N_α- Separation Axioms in Topological Spaces

Abstract

The aim of this work is to establish new forms of N_α-separation axioms by employing N_α-open sets in topological spaces, and we investigate some of their features. Moreover, some definitions and theorems are provided.

1 Introduction

A topological space (TS) is a fairly broad concept. More specificity is frequently desirable. Some studies in TS and their extensions in nonclassical TS are shown by many mathematicians [1,2,3,4,5,6,7,8,9,10,11,12,13]. One method is to define topological spaces with more constrained attributes using the separation axioms. In general, it is not true that a sequence in a topological space has only one limit. However, using the separation axioms, a type of space may be established in which the limit, if it exists at all, is unique. In 2015, N_a- open sets were initially examined by N.A Dawood and N.M Ali, see [14]; by using these sets, we study some classes of N_α- separation axioms and N_α- Ti separation for each i = 0, 1, 2, and look into some of their characteristics. Separation axioms have also been generalized to other generic topological spaces such as ordered topological spaces [15]. Ibrahim [16] presented and explored the features of a strong variant of α-open sets termed α_γ-open via operation in 2013. Khalaf and Ibrahim [17] extended their investigation of the features of operations defined on the collection of α-open sets introduced by Ibrahim [16], defining and discussing numerous properties of α_γ-regular, α-β-compact, and α_γ-connected spaces, as well as α-(γ, β)-continuous functions.

In this chapter, we use N_α- open sets in topological spaces to create new types of N_α-separation axioms and study some of their properties. There are also some definitions and theorems offered. Here in this work, all spaces X and Y are topological spaces, also the closure (interior resp.) of a subset A of X is denoted by cl (A) (int (A) resp.)

2 Fundamental Concepts

We will cover some fundamental principles that will be useful in our work.

Definition 1 [15]

Assume that X is a topological space (TS), a set A is named N_a- open set (N_α − OS) if for some α-open set B ≠ ∅ satisfies cl (B) ⊆ A. Also, its complement is named a N_a- closed set (N_α − CS). The collection of all N_α- open sets is referred as N_αO(X), and its complement by N_αC(X).

Remark 1 [15]

A set A is a (N_α − CS) if for some α-closed set ∅ ≠ B ≠ X satisfies ⊆ int (B).

Remark 2 [15]

• (i) X and ∅ are N_α- open sets (N_α − OS s) in any (TS) X.

• (ii) Any clopen set is (N_α − OS).

• (iii) Any set in discrete space is (N_α − OS).

Theorem 1 [15]

Let X₁, X₂ be topological spaces (TSs). Then A₁ and A₂ are (N_α − OS s) in X₁, X₂ resp., if and only if A₁ × A₂ is (N_α − OS) in X₁ × X₂.

Proposition 1 [15]

Let X be (TS). Then:

• (i) Finite union of (N_α − OS s) is (N_α − OS) also.

• (ii) Finite intersection of (N_α − CS s) is (N_α − CS) also.

Definition 2 [15]

The union of all N_a- open set of X contained in A is named N_a- interior of A and is denoted N_α^int(A), and the intersection of all (N_α − CS) containing A is called N_α- closure of A, refereed by N_α^cl(A).

Definition 3 [18]

We say A is generalized N_α – closed set (gN_α − CS) of a space X, if N_α^cl(A) ⊆ B whenever A ⊆ B and B is (N_α − OS).

The complement of (gN_α − CS) is generalized N_α- open set (gN_α − OS) in X.

Theorem 2 [18]

• (i) If A is (N_α − CS) in X, then it is (gN_α − CS).

• (ii) If A is (N_α − OS) in X, then it is (gN_α − OS).

Proposition 2 [15]

Suppose that (Y, t_y) is a subspace of a (TS) X with A ⊆ Y ⊆ X. Then:

• (i) If A ∈ N_aO(X), then A ∈ N_αO(Y).

• (ii) If A ∈ N_aO(Y), then A ∈ N_αO(X), where Y is clopen set in X

Definition 4 [19]

Let X₁, X₂ be (TSs) where, f : X₁ → X₂ is a mapping, then f is named:

• (i) N_α(N_α∗- continues) resp. if f⁻¹ (A) is (N_α − OS) in X₁ for each A open set ((N_α − OS))in X₂ resp.

• (ii) N_α(N_α∗- open) mapping if f(A) is (N_α − OS) in X₂ for each open set ((N_α − OS)) A in X₁ resp.

• (iii) gN_α-continuous (gN_α∗ -continuous) resp. mapping if for each open set ((gN_α − OS)) set A in Y respectively then f⁻¹(A) is (gN_α − OS) in X.

3 Some Characteristics of N_α-Separation N_α- Axioms

In this section, we study N_α^Ti- space X for each i = 0, 1, 2 and we discuss some of these spaces’ characteristics and remarks. We will prove certain theorems in the following cases when X is a finite space.

Definition 5

Assume that X is a (TS). We say X is a N_α^TO-space if for any x ≠ y in X, there exists (N_α − OS) A containing one of them but not other.

Theorem 3

Let X be a (TS). Then X is N_α^To-space if and only if N_α^Cl{x} ≠ N_α^Cl{y}.

Proof

Let N_α^Cl{x} ≠ N_α^Cl{y}, ∀x ≠ y in X. This implies N_α^Cl{x} ⊈N_α^Cl{y} or N_α^Cl{y}⊈ N_α^Cl{x}. Suppose N_α^Cl{x} ⊈ N_α^Cl{y}, hence X ∉ N_α^Cl{y}, thus x ∈ (N_α^Cl{y})^c, which is (N_α − OS) and y ∉ (N_α^Cl{y})^c. Thus, X is N_α^To-space, assume X is N_α^To-space; hence, for each x ≠ y in X, there exists (N_α − OS) G such that x ∈ G, y ∉ G or y ∈ G, x ∉ G. Hence, G^c is (Na − CS). x ∉ G^c, y ∈ G^cy ; hence, x ∉ N_α^Cl{y}, x ∈ N_α^Cl{x} ; this means x ∉ N_α^Cl{y}. Thus, N_α^Cl{x} ≠ N_α^{y}.

Definition 6

Let X be a (TS). Then X is named N_α^T1-space if each pair of distinct points x and y of X, there exist two N_α- open sets A, B containing x and y, respectively, such that y ∉ A, x ∉ B.

Proposition 3

Let X be a (TS). Then X is N_α^T1 -space if and only if {x} is (N_α − CS) ∀x ∈ X.

Proof

Assume that X is N_α^T1-space, to show that each {x} is (N_α − CS), this means we must show that X/{x} is (N_α − OS) for each singleton set {x} in X.

Let y ∈ X/{x}, then y ≠ x in X, since X is N_α^T1 space, then there exists (N_α − OS) G with y ∈ G and x ∉ G. This implies that $y\in G\subseteq \frac{X}{\left\{x\right\}};$ this implies X/{x} is (N_α − OS). Hence, {x} is (N_α − CS).

Conversely: Let {x} be (N_α − CS), ∀ x ∈ X, to prove X is N_α^T1 -space. Let x ≠ y in X, hence {x}, {y} are (N_α − CSs) hence {x}^c, {y}^c are (N_α − OSs) and y ∈ {x}^c, x ∉ {x}^c, x ∈ {y}^c, y ∉ {y}^c. Therefore, X is Na^T1-space.

Definition 7

Let X be a (TS). Then X is named N_α^T2-space if for any two distinct points x, y in X there exists two (N_α − OSs) X satisfy x ∈ A₁, y ∈ A₂ and A₁ ∩ A₂ = ∅

Proposition 4

If X is Na^T2- space, then A = {(x, y) : x = y, x, y ∈ X} is (Na − CS).

Proof

Assume that X is N_α^T2- space, to prove A is (N_α − CS), let (x, y) ∈ A^c ⊆ X × X/A, this mean x and y are two distinct points in X, where X is N_α^T1-space then for some A₁, A₂ ∈ N_αO(X) satisfy x ∈ A₁, y ∈ A₂ and A₁, A₂ are disjoint sets, hence (x, y) ∈ A₁ × A₂ ⊆ A^c, but A₁ × A₂ ∈ N_αO(X × X) (see Theorem 1), hence A^c is N_α- open set, thus A is (N_α − CS).

Proposition 5

If f, g : x → y are N_α∗- continuous and Y is N_α^T2 space, then the set A = {x : x ∈ X f(x) = g(x)} is (N_α − CS).

Proof

If x ∉ A, then x ∈ A^c this mean that f(x) ≠ g(x) in Y, since Y is N_α^T2-space, then there exist B₁, B₂ ∈ N_αO(Y) such that f(x) ∈ B₁, g(x) ∈ B₂ and B₁ ∩ B₂ = ∅, but f⁻¹(B₁), g⁻¹(B₂) ∈ N_αO(X) since f, g are N_α∗-continuous, hence x ∈ f⁻¹(B₁), x ∈ g⁻¹(B₂) hence x ∈ f⁻¹(B₁) ∩ g⁻¹(B₂), let B = f⁻¹(B₁) ∩ g⁻¹(B₂), where B is (N_α − OS). Now we shall prove B ⊆ A^c, i.e B ∩ A = ∅. Suppose that B ∩ A ≠ ∅ this mean y ∈ B ∩ A; thus, y ∈ A, y ∈ B. Hence, y ∈ f⁻¹(B₁), y ∈ g⁻¹(B₂), hence f(y) ∈ B₁, g(y) ∈ B₂, y ∈ A. Thus, f(y) = g(y), since y ∈ A, hence B₁ ∩ B₂ ≠  ∅ , which is a contradiction, thus B ⊆ A^c, thus A^c ∈ N_αO(X),, hence A ∈ N_αC(x).

Proposition 6

If X and Y are N_α^Ti- space, then X × Y is N_α^Ti- space ∀i = 0, 1, 2

Proof

Assume that X and Y are N_α^Ti- space. Put i = 0 and take (x₁, y₁) ≠ (x₂, y₂) in X × Y, then for any two distinct points x₁ and x₂ in X, there exists A₁ ∈ N_αO(X) such that x₁ ∈ A₁, x₂ ∉ A₁ or x₁ ∉ A₁, x₂ ∈ A₁, also y₁ ≠ y_2, then there exists A₂ ∈ N_αO(Y) such that y₁ ∈ A₂, y₂ ∉ A₂ or y₁ ∉ A₂, y₂ ∈ A₂ then (x₁, y₁) ∈ A₁ × A₂ (x₂, y₂) ∉ A₁ × A₂ or (x₁, y₁) ∉ A₁ × A₂(x₂, y₂) ∈ A₁ × A₂ but A₁ × A₂ is (N_α − OS) in X × Y (see Theorem 1). Hence X × Y N_α^Ti- space. Similarly, we can prove other states for i = 1, 2.

Proposition 7

If X is N_α^Ti, then it is N_α^Ti − 1 –space, where i = 2, 1.

Proof

The proof is consider from Definitions 5, 6, and 7.

Theorem 4

The inverse image of N_α^Ti -space under injective N_α∗- continuous mapping is also N_α^Ti space, where i = 0, 1, 2

We shall prove only when i = 2 and the other cases are similarly.

Proof

Let f : X → Y be injective, N_α∗- continuous mapping and x₁ ≠ x₂ in X, since f is injective then y₁ = f(x₁) ≠ f(x₂) = y₂ in Y where Y is N_α^T2 then there exist two disjoint N_α- open set A₁, A₂ in Y satisfy y₁ ∈ A₁, y₂ ∈ A₂, since f is N_α∗- continuous than f⁻¹(A₁), f⁻¹(A₂) are (N_α − OSs) in X such that x₁ ∈ f⁻¹(A₁), x₂ ∈ f⁻¹(A₂) and f⁻¹(A₁) ∩ f⁻¹(A₂) = ∅. Therefore, X is N_α^T2-space.

Theorem 5

If f : X → Y is injective N_α- continuous and Y is T₂ space, then X is N_α^T2 – space.

Proof

Similar to the proof of Theorem 4.

Definition 8

Let X be a (TS). Then X is called gN_α^Ti-space, where i = 0, 1, 2 if:

• (i) i = 0 if for any x ≠ y in X, there exists (gN_α − OS) A containing one of them but not other.

• (ii) i = 1 if for any x ≠ y in X, there exist two (gN_α − OSs) A, B containing x and y, respectively, satisfy y ∉ A, x ∉ B.

• (iii) i = 2 if for each pair of distinct point x, y in X there exist disjoint (gN_α − OSs) A, B such that x ∈ A, y ∈ B.

Proposition 8

Every N_α^Ti- space is gN_α^Ti space.

Proof

The proof is in hand, from Theorem 2 where every N_α- open set is gN_α- open set.

By Propositions 7 and 8 we have the following Diagram 1

Diagram 1 The relationship between N_α^Ti-spaces and gN_α^Ti spaces

Theorem 6

If f : X → Y is injective gN_α- continuous and Y is T₂ – space than X is gN_α − T₂ space.

Proof

Assume that x ≠ y in X, since f is injective, thus f(x) ≠ f(y) in Y where Y is T₂ space, then there exists disjoint open sets A, B satisfy f(x) ∈ A, f(y) ∈ B and A ∩ B = ∅, since f is gN_α- continuous, then f⁻¹(A), f⁻¹(B) are (gN_α − OSs) in X see (Definition 4(iii)) where x ∈ f⁻¹(A), y ∈ f⁻¹(B) and f⁻¹(A) ∩ f⁻¹(B) =  ∅ . Hence, X is gN_α − T₂ space.

4 Conclusion and Future Work

We use N_α-open sets in topological spaces to generate new sorts of N_α-separation axioms and investigate some of their features in this research. Some theorems are also provided. In future work, we will discuss in nonclassical (TS) such as neutrosophic/fuzzy/soft topological spaces.