malki (Al-Saadi and Al-Malki, 2023) presented a new generalized space. This space is characterized by all the nice features of $GT$ spaces in the sense of primal sets. Some operators are defined in this space. Their behaviours are studied carefully.

In this article, we give attention to the notion of “continuity”. So, we defined some type of generalized primal continuous function. We can summarize the results as follows:

If $g$ and $g^{‵}$ are generated by the neighbourhood system, then every $GPN$ -continuous function is $GP$ -continuous; see Theorem 24. The opposite of this is not always true; see Example 25. However, when we consider $ψ^{‵} = ψ_{g^{‵}}^{‵}$ , we can show that every $GP$ -continuous function is $GPN$ -continuous; see Theorem 26. Moreover, by using a generalized primal neighbourhood system, $ψ (c l^{♢}, g) \in Ψ (T)$ , we conclude that every $GPN$ -continuous is $GP$ - $θ$ -continuous; see Theorems 27 and 29. Also, every $GP$ -continuous function is almost $GP$ -continuous; see Theorem 31. To make the opposite of this relation true, we use the technique of Császár, which changes the generalized topology of other generalized topologies, which is weaker than it. So, Proposition 33 shows that the two notions coincide. All these results are presented in Section 2.

Section 3 discusses the relationship between a strong $GP$ -continuous function and several different types, as follows: Every strong $GP$ -continuous function is strongly $GP$ - $θ$ -continuous when the space $T^{‵}$ is strong; see Theorem 39. Example 40 shows that the converse is not true. Moreover, every strongly $GP$ - $θ$ -continuous is super $GP$ -continuous, see Theorem 42 and every super $GP$ -continuous is $GP$ -continuous, see Theorem 43. Example 44 shows that the converse need not be true. To prove the converse of Theorem 39, we presented the notion of “weakly $GP$ -closed function” or the notion of “ $GP$ -regular”. So, Theorem 47 shows that every strongly $GP$ - $θ$ -continuous is strong $GP$ -continuous if $p$ is a weakly $GP$ -closed function, while Theorem 50 shows that every strongly $GP$ - $θ$ -continuous is strong $GP$ -continuous if the $GPT$ space $T^{‵}$ is $GP$ -regular space.

To make it easier for the reader, we present these results in Fig. 1 as follows:

Finally, Corollary 51 shows that the notion of “strong $GP$ -continuous function” coincides with the notion “ $GP$ -continuous” when the space $T^{‵}$ is both strong and $GP$ -regular.

The relationship among several kinds of GP -continuous function.

CRediT authorship contribution statement

Hanan Al-Saadi: Methodology, Writing – original draft, Writing – review & editing. Huda Al-Malki: Methodology, Writing – original draft, Writing – review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References

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Introduction

Literature review
GPT spaces

Main results

G P -continuous function
GP - θ -continuous function
Almost GP -continuity

Strong GP -continuity

Weakly GP -closed and GP -open functions

GP -connected and GP -hyperconnected spaces

Conclusion

CRediT authorship contribution statement

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[1] Acharjee S., Özkoç M., Issaka F., . Primal topological spaces. 2022. ArXiv
[CrossRef]

[2] AL-Omari A., Acharjee S., Özkoç M., . A new operator of primal topological spaces. 2022. ArXiv
[CrossRef]

[3] Al-Omari A., Noiri T., . On $Ψ_{*}$ -operator in ideal $m$ -spaces. Bol. Soc. Parana. Math.. 2012;30:53-66.
[CrossRef] [Google Scholar]

[4] Al-Saadi H., Al-Malki H., . Generalized primal topological spaces. AIMS Math.. 2023;8:24162-24175.
[CrossRef] [Google Scholar]

[5] Al-Saadi H., Al-Malki H., . Categories of open sets in generalized primal topological spaces. Mathematics. 2024;12:207.
[CrossRef] [Google Scholar]

[6] Al-shami T., Ameen Z., Abu-Gdairi R., Mhemdi A., . On primal soft topology. Mathematics. 2023;11:2329.
[CrossRef] [Google Scholar]

[7] Choquet G., . Sur les notions de filtre et de grille. C. R. Acad. Sci. Paris. 1947;224:171-173.
[Google Scholar]

[8] Császár Á., . Generalized open sets. Acta Math. Hungar.. 1997;75:65--87.
[CrossRef] [Google Scholar]

[9] Császár Á., . Generalized topology, generalized continuity. Acta Math. Hungar.. 2002;96:351--357.
[CrossRef] [Google Scholar]

[10] Császár Á., . Extremely disconnected generalized topologies. Ann. Univ. Sci. Budapest.. 2004;47:91-96.
[Google Scholar]

[11] Császár Á., . Separation axioms for generalized topologies. Acta Math. Hungar.. 2004;104:63--69.
[CrossRef] [Google Scholar]

[12] Császár Á., . Generalized open sets in generalized topologies. Acta Math. Hungar.. 2005;106:53-66.
[CrossRef] [Google Scholar]

[13] Császár Á., . $δ$ - And $θ$ -modifications of generalized topologies. Acta. Math. Hungar.. 2008;120:275-279.
[CrossRef] [Google Scholar]

[14] Császár Á., . Remarks on quasi topologyies. Acta. Math. Hungar.. 2008;119:197-200.
[CrossRef] [Google Scholar]

[15] Hatir E., Jafari S., . On some new classes of sets and a new decomposition of continuity via grills. J. Adv. Math. Stud.. 2010;3:33-40.
[Google Scholar]

[16] Janković D., Hamlett T., . New topologies from old via ideals. Amer. Math. Monthly. 1990;97:295-310.
[CrossRef] [Google Scholar]

[17] Jayanthi D., . Contra continuity on generalized topological spaces. Acta Math. Hungar.. 2012;137:263-271.
[CrossRef] [Google Scholar]

[18] Kim Y., Min W., . $R (g, g ‵)$ -Continuity on generalized topological spaces. Commun. Korean Math. Soc.. 2012;27:809-813.
[CrossRef] [Google Scholar]

[19] Kuratowski K., . Topology. Elsevier. 2014;Vol. I
[Google Scholar]

[20] Min W., . A note on $θ (g, g^{‵})$ -continuity in generalized topological spaces. Acta Math. Hungar.. 2009;125:387-393.
[CrossRef] [Google Scholar]

[21] Min W., . Almost continuity on generalized topological spaces. Acta Math. Hungar.. 2009;125:121-125.
[Google Scholar]

[22] Min W., Kim Y., . Some strong forms of $(g, g^{‵})$ -continuity on generalized topological spaces. Honam Math. J.. 2011;33
[CrossRef] [Google Scholar]

[23] Modak S., . Grill-filter space. J. Indian Math. Soc.. 2013;80:313-320.
[Google Scholar]

[24] Roy B., Mukherjee M., . On a typical topology induced by a grill. Soochow J. Math.. 2007;33:771-786.
[Google Scholar]

[25] Roy B., Mukherjee M., Ghosh S., . On a new operator based on grill and its associated topology. Arab J. Math. Sc.. 2008;14:21-32.
[Google Scholar]

Strong GP -continuity and weakly GP -closed functions on GPT spaces

Abstract

Keywords

54A05

54C10

54A10

GP-continuous

GPN-continuous

Strong GP-continuous

Weakly GP-closed

GP-connected

GP-hyperconnected

1 Introduction

1.1 Literature review

1.2 GPT spaces

2 Main results

2.1 G P -continuous function

2.2 GP - θ -continuous function

2.3 Almost GP -continuity

3 Strong GP -continuity

3.1 Weakly GP -closed and GP -open functions

4 GP -connected and GP -hyperconnected spaces

5 Conclusion

CRediT authorship contribution statement

Declaration of competing interest

References

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Strong $GP$ -continuity and weakly $GP$ -closed functions on $GPT$ spaces

1.2 $GPT$ spaces

2.1 $G P$ -continuous function

2.2 $GP$ - $θ$ -continuous function

2.3 Almost $GP$ -continuity

3 Strong $GP$ -continuity

3.1 Weakly $GP$ -closed and $GP$ -open functions

4 $GP$ -connected and $GP$ -hyperconnected spaces