New types of nearly open sets in topological spaces and their applications

Abstract

In the current study, the idea of $S_{\psi }^{\phi }$-open sets is used to introduce some new types of nearly open sets in topological space, where $\psi =\{\theta ,\delta ,\tau \}$ and $\phi =\{\theta ,\delta ,\tau ,\alpha ,a\}$, thereby illustrating some of their basic qualities and the links between them. Some characterizations, topological properties, mappings, and separation axioms of these new concepts are also presented.

KEYWORDS:

• $S_{\psi }^{\phi }$-open
• $S_{\psi }^{\phi }$-continuity
• $S_{\psi }^{\phi }$-T_i-spaces
• $S_{\psi }^{\phi }$-regular
• $S_{\psi }^{\phi }$-normal spaces

Mathematical subject classifications:

• 54C99
• 54D10
• 54D15
• 54D30
• 54F05

1. Introduction

The concept of nearly open sets is painstakingly regarded as a crucial notion in topology, with vast applications in real life. Besides, various researchers have presented numerous studies with regard to this class of open sets, in which dissimilar modified forms of mappings, and separation axioms to mention a few are suggested. Indeed, among the most important types of nearly open sets within topological spaces are the so-called $\psi$-semi-open sets, which first included $\tau$-semiopen, which was studied by Levine [1], then followed by Park et al. [2] and Caldas et al. [3], who presented the concepts of δ-semiopen and θ-semiopen, respectively, using the concept of δ-open and θ-open [4]. Later on, the concept of Z_α-open [5] (resp. θ*-semiopen [6]) was presented by taking the union of α-open in [7] with δ-semiopen in [2] (rep. θ-semiopen in [3]). We also mention independent concepts of the $\psi$-semi-open sets, which are called the $\psi$-pre-open concepts containing $\tau$-preopen [8], δ-preopen [9], and θ-preopen [10]. The topological concepts in [11–19] were deduced by using the union among different types of $\psi$-semi-open and $\psi$-pre-open, where $\psi =\{\theta ,\delta ,\tau \}$.

However, the main ambition of the current study is to propose some new concepts of nearly open sets by utilizing the notion of $S_{\psi }^{\phi }$-open sets, with $\psi =\{\theta ,\delta ,\tau \}$ and $\phi =\{\theta ,\delta ,\tau ,\alpha ,a\}$. Moreover, some of the well-known results, including those presented by [1–3] and [5,6] would be deduced from the new notions which corollaries may be accounted for the results of this work. Eventually, many characterizations of these notions would be presented via some concepts of the separation axioms and the topological mappings.

The current work is structured as follows: in Section 2 is set to present some preliminaries about nearly open sets in topological spaces, while Section 3 will be investigating the new fundamental ideas of $S_{\psi }^{\phi }$-open and $S_{\psi }^{\phi }$-closed sets. In Section 4, we give the application of the introduced concepts and establish a comparison study with regard to each subclass of $S_{\psi }^{\phi }$-open and $S_{\psi }^{\phi }$-closed sets. The section 5 gives some concluding points, together with a look-forward message.

2. Preliminaries

The topological spaces $\mathcal{X}$ and $\mathcal{Y}$ are assumed to be continuously specified in the form ($\mathcal{X}$, $\xi$) and ($\mathcal{Y}$, $\eta$), respectively. For any set $\mathcal{B}$ within the space $\mathcal{X}$, the notations ${\text{C}}_{\tau }$($\mathcal{B}$), ${\mathrm{I}}_{\tau }$($\mathcal{B}$), and $\mathcal{X}$ \ $\mathcal{B}$ indicate, the closure, the interior, and the complement of $\mathcal{B}$, respectively.

Definition 2.1.

A point $\upsilon \in \mathcal{X}$ is said a δ-adherent (resp. a θ-adherent) point of the subset $\mathcal{B}$ ⊆ $\mathcal{X}$ if $\mathcal{B}\cap$ ${\mathrm{I}}_{\tau }$(${\text{C}}_{\tau }$($\mathcal{W}$)) $\ne \mathrm{\varnothing }$ (resp. $\mathcal{B}\cap$ ${\text{C}}_{\tau }$($\mathcal{W}$) $\ne \mathrm{\varnothing }$), for any open set $\mathcal{W}$ and $\upsilon$ ∈ $\mathcal{W}$.

The set of δ-adherent (resp. θ-adherent) points of $\mathcal{B}$ is denoted by ${\text{C}}_{\delta }$($\mathcal{B}$) (resp. ${\text{C}}_{\theta }$($\mathcal{B}$)).

The subset $\mathcal{B}$ ⊆ $\mathcal{X}$ is referred to as a δ-closed (resp. a θ-closed) in the case of $\mathcal{B}$ = ${\text{C}}_{\delta }$($\mathcal{B}$) (resp. $\mathcal{B}$ = ${\text{C}}_{\theta }$($\mathcal{B}$)). The complement of the δ-closed (resp. the θ-closed) set is known as an δ-open (resp. a θ-open) set, namely, if $\mathcal{B}$ ⊆ $\mathcal{X}$ is the δ-closed (resp. the θ-closed) set, then $\mathcal{X}$\ $\mathcal{B}$ = $\mathcal{X}\mathrm{\setminus }{\text{C}}_{\delta }$($\mathcal{B}$) $={\text{I}}_{\delta }$($\mathcal{X}\mathrm{\setminus }\mathcal{B}$) (resp. $\mathcal{X}$\ $\mathcal{B}$ = $\mathcal{X}\mathrm{\setminus }{\text{C}}_{\theta }$($\mathcal{B}$) $={\text{I}}_{\theta }$($\mathcal{X}\mathrm{\setminus }\mathcal{B}$)) is the δ-open (resp. the θ-open).

Definition 2.2.

If the set $\mathcal{B}$ ⊆ $\mathcal{X}$, then $\mathcal{B}$ is explicitly known as [1–3,5–23]:

1. a-open if $\mathcal{B}$ ⊆ ${\mathrm{I}}_{\tau }$(${\text{C}}_{\tau }$(${\text{I}}_{\delta }$($\mathcal{B}$))),

2. α-open if $\mathcal{B}$ ⊆ ${\mathrm{I}}_{\tau }$(${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$($\mathcal{B}$)))

3. θ-semiopen if $\mathcal{B}$ ⊆ ${\text{C}}_{\tau }$(${\text{I}}_{\theta }$($\mathcal{B}$)),

4. δ-semiopen if $\mathcal{B}$ ⊆ ${\text{C}}_{\tau }$(${\text{I}}_{\delta }$($\mathcal{B}$)),

5. semiopen if $\mathcal{B}$ ⊆ ${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$($\mathcal{B}$)),

6. θ*-semiopen if $\mathcal{B}$ ⊆ ${\text{C}}_{\tau }$(${\text{I}}_{\theta }$($\mathcal{B}$)) ∪ ${\mathrm{I}}_{\tau }$(${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$($\mathcal{B}$))),

7. Z_α-open if $\mathcal{B}$ ⊆ ${\text{C}}_{\tau }$(${\text{I}}_{\delta }$($\mathcal{B}$)) ∪ ${\mathrm{I}}_{\tau }$(${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$($\mathcal{B}$))),

8. preopen if $\mathcal{B}$ ⊆ ${\mathrm{I}}_{\tau }$(${\text{C}}_{\tau }$($\mathcal{B}$)),

9. δ-preopen if $\mathcal{B}$ ⊆ ${\mathrm{I}}_{\tau }$(${\text{C}}_{\delta }$($\mathcal{B}$)),

10. θ-preopen if $\mathcal{B}$ ⊆ ${\mathrm{I}}_{\tau }$(${\text{C}}_{\theta }$($\mathcal{B}$)),

11. Y-open if $\mathcal{B}$ ⊆ ${\text{C}}_{\tau }$(${\text{I}}_{\theta }$($\mathcal{B}$)) ∪ ${\mathrm{I}}_{\tau }$(${\text{C}}_{\tau }$($\mathcal{B}$)),

12. Z-open if $\mathcal{B}$ ⊆ ${\text{C}}_{\tau }$(${\text{I}}_{\delta }$($\mathcal{B}$)) ∪ ${\mathrm{I}}_{\tau }$(${\text{C}}_{\tau }$($\mathcal{B}$)),

13. b-open or sp-open or γ-open if $\mathcal{B}$ ⊆ ${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$($\mathcal{B}$)) ∪ ${\mathrm{I}}_{\tau }$(${\text{C}}_{\tau }$($\mathcal{B}$)),

14. M-open if $\mathcal{B}$ ⊆ ${\text{C}}_{\tau }$(${\text{I}}_{\theta }$($\mathcal{B}$)) ∪ ${\mathrm{I}}_{\tau }$(${\text{C}}_{\delta }$($\mathcal{B}$)),

15. e-open if $\mathcal{B}$ ⊆ ${\text{C}}_{\tau }$(${\text{I}}_{\delta }$($\mathcal{B}$)) ∪ ${\mathrm{I}}_{\tau }$(${\text{C}}_{\delta }$($\mathcal{B}$)),

16. Z*-open if $\mathcal{B}$ ⊆ ${\mathrm{I}}_{\tau }$(${\text{C}}_{\delta }$($\mathcal{B}$)) ∪ ${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$($\mathcal{B}$)),

17. β-open if $\mathcal{B}$ ⊆ ${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$(${\text{C}}_{\tau }$($\mathcal{B}$))),

18. β*-open if $\mathcal{B}$ ⊆ ${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$(${\text{C}}_{\tau }$($\mathcal{B}$))) ∪ ${\mathrm{I}}_{\tau }$(${\text{C}}_{\delta }$($\mathcal{B}$)),

19. e*-open or δ-β-open if $\mathcal{B}$ ⊆ ${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$(${\text{C}}_{\delta }$($\mathcal{B}$))).

The complement of (α, Z_α, a, Y, Z, γ, Z*, M, e)-open, (θ, δ, τ, θ*)-semiopen, and (τ, δ, θ)-preopen are called (α, Z_α, a, Y, Z, γ, Z*, M, e)-closed, (θ, δ, τ, θ*)-semiclosed and (τ, δ, θ)-preclosed, corresponding to the citation [7,5,19,17,15,14,18,16,13,3,2,24], and [8–10], respectively.

The intersection of every (a, α)-closed, $\psi$-semi-closed, and $\psi$-pre-closed sets including $\mathcal{B}$ are called the (a, α)-closure, $\psi$-pre-closure, and $\psi$-semi-closure of $\mathcal{B}$, respectively, where $\psi =\{\theta ,\delta ,\tau \}$, and they are denoted by (${\text{C}}_a\left(\mathcal{B}\right)$, $\text{C}\left(\mathcal{B}\right)$), $\text{S}{\text{C}}_{\psi }\left(\mathcal{B}\right)$, and $\text{P}{\text{C}}_{\psi }\left(\mathcal{B}\right)$, respectively. The (a, α)-interior, $\psi$-semi-interior, and $\psi$-pre-interior of $\mathcal{B}$ is the union of every (a, α)-open, $\psi$-semiopen, and $\psi$-preopen sets included in B, and are further represented by (${\text{I}}_a\left(\mathcal{B}\right)$, $\text{I}\left(\mathcal{B}\right)$), $\text{S}{\text{I}}_{\psi }\left(\mathcal{B}\right)$, and $\text{P}{\text{I}}_{\psi }\left(\mathcal{B}\right)$, respectively.

Lemma 2.1.

The following characteristics apply to $\mathcal{B}$ ⊆ $\mathcal{X}$ and $\psi =\{\theta ,\delta ,\tau \}$:

1. $\text{S}{\text{C}}_{\psi }\left(\mathcal{B}\right)=\mathcal{B}\cup {\mathrm{I}}_{\tau }$(${\text{C}}_{\psi }$($\mathcal{B}$)) and $\text{S}{\text{I}}_{\psi }\left(\mathcal{B}\right)$ = ${\text{C}}_{\tau }$(${\text{I}}_{\psi }$($\mathcal{B}$)),

2. $\text{P}{\text{C}}_{\psi }\left(\mathcal{B}\right)=\mathcal{B}\cup {\text{C}}_{\tau }$(${\text{I}}_{\psi }$($\mathcal{B}$)), $\text{P}{\text{I}}_{\psi }\left(\mathcal{B}\right)=\mathcal{B}\cap {\mathrm{I}}_{\tau }$(${\text{C}}_{\psi }$($\mathcal{B}$)) and $\text{P}{\text{I}}_{\psi }$($\mathcal{X}\mathrm{\setminus }$ $\mathcal{B}$) = $\mathcal{X}\mathrm{\setminus }$ $\text{P}{\text{C}}_{\psi }\left(\mathcal{B}\right)$,

3. $\text{C}\left(\mathcal{B}\right)$ = $\mathcal{B}\cup {\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$(${\text{C}}_{\tau }$($\mathcal{B}$))), and $\text{I}\left(\mathcal{B}\right)=\mathcal{B}\cap {\mathrm{I}}_{\tau }$(${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$($\mathcal{B}$))),

4. ${\text{C}}_a\left(\mathcal{B}\right)$ =  $\mathcal{B}$∪ ${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$(${\text{C}}_{\delta }$($\mathcal{B}$))), and ${\text{I}}_a\left(\mathcal{B}\right)$ = $\mathcal{B}\cap {\mathrm{I}}_{\tau }$(${\text{C}}_{\tau }$(${\text{I}}_{\delta }$($\mathcal{B}$))).

Lemma 2.2.

The succeeding properties are satisfied for each $\mathcal{B}$ ⊆ $\mathcal{X}$:

1. ${\text{I}}_{\theta }$(${\text{I}}_{\theta }$($\mathcal{B}$)) = ${\text{I}}_{\theta }$(${\text{I}}_{\delta }$($\mathcal{B}$)) = ${\text{I}}_{\theta }$(${\mathrm{I}}_{\tau }$($\mathcal{B}$)) = ${\text{I}}_{\theta }$(${\text{I}}_a$($\mathcal{B}$)) = ${\text{I}}_{\theta }$($\text{I}$($\mathcal{B}$)) = ${\text{I}}_{\delta }$(${\text{I}}_{\theta }$($\mathcal{B}$)) = ${\mathrm{I}}_{\tau }$(${\text{I}}_{\theta }$($\mathcal{B}$)) = ${\text{I}}_{\theta }$($\mathcal{B}$),

2. ${\text{I}}_{\delta }$(${\text{I}}_{\delta }$($\mathcal{B}$)) = ${\text{I}}_{\delta }$(${\mathrm{I}}_{\tau }$($\mathcal{B}$)) = ${\text{I}}_{\delta }$(${\text{I}}_a$($\mathcal{B}$)) = ${\text{I}}_{\delta }$($\text{I}$($\mathcal{B}$)) = ${\mathrm{I}}_{\tau }$(${\text{I}}_{\delta }$($\mathcal{B}$)) = ${\text{I}}_{\delta }$($\mathcal{B}$),

3. ${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$(${\text{I}}_a$($\mathcal{B}$))) = ${\text{C}}_{\tau }$(${\text{I}}_{\delta }$($\mathcal{B}$)).

Proof.

3. Assume that $\mathcal{B}$ ⊆ $\mathcal{X}$, by using the Lemma 2.1, we deduce ${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$(${\text{I}}_a$($\mathcal{B}$))) = ${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$($\mathcal{B}$ ∩ ${\mathrm{I}}_{\tau }$(${\text{C}}_{\tau }$(${\text{I}}_{\delta }$($\mathcal{B}$))))) ⊆ ${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$(${\text{C}}_{\tau }$(${\text{I}}_{\delta }$($\mathcal{B}$)))) ⊆ ${\text{C}}_{\tau }$(${\text{I}}_{\delta }$($\mathcal{B}$)). Also, we have ${\text{C}}_{\tau }$(${\text{I}}_{\delta }$($\mathcal{B}$)) ⊆ ${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$(${\text{I}}_a$($\mathcal{B}$))), therefore, ${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$(${\text{I}}_a$($\mathcal{B}$))) = ${\text{C}}_{\tau }$(${\text{I}}_{\delta }$($\mathcal{B}$)).

3. ${\mathit{S}}_{\mathit{\psi }}^{\mathit{\phi }}$-open sets

Definition 3.1.

The subset $\mathcal{B}$ of $\mathcal{X}$ is referred to as

1. $S_{\psi }^{\phi }$-open sets iff $\mathcal{B}$ ⊆ $\text{P}{\text{C}}_{\psi }$(${\text{I}}_{\phi }$($\mathcal{B}$)),

2. $S_{\psi }^{\phi }$-closed sets iff $\text{P}{\text{I}}_{\psi }$(${\text{C}}_{\phi }$($\mathcal{B}$)) ⊆ $\mathcal{B}$,

where $\psi =\{\theta ,\delta ,\tau \}$ and $\phi =\{\theta ,\delta ,\tau ,\alpha ,a\}$.

The collection of $S_{\psi }^{\phi }$-open and $S_{\psi }^{\phi }$-closed sets are referenced by $S_{\psi }^{\phi }$O($\mathcal{X}$) and $S_{\psi }^{\phi }$C($\mathcal{X}$), respectively.

Now, we show the derivation of some resulting sets in the Table 1 as follows:

Table 1. Some results and properties of $S_{\psi }^{\phi }$-open sets.

	$\phi$
$\psi$	$\mathrm{\Theta }$	$\delta$	$\tau$	$a$	$\alpha$
$\theta$	$\mathcal{B}=\text{S}{\text{I}}_{\theta }\left(\mathcal{B}\right)$	$\mathcal{B}={\text{I}}_{\delta }\left(\mathcal{B}\right)\cup \text{S}{\text{I}}_{\theta }\left(\mathcal{B}\right)$	$\mathcal{B}={\text{I}}_{\tau }\left(\mathcal{B}\right)\cup \text{S}{\text{I}}_{\theta }\left(\mathcal{B}\right)$	$\mathcal{B}={\text{I}}_a\left(\mathcal{B}\right)\cup \text{S}{\text{I}}_{\theta }\left(\mathcal{B}\right)$	$\mathcal{B}=\text{I}\left(\mathcal{B}\right)\cup \text{S}{\text{I}}_{\theta }\left(\mathcal{B}\right)$
	$\theta$-semiopen	—	—	—	${\theta }^{\ast }$-semiopen
$\delta$	$\mathcal{B}=\text{S}{\text{I}}_{\theta }\left(\mathcal{B}\right)$	$\mathcal{B}=\text{S}{\text{I}}_{\delta }$($\mathcal{B})$	$\mathcal{B}={\text{I}}_{\tau }\left(\mathcal{B}\right)\cup \text{S}{\text{I}}_{\delta }\left(\mathcal{B}\right)$	$\mathcal{B}=\text{S}{\text{I}}_{\delta }\left(\mathcal{B}\right)$	$\mathcal{B}=\text{I}\left(\mathcal{B}\right)\cup \text{S}{\text{I}}_{\delta }\left(\mathcal{B}\right)$
	$\theta$-semiopen	$\delta$-semiopen	—	$\delta$-semiopen	Zα-open
$\tau$	$\mathcal{B}=\text{S}{\text{I}}_{\theta }\left(\mathcal{B}\right)$	$\mathcal{B}=\text{S}{\text{I}}_{\delta }\left(\mathcal{B}\right)$	$\mathcal{B}=\text{S}{\text{I}}_{\tau }\left(\mathcal{B}\right)$	$\mathcal{B}=\text{S}{\text{I}}_{\delta }\left(\mathcal{B}\right)$	$\mathcal{B}=\text{S}{\text{I}}_{\tau }\left(\mathcal{B}\right)$
	$\theta$-semiopen	$\delta$-semiopen	$\tau$-semiopen	$\delta$-semiopen	$\tau$-semiopen

Case ψ = θ and φ = τ: if $\mathcal{B}$ ⊆ $\mathcal{X}$ is a $S_{\theta }^{\tau }$-open set, then from Definition 3.1 and Lemmas 2.1, 2.2 we obtain $\mathcal{B}$ ⊆ $\text{P}\text{C}$(${\mathrm{I}}_{\tau }$($\mathcal{B}$)) = ${\text{C}}_{\tau }$(${\text{I}}_{\theta }$(${\mathrm{I}}_{\tau }$($\mathcal{B}$))) ∪ ${\mathrm{I}}_{\tau }$($\mathcal{B}$) = ${\text{C}}_{\tau }$(${\text{I}}_{\theta }$($\mathcal{B}$)) ∪ ${\mathrm{I}}_{\tau }$($\mathcal{B}$). Since

$\mathcal{B}$⊆ $\mathcal{B}$ ∩ (${\mathrm{I}}_{\tau }$($\mathcal{B}$) ∪ ${\text{C}}_{\tau }$(${\text{I}}_{\theta }$($\mathcal{B}$))) = ${\mathrm{I}}_{\tau }$($\mathcal{B}$) ∪ $\text{S}\text{I}$($\mathcal{B}$) ⊆ $\mathcal{B}$. Then, $\mathcal{B}$ = ${\mathrm{I}}_{\tau }$($\mathcal{B}$) ∪ $\text{S}\text{I}$($\mathcal{B}$).

Case ψ = τ and φ = a: if $\mathcal{B}$ ⊆ $\mathcal{X}$ is a $S_{\tau }^a$-open set, then

$\mathcal{B}$⊆ $\text{P}{\text{C}}_{\tau }$(${\text{I}}_a$($\mathcal{B}$)) = ${\text{I}}_a$($\mathcal{B}$) ∪ ${\text{C}}_{\tau }$(${\mathrm{I}}_{\tau }$(${\text{I}}_a$($\mathcal{B}$))). From Lemma 2.2, we get $\mathcal{B}$⊆ $\mathcal{B}$∩ (${\text{I}}_a$($\mathcal{B}$)) ∪ ${\text{C}}_{\tau }$(${\text{I}}_{\delta }$($\mathcal{B}$)) = $\text{S}{\text{I}}_{\delta }$($\mathcal{B}$) ⊆ $\mathcal{B}$, therefore $\mathcal{B}$ is the δ-semiopen set.

Similarly, other results of $S_{\psi }^{\phi }$-open sets can be easily summarized in the following table:

Moreover, some of the relationships among different classes of the space $\mathcal{X}$ are illustrated in the following diagram 1 Figure 1.

Figure 1. The relationships among different classes of the space ($\mathcal{X}$, $\xi$).

As shown by [5,6,15,17] and the following examples, the contradictory diagrams are not necessarily true:

Example 3.1.

If $\mathcal{X}$ = {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_3$, ${\upsilon }_4$, $v_5$} and $\xi$ = {∅, {${\upsilon }_1$}, {${\upsilon }_2$}, {${\upsilon }_3$}, {${\upsilon }_1$, ${\upsilon }_2$}, {${\upsilon }_1$, ${\upsilon }_3$}, {${\upsilon }_2$, ${\upsilon }_3$}, {${\upsilon }_2$, ${\upsilon }_4$}, {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_3$}, {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_4$}, {${\upsilon }_2$, ${\upsilon }_3$, ${\upsilon }_4$}, {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_3$, ${\upsilon }_4$}, $\mathcal{X}$}. Then:

1. {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_5$} $\subset$ $\mathcal{X}$ is a $S_{\theta }^{\tau }$-open (resp. θ*-semiopen, $S_{\delta }^{\tau }$-open) set, and it is not $S^{\delta }$-open and open sets (resp. it is not $S_{\theta }^a$-open and α-open sets, it is not a δ-semiopen set),

2. {${\upsilon }_1$, ${\upsilon }_3$, ${\upsilon }_5$} $\subset$ $\mathcal{X}$ is a $S_{\theta }^{\delta }$-open (resp. $S_{\theta }^a$-open) set, and it is not a δ-open (resp. it is not a a-open) set,

3. {${\upsilon }_2$, ${\upsilon }_5$} $\subset$ $\mathcal{X}$ is a semiopen set and it is not a Z_α-open set.

Example 3.2.

If $\mathcal{X}$ = {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_3$, ${\upsilon }_4$, $v_5$} and $\xi$ = {∅, {${\upsilon }_1$}, {${\upsilon }_2$}, {${\upsilon }_1$, ${\upsilon }_2$}, {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_3$}, {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_4$}, {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_3$, ${\upsilon }_4$}, $\mathcal{X}$}. Then:

1. {${\upsilon }_1$} $\subset$ $\mathcal{X}$ is a $S_{\theta }^{\delta }$-open set but it not a θ-semiopen set.

2. {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_3$, ${\upsilon }_5$} $\subset$ $\mathcal{X}$ is a $S_{\theta }^a$-open (resp. θ*-semiopen) set but it is not a $S^{\delta }$-open (resp. it is not $S_{\theta }^{\tau }$-open) set,

3. {${\upsilon }_2$, ${\upsilon }_3$, ${\upsilon }_4$} $\subset$ $\mathcal{X}$ is a δ-semiopen (resp. $S_{\delta }^{\tau }$-open, Z_α-open) set but not a $S_{\theta }^a$-open (resp. it is not $S_{\theta }^{\tau }$-open, it is not θ*-semiopen) set,

4. {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_4$, ${\upsilon }_5$} $\subset$ $\mathcal{X}$ is a Z_α-open and it is not a $S_{\delta }^{\tau }$-open set.

Lemma 3.1.

For a topological space ($\mathcal{X}$, $\xi$), then following expressions are hold:

1. The intersection of the arbitrary $S_{\psi }^{\phi }$-closed sets is the $S_{\psi }^{\phi }$-closed set.

2. The union of arbitrary $S_{\psi }^{\phi }$-open sets is the $S_{\psi }^{\phi }$-open set.

In the following example, it has been demonstrated that if ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$are $S_{\psi }^{\phi }$-open sets, then $\mathcal{B}$₁ ∩ $\mathcal{B}$₂ is not $S_{\psi }^{\phi }$-open set for each $\psi =\{\theta ,\delta ,\tau \}$ and $\phi =\{\theta ,\delta ,\tau ,\alpha ,a\}$.

Example 3.3.

Suppose that $\mathcal{X}$ = {$v_1$, ${\upsilon }_2$, ${\upsilon }_3$, ${\upsilon }_4$, ${\upsilon }_5$} and $\xi$ =  {$\mathcal{X}$, ∅, {${\upsilon }_2$}, {${\upsilon }_3$}, {${\upsilon }_2$, ${\upsilon }_3$}, {${\upsilon }_2$, ${\upsilon }_4$}, {${\upsilon }_2$, ${\upsilon }_3$, ${\upsilon }_4$}}. Then, both the subsets {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_4$} and {${\upsilon }_1$, ${\upsilon }_3$} of $\mathcal{X}$ are $S_{\psi }^{\phi }$-open sets, but {$v_1$, $v_2$, $v_4$} ∩ {$v_1$, $v_3$} = {$v_1$} is not $S_{\psi }^{\phi }$-open sets.

Definition 3.2.

Let $\mathcal{B}$ be a subset of a topological space ($\mathcal{X}$, $\xi$):

1. The $S_{\psi }^{\phi }$-closure of $\mathcal{B}$ is the intersection of all $S_{\psi }^{\phi }$-closed sets that contain $\mathcal{B}$ and is represented by S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$).

2. The $S_{\psi }^{\phi }$-interior of $\mathcal{B}$, indicated by S${\text{I}}_{\psi }^{\phi }$($\mathcal{B}$), and is defined by the union of all $S_{\psi }^{\phi }$-open sets contained in $\mathcal{B}$.

Theorem 3.1.

For a space $\mathcal{X}$ and $\mathcal{B}$₁, $\mathcal{B}$₂ ⊆ $\mathcal{X}$, then the following properties are satisfied:

1. S${\text{C}}_{\psi }^{\phi }$($\mathcal{X}$) = $\mathcal{X}$ and S${\text{C}}_{\psi }^{\phi }$(∅) = ∅,

2. $\mathcal{B}$₁ ⊆ S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁),

3. If $\mathcal{B}$₁ ⊆ $\mathcal{B}$₂, then ${\text{SC}}_{\psi }^{\phi }$($\mathcal{B}$₁) ⊆ S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₂),

4. $v$ ∈ S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁) iff each $\mathcal{M}$∈ $S_{\psi }^{\phi }$O containing $v$, $\mathcal{M}$ ∩ $\mathcal{B}$₁ ≠ ∅,

5. $\mathcal{B}$₁ is $S_{\psi }^{\phi }$-closed iff $\mathcal{B}$₁ = S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁),

6. S${\text{C}}_{\psi }^{\phi }$(S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁)) = S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁),

7. S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁) ∪ S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₂) ⊆ S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁ ∪ $\mathcal{B}$₂),

8. S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁ ∩ $\mathcal{B}$₂) ⊆ S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁) ∩ S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₂).

Proof.

Sufficing to prove (6). By (2), we obtain $\mathcal{B}$₁ ⊆ S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁), and then S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁) ⊆ S${\text{C}}_{\psi }^{\phi }$(S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁)). Assuming that, $v$ ∈ S${\text{C}}_{\psi }^{\phi }$(S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁)). For every $\mathcal{M}$ ∈ $S_{\psi }^{\phi }$O and $v$ ∈ $\mathcal{M}$, $\mathcal{M}$ ∩ S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁) ≠ ∅. Let $v$ ∈ $\mathcal{M}$ ∩ S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁), then, for each point $v$ ∈ $\mathcal{W}$ ∈ $S_{\psi }^{\phi }$O, and $\mathcal{B}$₁ ∩ $\mathcal{W}$ ≠ ∅. Accordingly, $\mathcal{M}$ is a $S_{\psi }^{\phi }$-open set, and $v$ ∈ $\mathcal{M}$ with $\mathcal{B}$₁ ∩ $\mathcal{M}$ ≠ ∅, hence we have $v$ ∈ S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁). Therefore, ${\text{SC}}_{\psi }^{\phi }$(S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁)) ⊆ S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁).

Theorem 3.2.

For a space $\mathcal{X}$ and $\mathcal{B}$₁, $\mathcal{B}$₂ ⊆ $\mathcal{X}$, then the following properties are satisfied:

1. S${\text{I}}_{\psi }^{\phi }$($\mathcal{X}$) = $\mathcal{X}$ and S${\text{I}}_{\psi }^{\phi }$(∅) = ∅,

2. S${\text{I}}_{\psi }^{\phi }$($\mathcal{B}$₁) ⊆ $\mathcal{B}$₁,

3. If $\mathcal{B}$₁ ⊆ $\mathcal{B}$₂, then S${\text{I}}_{\psi }^{\phi }$($\mathcal{B}$₁) ⊆ S${\text{I}}_{\psi }^{\phi }$($\mathcal{B}$₂),

4. 4. $v$ ∈ S${\text{I}}_{\psi }^{\phi }$($\mathcal{B}$₁) iff there is a $S_{\psi }^{\phi }$-open set $\mathcal{W}$ and $v$ ∈ $\mathcal{W}$ ⊆ $\mathcal{B}$₁,

5. S${\text{I}}_{\psi }^{\phi }$(S${\text{I}}_{\psi }^{\phi }$($\mathcal{B}$₁)) = S${\text{I}}_{\psi }^{\phi }$($\mathcal{B}$₁),

6. 6. $\mathcal{B}$₁ is $S_{\psi }^{\phi }$-open iff $\mathcal{B}$₁ = S${\text{I}}_{\psi }^{\phi }$($\mathcal{B}$₁),

7. 7. S${\text{I}}_{\psi }^{\phi }$($\mathcal{B}$₁ ∩ $\mathcal{B}$₂) ⊆ S${\text{I}}_{\psi }^{\phi }$($\mathcal{B}$₁) ∩ ${\text{SI}}_{\psi }^{\phi }$($\mathcal{B}$₂),

8. S${\text{I}}_{\psi }^{\phi }$($\mathcal{B}$₁ ∪ $\mathcal{B}$₂) ⊇ S${\text{I}}_{\psi }^{\phi }$($\mathcal{B}$₁) ∪ S${\text{I}}_{\psi }^{\phi }$($\mathcal{B}$₂).

Proof.

The proof is clear.

Theorem 3.3.

For $\mathcal{B}$ ⊆ $\mathcal{X}$, we get

1. S${\text{C}}_{\psi }^{\phi }$($\mathcal{X}\mathrm{\setminus }$ $\mathcal{B}$) = $\mathcal{X}\mathrm{\setminus }$ S${\text{I}}_{\psi }^{\phi }$($\mathcal{B}$),

2. S${\text{I}}_{\psi }^{\phi }$($\mathcal{X}\mathrm{\setminus }$ $\mathcal{B}$) = $\mathcal{X}\mathrm{\setminus }$ S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$).

Theorem 3.4.

For $\mathcal{B}$ ⊆ $\mathcal{X}$, the below stated properties are equivalent:

1. $\mathcal{B}$ is a $S_{\psi }^{\phi }$-open (resp. $S_{\psi }^{\phi }$-closed) set,

2. there exists $\mathcal{K}$ is φ-open (resp. φ-closed) set such that $\mathcal{K}$ ⊆ $\mathcal{B}$⊆ $\text{P}{\text{C}}_{\psi }$($\mathcal{K}$) (resp. $\text{P}{\text{I}}_{\psi }$($\mathcal{K}$) ⊆ $\mathcal{B}$ ⊆ $\mathcal{K}$),

3. $\text{P}{\text{C}}_{\psi }$($\mathcal{B}$) = $\text{P}{\text{C}}_{\psi }$($\mathrm{I}$($\mathcal{B}$)) (resp. $\text{P}{\text{I}}_{\psi }$($\mathcal{B}$) = $\text{P}{\text{I}}_{\psi }$($\text{C}$($\mathcal{B}$))).

Proof.

(1)$\Rightarrow$(2). It follows from putting $\mathcal{K}$ = $\mathrm{I}$($\mathcal{B}$),

(2)$\Rightarrow$(3). It is clear,

(3)$\Rightarrow$(1). Since $\mathcal{B}$ ⊆ $\text{P}{\text{C}}_{\psi }$($\mathcal{B}$) = $\text{P}{\text{C}}_{\psi }$($\mathrm{I}$($\mathcal{B}$)). Then, $\mathcal{B}$ is a $S_{\psi }^{\phi }$-open set.

Proposition 3.1.

If $\mathcal{B}$₁∈ $S_{\psi }^{\phi }$O (resp. $S_{\psi }^{\phi }$C) and $\mathcal{B}$₁ ⊆ $\mathcal{B}$₂ ⊆ $\text{P}{\text{C}}_{\psi }$($\mathcal{B}$₁) (resp. $\text{P}{\text{I}}_{\psi }$($\mathcal{B}$₁) ⊆ $\mathcal{B}$₂ ⊆ $\mathcal{B}$₁), then $\mathcal{B}$₂ ∈ $S_{\psi }^{\phi }$O (resp. $S_{\psi }^{\phi }$C).

4. Some applications of ${\mathit{S}}_{\mathit{\psi }}^{\mathit{\phi }}$-open and ${\mathit{S}}_{\mathit{\psi }}^{\mathit{\phi }}$-closed sets

Definition 4.1.

Let $g$ be a mapping from ($\mathcal{X}$, $\xi$) to ($\mathcal{Y}$, $\eta$), the $g$ is called $S_{\psi }^{\phi }$-continuous if $g^{-1}$($\mathcal{W}$) ∈ $S_{\psi }^{\phi }$O($\mathcal{X}$), for every $\mathcal{W}$ ∈ $\eta$.

Definition 4.2.

A function $g$: $\mathcal{X}$ → $\mathcal{Y}$ is referred to as super-cont. [25] (resp. strongly θ-cont. [26], a-cont. [19], α-cont. [27], pre-cont. [8], θ-semi-cont. [3], δ-semi-cont. [28], θ*-semi-cont. [6], Z_α-cont. [5], semi-cont. [1], γ-cont. [14], Y-cont. [17] and Z-cont. [15]), if $g^{-1}$($\mathcal{W}$) is δ-open (resp. θ-open, a-open, α-open, preopen, θ-semiopen, δ-semiopen, θ*-semiopen, Z_α-open, semiopen, γ-open, Y-open, and Z-open) of $\mathcal{X}$, respectively, for any $\mathcal{W}$ ∈ $\eta$.

The illustrative diagram 2 below holds for the function $g$: $\mathcal{X}$ → $\mathcal{Y}$ Figure 2:

Figure 2. The relationships among different classes of the function $g$: $\mathcal{X}$ → $\mathcal{Y}$.

As shown in the examples given in [5,6,15,17], and the examples below, the illustration implications are not reversible.

Example 4.1.

Referring to Example 3.1, we introduce a mapping $g_j$: ($\mathcal{X}$, $\xi$) → (${\mathcal{Y}}_j$, ${\eta }_j$), $(j=1,2,3)$ as follows:

1. If $g_1$(${\upsilon }_1$) = $g_1$(${\upsilon }_2$) = $g_1$(${\upsilon }_5$) = ${\upsilon }_3$, $g_1$(${\upsilon }_3$) = ${\upsilon }_4$, $g_1$(${\upsilon }_4$) = ${\upsilon }_5$, then ${\mathcal{Y}}_1$ = {${\upsilon }_3$, ${\upsilon }_4$, ${\upsilon }_5$} and ${\eta }_1$ = {∅, {${\upsilon }_3$}, {${\upsilon }_4$},{${\upsilon }_3$, ${\upsilon }_4$},{${\upsilon }_3$, ${\upsilon }_5$}, ${\mathcal{Y}}_1$}. Thus, for any open set ${\mathcal{W}}_1$∈ ${\eta }_1$, the inverse mapping ${g_1}^{-1}$(${\mathcal{W}}_1$) is $S_{\theta }^{\tau }$-cont. (resp. θ*-semi-cont., $S_{\delta }^{\tau }$-cont.), but it is not $S^{\delta }$-cont. and it is not cont. (resp. it is not $S^a$-cont. and it is not α-cont., it is not δ-semi-cont.),

2. If $g_2$(${\upsilon }_1$) = $g_2$(${\upsilon }_3$) = $g_2$(${\upsilon }_5$) = ${\upsilon }_3$ and $g_2$(${\upsilon }_2$) = $g_2$(${\upsilon }_4$) = ${\upsilon }_5$, then ${\mathcal{Y}}_2$ = {${\upsilon }_3$, $v_5$} and ${\eta }_2$ =  {∅, {${\upsilon }_3$},{${\upsilon }_5$}, ${\mathcal{Y}}_2$}. For every open set ${\mathcal{W}}_2$∈ ${\eta }_2$, the inverse mapping ${g_2}^{-1}$(${\mathcal{W}}_2$) is $S^{\delta }$-cont. (resp. $S^a$-cont.) but it is not super-cont. (resp. it is not a-cont.),

3. If $g_3$(${\upsilon }_1$) = $g_3$(${\upsilon }_3$) = $g_3$(${\upsilon }_4$) = ${\upsilon }_5$; $g_3$(${\upsilon }_2$) = $g_3$(${\upsilon }_5$) = ${\upsilon }_2$, then ${\mathcal{Y}}_3$ = {${\upsilon }_2$, $v_5$} and ${\eta }_2$ =  {∅, {${\upsilon }_2$},{${\upsilon }_5$}, ${\mathcal{Y}}_3$}, thus ${g_3}^{-1}$(${\mathcal{W}}_3$) is semi-cont. and it is not Z_α-cont., for every open set ${\mathcal{W}}_3$∈ ${\eta }_3$.

Example 4.2.

According to Example 3.2, $g_k$: ($\mathcal{X}$, $\xi$) → (${\mathcal{Y}}_k$, ${\eta }_k$), $(k=4,5,6$) is expressed as:

1. If $g_4$: ($\mathcal{X}$, $\xi$) → ($\mathcal{X}$, $\xi$) is the identity function, then ${g_4}^{-1}$(${\mathcal{W}}_4$) is $S^{\delta }$-cont. and it is not θ-semi-cont., for each open set ${\mathcal{W}}_4$∈ $\xi$,

2. If $g_5$(${\upsilon }_1$) = ${\upsilon }_1$, $g_5$(${\upsilon }_2$) = ${\upsilon }_2$, $g_5$(${\upsilon }_3$) = ${\upsilon }_4$ and $g_5$(${\upsilon }_4$) = $g_5$(${\upsilon }_5$) = ${\upsilon }_3$, then ${\mathcal{Y}}_5$ = {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_3$, ${\upsilon }_4$} and ${\eta }_5$ =  {∅, {${\upsilon }_1$}, {${\upsilon }_2$}, {${\upsilon }_1$, ${\upsilon }_2$}, {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_3$}, {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_4$}, ${\mathcal{Y}}_5$}. Thus, ${g_5}^{-1}$(${\mathcal{W}}_5$) is Z_α-cont. but it is not $S_{\delta }^{\tau }$-cont., for each open set ${\mathcal{W}}_5$∈ ${\eta }_5$,

3. If $g_6$(${\upsilon }_1$) = ${\upsilon }_1$, $g_6$(${\upsilon }_2$) = ${\upsilon }_2$, $g_6$(${\upsilon }_3$) = $g_6$(${\upsilon }_5$) = ${\upsilon }_3$ and $g_6$(${\upsilon }_4$) = ${\upsilon }_5$, then ${\mathcal{Y}}_6$ = {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_3$, $v_5$} and ${\eta }_6$ = {∅, {${\upsilon }_1$}, {${\upsilon }_2$}, {${\upsilon }_1$, ${\upsilon }_2$}, {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_3$}, {${\upsilon }_1$, ${\upsilon }_2$, ${\upsilon }_5$}, ${\mathcal{Y}}_6$}. The inverse mapping ${g_6}^{-1}$(${\mathcal{W}}_6$) is a $S^a$-cont. (resp. θ*-semi-cont.) and on corresponding, it is not $S^{\delta }$-cont. (resp. it is not $S_{\theta }^{\tau }$-cont.), for each open set ${\mathcal{W}}_6$∈ ${\eta }_6$,

4. If $g_7$(${\upsilon }_1$) = $g_7$(${\upsilon }_5$) = ${\upsilon }_5$ and $g_7$(${\upsilon }_2$) = $g_7$(${\upsilon }_3$) = $g_7$(${\upsilon }_4$) = ${\upsilon }_2$, then ${\mathcal{Y}}_7$ = {${\upsilon }_2$, ${\upsilon }_5$} and ${\eta }_7$ = {∅, {${\upsilon }_2$}, {${\upsilon }_5$}, ${\mathcal{Y}}_7$}. For each open set ${\mathcal{W}}_6$∈ ${\eta }_6$, ${g_7}^{-1}$(${\mathcal{W}}_7$) is a δ-semi-cont. (resp. $S_{\delta }^{\tau }$-cont., Z_α-cont.) mapping and it is not $S^a$-cont. (resp. it is not $S_{\theta }^{\tau }$-cont., it is not θ*-semi-cont.).

Example 4.3.

Let $\mathcal{X}$= {${\upsilon }_{+}$, ${\upsilon }_{,}$, ${\upsilon }_{-}$, ${\upsilon }_{.}$, ${\upsilon }_{/}$}, $\xi$ = {∅, {${\upsilon }_{-}$, ${\upsilon }_{.}$},{${\upsilon }_{+}$, ${\upsilon }_{,}$}, {${\upsilon }_{+}$, ${\upsilon }_{,}$, ${\upsilon }_{-}$, ${\upsilon }_{.}$}, $\mathcal{X}$}, and $g$ be a mapping from ($\mathcal{X}$, $\xi$) to $\left(\mathcal{Y},\eta \right)$ defined as: $g$(${\upsilon }_{+}$) = ${\upsilon }_{-}$, $g$(${\upsilon }_{,}$) = $g$(${\upsilon }_{-}$) = ${\upsilon }_{.}$ and $g$(${\upsilon }_{.}$) = $g$(${\upsilon }_{/}$) = ${\upsilon }_{/}$. Then, $\mathcal{Y}$ =  {${\upsilon }_{-}$, ${\upsilon }_{.}$, ${\upsilon }_{/}$} and $\eta$ = {∅, {${\upsilon }_{.}$}, {${\upsilon }_{.}$, ${\upsilon }_{/}$}, {${\upsilon }_{-}$, ${\upsilon }_{.}$}, $\mathcal{Y}$}. The inverse mapping $g^{-1}$($\mathcal{W}$) is Y-cont. (resp. Z-cont.) and it is not θ*-semi-cont. (resp. it is not Z_α-cont.), for every open set for each open set $\mathcal{W}$∈ $\eta$.

Theorem 4.1.

Let $g$ be a mapping from $\mathcal{X}$ to $\mathcal{Y}$. Then the following properties are equivalent:

1. $g$ is a $S_{\psi }^{\phi }$-cont. mapping,

2. for any point $\upsilon$ ∈ $\mathcal{X}$ and U∈ $\xi$ containing $g$($\upsilon$), there $\upsilon$ ∈ $\mathcal{G}$ ⊆ $S_{\psi }^{\phi }$O($\mathcal{X}$) with $g$($\mathcal{G}$) ⊆ $\mathcal{U}$,

3. $g^{-1}$($\mathcal{B}$₂) ∈$S_{\psi }^{\phi }$O($\mathcal{X}$), for each open $\mathcal{B}$₂,

4. $\text{P}{\text{I}}_{\psi }$($\text{C}$($g^{-1}$($\mathcal{B}$₂))) ⊆ $g^{-1}$(${\text{C}}_{\tau }$($\mathcal{B}$₂)), for $\mathcal{B}$₂ ⊆ $\mathcal{Y}$,

5. $g^{-1}$(${\text{I}}_{\tau }$($\mathcal{B}$₂)) ⊆ $\text{P}{\text{C}}_{\psi }$($\text{I}$($g^{-1}$($\mathcal{B}$₂))), for $\mathcal{B}$₂ ⊆ $\mathcal{Y}$,

6. S${\text{C}}_{\psi }^{\phi }$($g^{-1}$($\mathcal{B}$₂)) ⊆ $g^{-1}$(${\text{C}}_{\tau }$($\mathcal{B}$₂)), for $\mathcal{B}$₂ ⊆ $\mathcal{Y}$,

7. $g$(S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁)) $\subseteq$ ${\text{C}}_{\tau }$($g$($\mathcal{B}$₁)), for $\mathcal{B}$₁ ⊆ $\mathcal{X}$,

8. $g^{-1}$(${\text{I}}_{\tau }$($\mathcal{B}$₂)) ⊆ S${\text{I}}_{\psi }^{\phi }$($g^{-1}$($\mathcal{B}$₂)), for $\mathcal{B}$₂ ⊆ $\mathcal{Y}$.

Proof.

The relation (1)$\iff$(2) and (1)$\iff$(3) are clear,

(3)$\Rightarrow$(4). Asume that $\mathcal{B}$₂ is a subset of a space $\mathcal{Y}$. It follows from (3) $g^{-1}$(${\text{C}}_{\tau }$($\mathcal{B}$₂)) is the$S_{\psi }^{\phi }$-closed set. By using Theorem 3.4, we have $g^{-1}$(${\text{C}}_{\tau }$($\mathcal{B}$₂))$\supseteq$ $\text{P}{\text{I}}_{\psi }$($g^{-1}$(${\text{C}}_{\tau }$($\mathcal{B}$₂))) = $\text{P}{\text{I}}_{\psi }$($\text{C}$($g^{-1}$(${\text{C}}_{\tau }$($\mathcal{B}$₂)))) $\supseteq$ $\text{P}{\text{I}}_{\psi }$($\text{C}$($g^{-1}$($\mathcal{B}$₂))),

(4)$\Rightarrow$(5). It is shown by putting $\mathcal{Y}$ $\mathrm{\setminus }$ $\mathcal{B}$₂ instead of $\mathcal{B}$₂ in the item (4), then $\text{P}{\text{I}}_{\psi }$($\text{C}$($g^{-1}$ ($\mathcal{Y}$ $\mathrm{\setminus }$ $\mathcal{B}$₂))) ⊆ $g^{-1}$(${\text{C}}_{\tau }$($\mathcal{Y}$ $\mathrm{\setminus }$ $\mathcal{B}$₂)). Therefore $g^{-1}$(${\text{I}}_{\tau }$($\mathcal{B}$₂)) ⊆ $\text{P}{\text{C}}_{\psi }$($\text{I}$($g^{-1}$($\mathcal{B}$₂))), for each $\mathcal{B}$₂ $\subseteq$ $\mathcal{Y}$,

(5)$\Rightarrow$(1). It is clear,

(3)$\Rightarrow$(6). Assume that $\mathcal{B}$₂ $\subseteq$ $\mathcal{Y}$ and $g^{-1}$(${\text{C}}_{\tau }$($\mathcal{B}$₂))∈ $S_{\psi }^{\phi }$C($\mathcal{X}$). Hence S${\text{C}}_{\psi }^{\phi }$($g^{-1}$($\mathcal{B}$₂)) $\subseteq$ S${\text{C}}_{\psi }^{\phi }$($g^{-1}$(${\text{C}}_{\tau }$($\mathcal{B}$₂))) = $g^{-1}$(${\text{C}}_{\tau }$($\mathcal{B}$₂)),

(6)$\Rightarrow$(7). For $\mathcal{B}$₁$\subseteq$ $\mathcal{X}$, we get $g$($\mathcal{B}$₁) $\subseteq$ $\mathcal{Y}$ and $g^{-1}$(${\text{C}}_{\tau }$($g$($\mathcal{B}$₁))) $\supseteq$ S${\text{C}}_{\psi }^{\phi }$($g^{-1}$($g$($\mathcal{B}$₁))) $\supseteq$ S${\text{C}}_{\psi }^{\phi }$(($\mathcal{B}$₁)). Thus, we deduce that ${\text{C}}_{\tau }$($g$($\mathcal{B}$₁)) $\supseteq$ $g$ $g^{-1}$(${\text{C}}_{\tau }$($g$($\mathcal{B}$₁))) $\supseteq$ $g$(S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$₁)),

(7)$\Rightarrow$(1). Suppose that $T$ $\subseteq$ $\mathcal{Y}$ is open and $\mathcal{F}$ = $\mathcal{Y}$ $\mathrm{\setminus }$ $T$ is closed in a space $\mathcal{Y}$, then $g^{-1}$($\mathcal{P}$) = $\mathcal{X}$ $\mathrm{\setminus }$ $g^{-1}$($T$). Consequently, $g$(S${\text{C}}_{\psi }^{\phi }$($g^{-1}$($\mathcal{F}$))) $\subseteq$ ${\text{C}}_{\tau }$($g$($g^{-1}$($\mathcal{F}$))) $\subseteq$ ${\text{C}}_{\tau }$($\mathcal{F}$) = $\mathcal{F}$, we obtain S${\text{C}}_{\psi }^{\phi }$($g^{-1}$($\mathcal{F}$)) ⊆$g^{-1}$($\mathcal{P}$). But, $g^{-1}$(F) = $\mathcal{X}$ \ $g^{-1}$($T$)∈ $S_{\psi }^{\phi }$ C($\mathcal{X}$). Therefore, $g$ is $S_{\psi }^{\phi }$-continuous,

(1)$\Rightarrow$(8). Supose that $\mathcal{B}$₂ $\subseteq$ $\mathcal{Y}$ and $g^{-1}$(${\text{I}}_{\tau }$($\mathcal{B}$₂)) is $S_{\psi }^{\phi }$-open. But, $g^{-1}$(${\text{I}}_{\tau }$($\mathcal{B}$₂)) = S${\text{I}}_{\psi }^{\phi }$($g^{-1}$(${\text{I}}_{\tau }$($\mathcal{B}$₂))) ⊆ S${\text{I}}_{\psi }^{\phi }$($g^{-1}$($\mathcal{B}$₂)). Therefore, $g^{-1}$(${\text{I}}_{\tau }$($\mathcal{B}$₂)) ⊆ S${\text{I}}_{\psi }^{\phi }$($g^{-1}$($\mathcal{B}$₂)).

Definition 4.3.

A topological space, say ($\mathcal{X}$, $\xi$) is said to be:

1.  $S_{\psi }^{\phi }$-T₂ if for each ${\upsilon }_1$ ≠ ${\upsilon }_2$ of $\mathcal{X}$, then, there is $\mathcal{M}$, $\mathcal{N}{S}_{\psi }^{\phi }\text{O}\left(\mathcal{X}\right)$ with $\mathcal{M}$ ∩ $\mathcal{N}$ = ∅ and ${\upsilon }_1$ ∈ $\mathcal{M}$ and ${\upsilon }_2$ ∈ $\mathcal{N}$.

2. $S_{\psi }^{\phi }$-T₁ if for each ${\upsilon }_1$ ≠ ${\upsilon }_2$ of $\mathcal{X}$, then, there is $S_{\psi }^{\phi }$-open sets ${\upsilon }_1$ ∈ $\mathcal{M}$ and ${\upsilon }_2$ ∈ $\mathcal{N}$ with ${\upsilon }_2$ ∉ $\mathcal{M}$ and ${\upsilon }_1$ ∉ $\mathcal{N}$.

3. $S_{\psi }^{\phi }$-T₀ if for each ${\upsilon }_1$ ≠ ${\upsilon }_2$ of $\mathcal{X}$, there is $\mathcal{M}$ $S_{\psi }^{\phi }\text{O}\left(\mathcal{X}\right)$ such that either ${\upsilon }_1$ ∈ $\mathcal{M}$, ${\upsilon }_2$ ∉ $\mathcal{M}$ or ${\upsilon }_2$ ∉ $\mathcal{M}$, ${\upsilon }_1$ ∈ $\mathcal{M}$.

Theorem 4.2.

If $g$ is a $S_{\psi }^{\phi }$-continuous and injection function from $\mathcal{X}$ into $\mathcal{Y}$ and $\mathcal{Y}$ is T_i, then $\mathcal{X}$ is $S_{\psi }^{\phi }$-T_i for i = 0,1,2.

Proof.

The proof will be implemented only in the case i = 2.

Let ${\upsilon }_{+}$ and ${\upsilon }_{,}$ be two points in $\mathcal{X}$ with ${\upsilon }_{+}$ ≠ ${\upsilon }_{,}$. Accordingly, we obtain that $g$(${\upsilon }_{+}$)$\ne g$(${\upsilon }_{,}$). On standing on the hypothesis that $\mathcal{Y}$ is a T₂ space, hence there are $\mathcal{M}$, $\mathcal{N}{S}_{\psi }^{\phi }\text{O}\left(\mathcal{X}\right)$, $\mathcal{M}$ ∩ $\mathcal{N}$ = ∅ with $g$(${\upsilon }_{+}$)∈ $\mathcal{M}$ and $g$(${\upsilon }_{,}$) ∈ $\mathcal{N}$. But the function $g$ is $S_{\psi }^{\phi }$-cont., then there are $\mathcal{U}$, $\mathcal{V}{S}_{\psi }^{\phi }\text{O}\left(\mathcal{X}\right)$, and ${\upsilon }_{+}$ ∈ $\mathcal{U}$, ${\upsilon }_{,}$ ∈ $\mathcal{V}$, with $g$($\mathcal{U}$) ⊆ $\mathcal{M}$ and $g$($\mathcal{V}$) ⊆ $\mathcal{N}$. Hence, $\mathcal{U}$ ∩ $\mathcal{V}$ = ∅. As a result, the space $\mathcal{X}$ is the $S_{\psi }^{\phi }$-T₂-space.

Lemma 4.1.

$\mathcal{B}$ ⊆ $\mathcal{X}$ is $S_{\psi }^{\phi }$-open if and only if $\mathcal{B}$ ∩ $\mathcal{M}$∈$S_{\psi }^{\phi }$O($\mathcal{X}$), for every θ-open set $\mathcal{M}$ of a space $\mathcal{X}$.

Theorem 4.3.

Assume $g$₁ and $g$₂ are two mappings from $\mathcal{X}$ into $\mathcal{Y}$. If $g$₁ is $S_{\psi }^{\phi }$-continuous, $g$₂ is θ-continuous along with the space ($\mathcal{Y}$, $\eta$), which is a T₂-space, then the set {$\upsilon$ ∈ $\mathcal{X}$: $g$₁($\upsilon$) = $g$₂($\upsilon$)} is $S_{\psi }^{\phi }$-closed sets in a space $\mathcal{X}$.

Proof.

Let us consider $\upsilon$ $\notin$ $\mathcal{B}$ and $\mathcal{B}$ = {$\upsilon$ $\in$ $\mathcal{X}$: $g$₁($\upsilon$) = $g$₂($\upsilon$)}. By hypothesis, the space $\mathcal{Y}$ is a T₂-space, which further means there are $\mathcal{M}$, $\mathcal{N}$ ∈ $S_{\psi }^{\phi }$O($\mathcal{Y}$) with $g$₁($\upsilon$) ∈ $\mathcal{M}$, $g$₂($\upsilon$) ∈ $\mathcal{N}$ and $\mathcal{M}$ ∩ $\mathcal{N}$ = ∅. But $g$ is $S_{\psi }^{\phi }$-continuous mapping, thus, there is $\mathcal{U}{S}_{\psi }^{\phi }\text{O}\left(\mathcal{X}\right)$ and $\upsilon$ $\in \mathcal{U}$ ⊆ $\mathcal{X}$ with $g$₁($\mathcal{U}$) ⊆ $\mathcal{M}$. Also, $g$₂ is θ-continuous mapping, then there is a θ-open set $\mathcal{V}$ and $\upsilon$ $\in \mathcal{V}$ ⊆ $\mathcal{X}$ such that $g$₂($\mathcal{V}$) ⊆ $\mathcal{N}$. Let, $\mathcal{Z}$ = $\mathcal{M}$ ∩ $\mathcal{N}$, then by using Lemma 4.1, $\mathcal{Z}$ is $S_{\psi }^{\phi }$-open set including $\upsilon$ with $g$₁($\mathcal{Z}$) ∩ $g$₂($\mathcal{Z}$) ⊆ $\mathcal{M}$ ∩ $\mathcal{N}$ = ∅. Thus, $\mathcal{Z}$ ∩ $\mathcal{B}$ = ∅, and $\upsilon$ ∉ S${\text{C}}_{\psi }^{\phi }$($\mathcal{B}$). Therefore, $\mathcal{B}$ is $S_{\psi }^{\phi }$-closed in $\mathcal{X}$.

Definition 4.4.

A space $\mathcal{X}$ is called a $S_{\psi }^{\phi }$-regular if for any $S_{\psi }^{\phi }$-closed set $\mathcal{R}$ and any point $\upsilon$ ∉ $\mathcal{R}$, then there are $\mathcal{M}$, $\mathcal{N}{S}_{\psi }^{\phi }\text{O}\left(\mathcal{X}\right)$ and $\mathcal{M}$ ∩ $\mathcal{N}$ = ∅ with $\mathcal{R}$ ⊆ $\mathcal{M}$ and $\upsilon$ ∈ $\mathcal{N}$.

Theorem 4.4.

If $\mathcal{X}$ is $S_{\psi }^{\phi }$-regular space and $g$ is mapping from $\mathcal{X}$ into $\mathcal{Y}$, and $S_{\psi }^{\phi }$-continuous injective open function, then $\mathcal{Y}$ is regular space.

Proof.

Let $\mathcal{R}$ ⊆ $\mathcal{Y}$ be closed set and ${\upsilon }_2$ ∉ $\mathcal{R}$. Let us take ${\upsilon }_2$ = $g$(${\upsilon }_1$). From hypothesis, $g$ is $S_{\psi }^{\phi }$-continuous mapping, hence $g^{-1}$($\mathcal{R}$) is a $S_{\psi }^{\phi }$-closed set. On taking $\mathcal{U}$=  $g^{-1}$($\mathcal{R}$), we obtain ${\upsilon }_1$∉ $\mathcal{U}$. Since $\mathcal{X}$ is $S_{\psi }^{\phi }$-regular, then, there are $\mathcal{M}$, $\mathcal{N}{S}_{\psi }^{\phi }\text{O}\left(\mathcal{X}\right)$ and $\mathcal{M}$ ∩ $\mathcal{N}$ = ∅ such that $\mathcal{U}$ ⊆ $\mathcal{M}$ and ${\upsilon }_1$ ∈ $\mathcal{N}$. Thus R = $g$($\mathcal{U}$) ⊆ $g$($\mathcal{M}$) and ${\upsilon }_2$ = $g$(${\upsilon }_1$) ∈ $g$($\mathcal{N}$) such that $g$($\mathcal{M}$) and $g$($\mathcal{N}$) are open sets and $g$($\mathcal{M}$) ∩ $g$($\mathcal{N}$) = ∅. Hence, according to that the space $\mathcal{Y}$ is regular.

Definition 4.5.

A topological space ($\mathcal{X}$, $\xi$) is said to be $S_{\psi }^{\phi }$-normal if for every pair of disjointed $S_{\psi }^{\phi }$-closed subsets R₁ and R₂ of $\mathcal{X}$, there are $\mathcal{M}$, $\mathcal{N}{S}_{\psi }^{\phi }\text{O}\left(\mathcal{X}\right)$ and $\mathcal{M}$ ∩ $\mathcal{N}$ = ∅ such that $\mathcal{R}$₁ ⊆ $\mathcal{M}$ with $\mathcal{R}$₂ ⊆ $\mathcal{N}$.

Theorem 4.5.

If ($\mathcal{X}$, $\xi$) is $S_{\psi }^{\phi }$-normal space and a mapping $g$ from ($\mathcal{X}$, $\xi$) into ($\mathcal{Y}$, $\eta$) is $S_{\psi }^{\phi }$-continuous injective open mapping, then ($\mathcal{Y}$, $\eta$) is normal space.

Proof.

Assume that $\mathcal{R}$₁ and $\mathcal{R}$₂ are closed and subsets of a space $\mathcal{Y}$ with $\mathcal{R}$₁ ∩ $\mathcal{R}$₂ = ∅. Since $g$ is $S_{\psi }^{\phi }$-continuous, then both $g^{-1}$($\mathcal{R}$₁) and $g^{-1}$($\mathcal{R}$₂) are $S_{\psi }^{\phi }$-closed. Let us take $\mathcal{M}$ =  $g^{-1}$($\mathcal{R}$₁) and $\mathcal{N}$ =  $g^{-1}$($\mathcal{R}$₂), followed that $\mathcal{M}$ ∩ $\mathcal{N}$ = ∅. But $\mathcal{X}$ is $S_{\psi }^{\phi }$-normal, then there exist $\mathcal{U}$, $\mathcal{V}{S}_{\psi }^{\phi }\text{O}\left(\mathcal{X}\right)$ and $\mathcal{U}$ ∩ $\mathcal{V}$ = ∅ with $\mathcal{M}$ ⊆ $\mathcal{U}$ and $\mathcal{N}$ ⊆ $\mathcal{V}$. Thus, $\mathcal{R}$₁ = $g$($\mathcal{M}$) ⊆ $g$($\mathcal{U}$), $\mathcal{R}$₂ = $g$($\mathcal{N}$) ⊆ $g$($\mathcal{V}$) and $g$($\mathcal{U}$) ∩ $g$($\mathcal{V}$) = ∅. Since $g$ is open function, then $g$($\mathcal{U}$) and $g$($\mathcal{V}$) are open sets. Therefore, $\mathcal{Y}$ is normal space.

5. Conclusion

In conclusion, some new types of nearly open sets that are based upon the definition of $S_{\psi }^{\phi }$-open sets have been established in the present work, where $\psi$Îq, δ, τ and $\phi$Îq, δ, τ, α, a. Moreover, some theoretical applications of mapping and other properties have been discussed. In particular, the findings of the present study could be used to improve the results of the rough sets [29,30], which were massively used in several statistical applications. More so, the present study can equally be used to generalize the cases of soft sets in soft topological spaces [31]; besides, this paper may be considered as a starting point for many works in life applications, thereby extending some results in [32–34].

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was funded by Taif University, Taif, Saudi Arabia (TU-DSPP-2024-231).