1. Introduction

The sets play a very important role in General Topology. There are many types of these sets like β-open sets, regularopen sets, semi regularopen sets, b-open sets and others were discussed in soft setting and fuzzy setting see ([4], [5], [14], [15]). Also, some of these sets were applied to investigate new spaces, new maps and new separation axioms see ([9]-[13]). In 1968, Velicko [18], investigated the definitions of δ -open and θ -open, where A is said to be δ -open (resp., θ -open) if for any , there exists an open set B satisfies (resp., ). Di Maio and Noiri [8] showed that in a space X is said to be semi-θ -open if for each , there exists a semi-open set B satisfies , where D is a semi closure of B. The family of all is denoted by An α-open sets are one of these sets. It was first studied in 1965 by O. Njasted. These sets form a topology on X which is finer than τ, see [17], [3]. The notion of Nα -open set was first studied in 2015 using the concept of an α – open set, it was studied by N. A. Dawood and N. M. Ali, see [6]. Here in this paper an attempt has been made to employ the notion of Nα -open set to show a new kind of sets that called -Open set and investigate it's properties. In this paper (X,τ) or a space X always denote a topological space on which no separation axioms are assumed unless explicitly stated. Assume in a space X, cl(A) and int(A) refer to the closure and interior of A respectively.

2. Preliminary Definitions

Definition (2.1) [17], [16], [18]

Let A be a set in space X. We say A is an α-open (resp., preopen and regularopen) if A(int(cl(int(A))),(resp., A int(cl(A))and A=int(cl(A)). Also, their complements are an α- closed (resp., preclosed and regularclosed)). The family of all α- open,(resp. preopen and regularopen) is denoted by _αO(X),(resp., PO(X) and RO(X)).

Definition (2.2) [6]

Let A be a set in a space X. We say A is an N_α-open if there exists non-empty an α-open set B such that cl(B) A, and it's complement is called an N_α -closed. Also, the family of all N_α- open (resp., N_α-closed) sets are denoted by N_αO(X), (resp., N_αC(X)).

Remark (2.3) [6]

(i) X and ϕ are N_α – open sets in every topological space.

(ii) Every subset in discrete space is N_α –open set.

(iii) In the usual topological space every interval is N_α- Open set.

(iv) Every closed and open (resp., α-open, preopen and -open) is N_α-open set.

(v) If A is N_α-open set. Then .

Proposition (2.4) [6]

If A is a closed set of a space X. Then A is N_α – open set if and only if .

Proposition (2.5) [6]

Finite union of N_α- open sets is N_α- open.

Theorem (2.6) [1]

The following statements are equivalent:

(i) X is Alexindroff space.

(ii) For any an open (resp.,closed) sets then their intersection(resp., union) is an open(resp., closed) also.

Definition (2.7) [7]

Let X be a space. If every an open set of X is a closed. Then X is called Locally Indiscrete space.

Proposition (2.8) [2]

For any set A of a space X. Then A is clopen if and only if preopen and closed.

Proposition (2.9) [6]

Let Y be a subspace of a space X such that A Y X. Then:

(i) If A∈ N_αO (X)(resp.N_αC (X)), then A∈ N_αO(Y)(resp. N_αC(Y)).

(ii) If A∈ N_αO(Y)(resp., N_αC (Y)), then A∈ N_αO(X)(resp. N_αC(X)) where Y is clopen set in X.

Theorem (2. 10) [6]

A space X is N_α^** - regular, if and only if all N_α-open set A containing x there exists an open set G containing x such that x∈G cl(G) A.

Theorem (2.11) [6]

(i) If a space X is N_α^** - regular. Then every N_α-open set is an open set.

(ii) If a space X is N_α^** - regular and Alexindroof. Then A is N_α-open if and only if is clopen.

3. N_αc - Open Set and Their Basic Properties

A new kind of sets called N_αc-open using the concept of N_α- open set is studied in this section. Some definitions, propositions and counterexamples are given.

Definition (3.1)

An N_α-open set A of a space X is N_αc- open if for all x∈ A there exists closed set F such that x∈ FA, it's complement is called N_αc- closed set. The family of all N_αc-open (resp.N_αc- closed) is denoted by N_αcO(X)(resp.,(N_αcC(X)).

Proposition (3.2)

A∈ N_αcO(X) if and only if A∈N_αO(X) as well as a union of closed sets.

Proof: Obvious

The above definition refer to every N_αc- open is N_α- open set, but the converse need not be true in general. See the following example:

Example (3.3)

Consider X ={a,b,c,d,e,} and τ = {X,{a},{b,c}, {a,b,c},ϕ}. Let A={a,b,d,e}∈ N_αO(X) but A ∉ N_αcO(X) since only {b,c,d,e} is closed set and contains b. However, {b,c,d,e} is not contained in A.

Corollary (3.4)

(i) Every closed N_α-open set is N_αc –open set.

Proof: Follows from Proposition (3.2).

(ii) Every closed and α-open (resp. -open) set is N_αc-open set.

Proof: Follows from Remark (2.3) (iv)) and Corollary (3.4(i)).

(iii) X and ϕ are N_αc –open sets in every topological space.

Proof: Follows from Remark (2.3) and Corollary (3.4(i)).

Proposition (3.5)

If A∈ N_αcO(X) then, .

Proof: If A∈ N_αcO(X) then A∈N_αO(X) this implies see Remark (2.3). The following example explains that the converse of Proposition of (3. 5) is not true.

Example (3.6)

Consider X = {a, b, c} with . Let A={b,c} then, but A∉N_αcO(X) because A∉N_αO (X).

Proposition (3.7)

If A is a closed set of a space X. Then A∈ N_αcO(X) if and only if

Proof: The first part follows from Proposition (3.5). The other part assume that , implies that A is N_α- open set see Proposition(2.4). Therefore, A∈ N_αcO(X) see Corollary(3.4).

Proposition (3.8)

If A∈ N_αcO(X), then cl(A)∈ N_αcO(X).

Proof: Suppose that A∈ N_αcO(X) then, by Proposition (3.5) where int(A) is an open set so it is an α –open and it's closure contain in cl(A) this implies cl(A)∈N_αO(X) thus by Corollary.(3.4) cl(A)∈N_αcO(X).

Proposition (3.9)

Finite union of N_αc- open sets is N_αc – open.

Proof: Let A_i be N_αc-open set, i=1, 2,…n. Thus A_i is N_α-open set, hence by Proposition (2.5) Let then there exists such that x∈A_i for some i= 1,2, …n, thus there exists closed set F such that x∈ F A_i, hence A_i. Thus .

Definition (3.10)

Let A X. The union of all N_αc - open sets contain in A is called N_αc -interior of A briefly N_αc-int(A).

Proposition (3.11)

Let A be a set of a space X. Then:

(i) N_αc int(A) A.

(ii) If A B then, N_αc int(A) N_αc int(B).

(iii) If A is N_αc – open set then, A = N_αc int(A).

Proof: Obvious.

Corollary (3.12)

A finite set A∈ N_αcO(X) if and only if for all x∈A there exists N_αc –open set B such that x∈B A.

Proof: Let A= {x_1,x₂,..., x_n}, suppose there exists B_i∈ N_αcO(X) such that x_i∈ B_iA, this implies thus by Proposition (3.9) A∈N_αcO(X). Conversely; suppose A∈N_αcO(X), hence by Proposition (3.11 (iii)), A= N_αc int(A), thus the proof is complete.

The intersection of two N_αc –open sets need not be N_αc-open set. See the next example:

Example (3.13)

Consider X={a,b,c,d} with We have A={a,c,d}, B={a,b,c} are N_αc-open sets, but their intersection {a,c} is not N_αc- open set since {a,c} is not N_α- open set.

Proposition (3.14)

If the collection of all N_αO(X) sets is form a topology on X, then N_αcO(X) is also.

Proof: We shall prove only the finite intersection of N_αcO(X) is also. Let A, B∈ N_αcO(X), then A,B∈ N_αO(X) so, AB∈ N_αO(X). Suppose x∈ A B, this implies x∈ A, x∈ B, then there exist closed sets E and F such that x∈E A, x∈F B this implies x∈ E FA B. Thus AB∈ N_αcO(X).

Definition (3.15)

Let AX. The intersection of all N_αc – closed sets containing A is called N_{αc –} closure of A briefly N_αc cl(A).

Proposition (3.16)

Let X be a space, ABX. Then;

(i)

(ii) If A is N_αc – closed set then A= N_αc cl(A). If X is a finite set then, A is N_αc – closed set if and only if A = N_αc cl(A).

Proof: Obvious.

Proposition (3.17)

Let A be a set of a space X. Then, x∈ N_αccl(A) if and only if for each N_αc- open set U_x containing x.

Proof: Suppose that for some U_x∈ N_αcO(X) this implies where U^c∈ N_αcC(X) not containing x this implies x ∉ N_αccl(A) which is a contradiction to hypothesis. Thus for any U_x∈ N_αcO(X). Conversely is a similarly.

Proposition (3.18)

Every regular closed set is N_αc – open.

Proof: Suppose that is regular closed so it is closed set since then where int(A) is an α – open set hence A∈ N_αO(X), thus by Corollary (3.4) A∈ N_αcO(X).

Lemma (3.19) [14]

Let X₁, X₂, be topological spaces. Then A₁, A₂, are N_α - open sets in X₁, X₂ respectively if and only if A₁ × A₂ is N_α - open set in X₁×X₂.

Proposition (3.20)

Let X₁, X₂ be topological spaces. Then A₁, A₂, are N_αc - open sets in X₁, X₂ respectively if and only if A₁ × A₂ is N_αc - open set in X₁× X₂.

Proof: Assume A₁, A₂∈ N_αc O(X₁), N_αc O(X₂) respectively, then by Proposition (3. 2)) A₁, A₂∈ N_α O(X₁), N_α O(X₂) respectively hence A₁× A₂∈ N_α O(X₁×X₂) see Lemma (3. 19), on the other hand for each _X1∈ A₁, _X2∈ A₂ there exist closed sets F₁, F₂, such that x₁∈ F₁ , A₁, x₂∈ F₂A₂. Thus (x₁,x₂)∈ F₁ × F₂A₁×A₂, where F₁ × F₂ is closed set in X₁ × X₂. Hence A₁ × A₂∈ N_αc O(X₁ × X₂). Conversely; if A₁ × A₂∈ N_αc O(X₁ × X₂) then, by Proposition (3.2) A₁ × A₂∈ N_α O(X₁ × X₂), hence by Lemma (3. 19) A₁ ∈ N_αO(X₁), A₂∈ N_α O(X₂). On the other hand for each (x₁, x₂) ∈ A₁×A₂ there exists closed set F₁× F₂ in X₁ × X₂ such that (x₁, x₂)∈ F₁×F₂ A₁×A₂, thus x₁∈ F₁₁, x₂∈ F₂ A₂, where F₁, F₂ are closed sets in X₁, X₂ respectively. Thus A₁, A₂∈ N_αc O(X₁), N_αc O(X₂) respectively.

Proposition (3.21)

Let X₁, X₂ be topological spaces. Then N_αc cl(A₁ × A₂) = N_αc cl(A₁) × N_αc cl(A₂), where A₁ X₁, A₂ X₂.

Proof: Let (x₁, x₂)∈ N_αc cl(A₁ × A₂), then by Proposition (3. 17)) for each N_αc- - open set in X₁×X₂ say G=G₁ ×G₂ containing (x₁, x₂) then Then where G₁,G₂∈N_αcO(X₁), N_αcO(X₂) resp., see Proposition (3.20) which means and . Thus x₁∈ N_αc cl(A₁), x₂∈ N_αc cl(A₂). Hence (x₁, x₂)∈ N_αc cl(A₁) × N_αc cl(A₂).

Conversely; Let (x₁, x₂)∈ N_αc cl(A₁) _× N_αc cl(A₂), then x₁∈ N_αc cl(A₁), x₂∈ N_αccl(A₂), hence for each N_αc – open sets G₁,G₂ containing x₁, x₂ respectively, then thus , this implies (A₁ × A₂) (G₁ × G₂), but G₁ ×G₂∈ N_αc O(X₁ × X₂) see Proposition (3. 20). Thus (x₁, x₂)∈ N_αc cl(A₁ × A₂).

Proposition (3.22)

Let Y be a subspace of a space X such that AYX. Then:-

(i) If A∈ N_αcO(X) then, A∈ N_αcO(Y).

(ii) If A∈ N_αcO(Y) then, A∈ N_αcO(X), where Y is clopen set in X.

Proof:

(i) Suppose that A∈ N_αcO(X), hence A∈ N_αO(X), thus by Proposition (2.9) A∈ N_αO(Y), let x∈A, hence there closed set F in X s.t x∈ FA for each x∈A, hence F is closed in Y, thus A∈ N_αcO(Y).

(ii) Suppose that A∈ N_αcO(Y) hence A∈ N_αO(Y) since Y is clopen set in X then by Proposition (2.9) A∈ N_αO(X). Let x∈ A, thus closed set F in Y s.t x∈ F A so F is closed set in X. Thus A∈ N_αcO(X).

Corollary (3.23)

Let Y be a subspace of a space X where A Y X such that Y is an α - closed set in X then A∈ N_αcC(X) if and only if A∈ N_αcC(Y)

Proof: Follows from Proposition (2.9).

Now, we shall discuss the relationships between our concept with some other classes of sets.

Remarks (3.24)

The following sets are independent with N_αc-open sets

(i) Open sets are incomparable with N_αcO(X)

Example (1): Let (R, τ_u) be the usual topology on real number R, we observe [a,b]∈ N_αcO(R) but [a,b]∉ τ_u. On the other hand every an open set is incomparable with N_αO(X), see [14].

(ii) Closed sets are incomparable with N_αcO(X) i. e

Example (2): In previous example . Assume that A=(0,1] then A∈ N_αcO(R) because A∈ N_αO(X) see Remark(2.3) so, hence A∈ N_αcO(X) see Proposition(3.2) but A is not closed set in . On the other hand consider X={a,b,c,d} with , let A={a,c}, then A is closed set in X but A ∉ N_αcO(X) because A ∉ N_αO(X).

(iii) _αOpen sets are incomparable with N_αcO(X)

Example (3): See Ex. (2) in (X, τ) let A={a,b,d} then A∈ _αO(X) but A ∉ N_αcO(X) because A ∉ N_αO(X), also if A =[a,b] in (R, τ_U), then A∈ N_αcO(X) but A ∉ _αO(X).

(iv) Preopen sets are incomparable with N_αcO(X)

Example (4): See Ex. 2. in (X, τ). Assume that A={b,c,d}, then A∈ PO(X) but A∉ N_αcO(X). Also we have A=[a,b] in (R,τ_U) where A∈ N_αcO(X) but A∉ PO(X).

(v) -Open sets are incomparable with N_αcO(X)

Example (5): In Ex. 2. We notice in (X_,τ), if A= {b}, then but A∉ N_αcO(X). Also if A={a,c,d}, then A∈ N_αcO(X) but A∉ O(X).

Proposition (3.25)

Let AX. Then:

(i) If A∈θ O(X), then A∈ N_αcO(X).

Proof: Follows from Definition of θ – open sets and τ _αO(X).

(ii) If A is clopen set, then A∈ N_αcO(X).

Proof: Direct by Remark (2.3) and Corollary (3.4(i)). The converse of (i), (ii) is not true. See the following example:

Example 6: See example (2) in Remarks (3.24) of a space X. Suppose that A={a,c,d}. Then A∈ N_αcO(X) but it is neither θ - open nor clopen set.

Proposition (3.26)

Every closed preopen is N_αc-open set.

Proof: It follows by Proposition (2.8) and Proposition. (3.25(ii)).

From above discussions, we have the following Diagram:

Now, we can prove the converse of previous results such as (3. 2) and (3. 25) are true if we add some conditions. See the next results:-

Proposition (3.27)

N_αcO(X) = N_αO(X) where a space X is T₁-space.

Proof: We show only N_αO(X) N_αcO(X). Assume that A is N_α -open set, let x∈ A then {x} is closed set, thus x∈{x}A. Thus A∈ N_αcO(X).

Proposition (3.28)

If a space X is N_α^** - regular space. Then we have the following results:-

(i) N_αO(X) = N_αcO(X).

(ii) θO(X) = N_αcO(X).

(iii) CO(X) = N_αcO(X) where X is Alexindroff space.

(iv) τ= N_αcO(X) where X is Locally Indiscrete space.

Proof: (i) We have N_αcO(X)) N_αO(X). To prove N_αO(X) N_αcO(X). Let A be N_α-open set, since X is N_α^** - regular space then for all x∈ A an open set G s.t x∈ G cl(G) A see Theorem (2.10). Hence A∈N_αcO(X) see Proposition (3.2). Thus N_αO(X) = N_αcO(X).

(ii) We have θO(X)) N_αcO(X) by Proposition (3.25). Assume A∈ N_αcO(X), then A∈ N_α O(X). Since X is N_α^**-regular space, then for all x∈ A there exists an open set G such that x∈ G cl(G) A this implies A∈ θO(X). Thus θO(X)= N_αcO(X).

(iii) We have every clopen set is N_αc-Open see Proposition (3.25). We shall prove N_αcO(X) CO(X). Since A∈ N_αcO(X), then A∈ N_αO(X). Since X is Alexindroof space then by Theorem (2.11) A∈ CO(X). Thus CO(X) = N_αcO(X).

(v) Suppose that A is nonempty an open set. Since X is Locally Indiscrete space, then A is closed, thus A is clopen set so it is N_α – open set see Remark (2. 3). Thus A∈ N_αcO (X) see Corollary (3.4). Let A∈ N_αcO (X), then A∈ N_αO (X). Since X is N_α ** - regular space, then A is an open set see Theorem (2.11). Hence N_αc O(X) = τ.

Proposition (3.29)

Each -Open set is N_αc – Open where a space X is Locally Indiscrete.

Proof: Suppose that A∈ O(X) so it is an open set. We have X is Locally Indiscrete hence A is closed, this implies A is clopen set then by Remark (2. 3) A is N_α - open set, thus by Corollary (3.4) A∈ N_αcO(X).

Proposition (3.30)

N_αcO (X) O(X) where X is a discrete space.

Proof: Assume that A∈ N_αc O (X), hence A∈ N_αO (X) and for all x∈ A there exists closed set F such that x∈ FA. Since X is discrete space, then every closed set is an open, this implies x∈ F = cl(F) = int (cl(F)A. Hence A∈ O(X).

Proposition (3.31)

Each an open set in a regular space X is N_αc-open set.

Proof: Let A∈ τ. Since X is a regular space then there exists an open set G such that x∈ G cl(G) A, for each x∈A. We get cl(G) A, where G is an open set so it is an α - open thus A∈ N_αO (X) and A= cl(G) for each x∈A. Thus by Proposition (3. 2)) A∈ N_αcO(X).

4. Conclusions

In this paper we have introduced the notion of Nα_c -open sets. Next, some of their basic properties are discussed and studied. Moreover, some relationships between our concepts with some classes of sets are investigated. This new notion will help us to study a new kind of N_αc – continuous mappings and to study a new kind of N_αc – separation axioms.