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On Neutrosophic Generalized Semi Generalized Closed Sets

On Neutrosophic Generalized Semi Generalized Closed Sets

Qays Hatem Imran, Ali H. M. Al-Obaidi, Florentin Smarandache, Said Broumi

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Qays Hatem Imran, Ali H. M. Al-Obaidi, Florentin Smarandache, Said Broumi (2022). On Neutrosophic Generalized Semi Generalized Closed Sets. International Journal of Neutrosophic Science 18(3), 10-20; DOI: 10.54216/IJNS.18030101

Abstract

The article considers a new generalization of closed sets in neutrosophic topological space. This generalization is called neutrosophic ℊ��ℊ-closed set. Moreover, we discuss its essential features in neutrosophic topological spaces. Furthermore, we extend the research by displaying new related definitions such as neutrosophic ℊ��ℊ-closure and neutrosophic ℊ��ℊ-interior and debating their powerful characterizations and relationships.

Keywords: ������ℊ��ℊ����, ������ℊ��ℊ����, ������ℊ��ℊ-closure and ������ℊ��ℊ-interior.

1. Introduction

The neutrosophic set theory was contributed by Smarandache in [1,2]. The neutrosophic topological space (simply ����������) was offered by Salama et al. in [3]. The definition of semi--open sets in neutrosophic topological spaces was displayed by Imran et al. in [4]. The neutrosophic generalized homeomorphism was submitted by PAGE et al. in [5]. The class of generalized neutrosophic closed sets was given by Dhavaseelan et al. in [6]. The concepts of neutrosophic generalized ��ℊ-closed sets and neutrosophic generalized ��ℊ-continuous functions were provided by Imran et al. in [7]. The objective of this article is to show the sense of neutrosophic ℊ��ℊ-closed set (briefly ������ℊ��ℊ����) and investigate their main characteristics in ����������. Moreover, we argue neutrosophic ℊ��ℊ-closure (in word ������ℊ��ℊ-closure) and neutrosophic ℊ��ℊ-interior (fleetingly ������ℊ��ℊ-interior) with revealing several of their vital spots.

2. Preliminaries

In this work, ��,�� (or simply ��) always mean ����������. Let �� be a neutrosophic set in a ���������� ��,�� , we denote the neutrosophic closure, the neutrosophic interior, and the neutrosophic complement of �� by ����������(��), ������������(��) and ���� = 1������ − ��, respectively.

Definition 2.1: [3]

The family �� of neutrosophic subsets of a non-empty neutrosophic set �� ≠ ∅ is called a neutrosophic topology (in short, ��������) on �� if it satisfies the below axioms:

(i) 0������, 1������ ∈ ��, (ii) ��1⨅��2 ∈ �� being ��1,��2 ∈ ��, (iii) ⨆���� ∈ �� for arbitrary family {����|�� ∈ ��}⊑ ��. In this case, we signified ���������� by ��,�� or ��. Moreover, the neutrosophic set in �� is named neutrosophic open (in short, ����������). Furthermore, for any ���������� ��, then ���� is titled neutrosophic closed set (briefly, ����������) in ��.

Definition 2.2:

Let �� be a neutrosophic subset of a ����������(��,��), then it is called to be: (i) a neutrosophic semi-open set and denoted by ������������ if �� ⊑ ����������(������������ �� ). [8] (ii) a neutrosophic semi-closed set and denoted by ������������ if ������������(���������� �� )⊑ ��. The intersection of entire ��������������, including �� is named a neutrosophic semi-closure, and it is symbolized by ������������(��).[8] (iii) a neutrosophic ��-open set and denoted by ������������ if �� ⊑ ������������(����������(������������ �� )). [9] (iv) a neutrosophic ��-closed set and denoted by ������������ if ������������(������������(���������� �� ))⊑ ��. The intersection of the whole �������������� including �� is named neutrosophic ��-closure, and it is symbolized by ������������(��).[9]

Definition 2.3:

Let �� be a neutrosophic subset of a ����������(��,��), and let �� be a a ���������� in (��,��) such that �� ⊑ �� then �� is called to be: (i) a neutrosophic generalized closed set, and it is denoted by ������ℊ���� if ����������(��)⊑ ��. The complement of a ������ℊ���� is a ������ℊ���� in (��,��). [10] (ii) a neutrosophic ��ℊ-closed set, and it is denoted by ��������ℊ���� if ������������(��)⊑ ��. The complement of a ��������ℊ���� is a ��������ℊ���� in (��,��). [11] (iii) a neutrosophic ℊ��-closed set, and it is denoted by ������ℊ������ if ������������(��)⊑ ��. The complement of a ������ℊ������ is a ������ℊ������ in (��,��). [12] (iv) a neutrosophic ��ℊ-closed set, and it is denoted by ��������ℊ���� if ������������(��)⊑ ��. The complement of a ��������ℊ���� is a ��������ℊ���� in (��,��). [13] (v) a neutrosophic ℊ��-closed set, and it is denoted by ������ℊ������ if ������ℊ����(��)⊑ ��. The complement of a ������ℊ������ is a ������ℊ������ in (��,��). [14]

Proposition 2.4:[9,10]

In a ����������(��,��), then the next arguments stand, and the opposite of every argument is not valid: (i) Each ���������� (resp. ����������) is a ������������ (resp. ������������). (ii) Each ���������� (resp. ����������) is a ������ℊ���� (resp. ������ℊ����).

(iii) Each ������������ (resp. ������������) is a ������������ (resp. ������������).

Proposition 2.5:[11,12]

In a ����������(��,��), then the next arguments stand, and the opposite of every argument is not valid: (i) Each ������ℊ���� (resp. ������ℊ����) is a ��������ℊ���� (resp. ��������ℊ����). (ii) Each ������������ (resp. ������������) is a ������ℊ������ (resp. ������ℊ������). (iii) Each ������ℊ������ (resp. ������ℊ������) is a ��������ℊ���� (resp. ��������ℊ����).

Proposition 2.6:[13-15]

In a ����������(��,��), then the next arguments stand, and the opposite of every argument is not valid: (i) Each ������ℊ���� (resp. ������ℊ����) is a ������ℊ������ (resp. ������ℊ������). (ii) Each ������������ (resp. ������������) is a ��������ℊ���� (resp. ��������ℊ����). (iii) Each ��������ℊ���� (resp. ��������ℊ����) is a ������ℊ������ (resp. ������ℊ������). (iv) Each ������ℊ������ (resp. ������ℊ������) is a ������ℊ������ (resp. ������ℊ������).

3. Neutrosophic Generalized ����-Closed Sets

In this sector, we present and analyse the neutrosophic generalized ��ℊ-closed sets and some of their features.

Definition 3.1:

Suppose that �� is a neutrosophic set in a ����������(��,��) and assume that �� is a ��������ℊ���� in (��,��) where �� ⊑ ��. The set �� is termed as a neutrosophic generalized ��ℊ-closed set, and it is signified by ������ℊ��ℊ���� if ����������(��)⊑ ��. The collection of all ������ℊ��ℊ������ in a ����������(��,��) is signified by ������ℊ��ℊ��(��).

Theorem 3.2:

In a ����������(��,��), the subsequent arguments are valid: (i) Each ���������� is a ������ℊ��ℊ����. (ii) Each ������ℊ��ℊ���� is a ������ℊ����.

Proof:

(i) Let ������������ and ��������ℊ������ be in a ����������(��,��) where �� ⊑ ��. Then ���������� �� =�� ⊑ ��. Therefore �� is a ������ℊ��ℊ����. (ii) Let ������ℊ��ℊ���� �� and ���������� �� be in a ����������(��,��) where �� ⊑ ��. Because each ���������� is a ��������ℊ����, we get ���������� �� ⊑ ��. Consequently, �� is a ������ℊ����.  The reverse of the above theorem is inaccurate, as displayed in the subsequent instances.

Example 3.3:

Suppose that ��= {��1,��2} is a set and assume that �� = {0������ ,��1,��2, 1������ } is a �������� defined on ��. Suppose that we have the sets ��1 = ��, 0.6,0.7 , 0.1,0.1 , (0.4,0.2) and ��2 = ��, 0.1,0.2 , 0.1,0.1 , (0.8,0.8) are given. Then the neutrosophic set ��3 = ��, 0.2,0.2 , 0.1,0.1 , (0.6,0.7) is a ������ℊ��ℊ����. However, this latter set is not a ����������.

Example 3.4:

Suppose that ��= {��1,��2,��3} is a set and assume that �� = {0������ ,��1,��2, 1������ } is a �������� defined on ��. Suppose that we have the following sets ��1 = ��, 0.5,0.5,0.4 , 0.7,0.5,0.5 , (0.4,0.5,0.5) and ��2 =

��, 0.3,0.4,0.4 , 0.4,0.5,0.5 , (0.3,0.4,0.6) are given. Then the neutrosophic set ��3 = ��, 0.4,0.6,0.5 , 0.4,0.3,0.5 , (0.5,0.6,0.4) is a ������ℊ����. However, this latter set is not a ������ℊ��ℊ����.

Theorem 3.5:

In a ����������(��,��), the subsequent arguments are valid: (i) Each ������ℊ��ℊ���� is a ��������ℊ����. (ii) Each ������ℊ��ℊ���� is a ������ℊ������. (iii) Each ������ℊ��ℊ���� is a ��������ℊ����. (iv) Each ������ℊ��ℊ���� is a ������ℊ������.

Proof:

(i) Let ������ℊ��ℊ���� �� and ���������� �� be in a ����������(��,��) where �� ⊑ ��. Because each ���������� is a ��������ℊ����, we get ������������(��)⊑ ���������� �� ⊑ ��. The latter implies ������������(��)⊑ ��. Consequently, �� is a ��������������. (ii) Let ������ℊ��ℊ���� �� and �������������� be in a ����������(��,��) where �� ⊑ ��. Because each ������������ is a ������������, which is a ��������ℊ����, we get ������������(��)⊑ ���������� �� ⊑ ��. The latter implies ������������(��)⊑ ��. Consequently, �� is a ������ℊ������. (iii) Let ������ℊ��ℊ������ and �������������� be in a ����������(��,��) where �� ⊑ ��. Because each ������������ is a ��������ℊ����, we get ������������(��)⊑ ���������� �� ⊑ ��. The latter implies ������������(��)⊑ ��. Consequently, �� is a ��������ℊ����. (iv) Let ������ℊ��ℊ���� �� and ���������� �� be in a ����������(��,��) where �� ⊑ ��. Because each ���������� is a ��������ℊ����, we get ������������(��)⊑ ���������� �� ⊑ ��. That implies ������������(��)⊑ ��. Consequently, �� is a ������ℊ������. The reverse of the above theorem is inaccurate, as displayed in the subsequent instances.

Example 3.6:

Let ��= {��1,��2} be a set and assume that �� = {0������ ,��1,��2, 1������ } is a �������� defined on ��. Suppose that we have the following sets ��1 = ��, 0.5,0.6 , 0.3,0.2 , (0.4,0.1) and ��2 = ��, 0.4,0.4 , 0.4,0.3 , (0.5,0.4) are given. Then the neutrosophic set ��3 = ��, 0.5,0.4 , 0.4,0.4 , (0.4,0.5) is a ��������ℊ���� and hence ������ℊ������ but not a ������ℊ��ℊ����.

Example 3.7:

Let ��= {��1,��2} and let �� = {0������ ,��1, 1������ } be a �������� on ��. Take ��1 = ��, 0.3,0.4,0.6 , 0.6,0.6,0.4 . Then the neutrosophic set ��2 = ��, 0.3,0.2,0.5 , 0.6,0.6,0.8 is a ��������ℊ���� but not a ������ℊ��ℊ����.

Example 3.8:

Let ��= {��1,��2} and let �� = {0������ ,��1, 1������ } be a �������� on ��. Where ��1 = ��, 0.3,0.2,0.3 , 0.8,0.6,0.7 . Then the neutrosophic set ��2 = ��, 0.3,0.2,0.6 , 0.8,0.9,0.8 is a ������ℊ������. However, this latter set is not a ������ℊ��ℊ����.

Remark 3.9:

The ������ℊ��ℊ���� are independent of ������������ and ������������.

Definition 3.10:

A neutrosophic subset �� of a ����������(��,��) is called a neutrosophic generalized ��ℊ-open set (in short, ������ℊ��ℊ����) iff 1������ − �� is a ������ℊ��ℊ����. The collection of entire ������ℊ��ℊ������ of a ����������(��,��) is signified by ������ℊ��ℊ��(��).

Proposition 3.11:

Let �� be a ���������� in ����������(��,��), then this set �� is ������ℊ��ℊ���� in the space (��,��).

Proof:

Let �� be a ���������� in a ����������(��,��), then 1������ − �� is a ���������� in (��,��). According to theorem (3.2), point (i), 1������ − �� is a ������ℊ��ℊ����. Therefore, �� is a ������ℊ��ℊ���� in (��,��).

Proposition 3.12:

Let �� be a ������ℊ��ℊ���� in ����������(��,��), then this set �� is ������ℊ���� in the space (��,��).

Proof:

Let �� be a ������ℊ��ℊ���� in a ����������(��,��), then 1������ − �� is a ������ℊ��ℊ���� in (��,��). According to theorem (3.2), point (ii), 1������ − �� is a ������ℊ����. Therefore, �� is a ������ℊ���� in (��,��).

Theorem 3.13:

In a ����������(��,��), the subsequent arguments are valid: (i) Each ������ℊ��ℊ���� is a ��������ℊ���� and ������ℊ������. (ii) Each ������ℊ��ℊ���� is a ��������ℊ���� and ������ℊ������.

Proof:

Similar to above proposition. 

Proposition 3.14:

If �� and �� are ������ℊ��ℊ������ in a ����������(��,��), then ��⨆�� is a ������ℊ��ℊ����.

Proof:

Let �� and �� be two ������ℊ��ℊ������ in a ����������(��,��) and let �� be any ��������ℊ���� in �� such that �� ⊑ �� and �� ⊑ ��. Then we have ��⨆�� ⊑ ��. Since �� and �� are ������ℊ��ℊ������ in ��, ����������(��)⊑ �� and ����������(��)⊑ ��. Now, ���������� ��⨆�� =����������(��)⨆����������(��)⊑ �� and so ���������� ��⨆�� ⊑ ��. Hence ��⨆�� is a ������ℊ��ℊ���� in ��.

Proposition 3.15:

If �� is a ������ℊ��ℊ���� in a ����������(��,��), then ���������� �� − �� does not include non-empty ���������� in (��,��).

Proof:

Let �� be a ������ℊ��ℊ���� in a ����������(��,��) and let �� be any ���������� in (��,��) such that �� ⊑ ���������� �� − ��. Since �� is a ������ℊ��ℊ����, we have ���������� �� ⊑1������ − ��. This implies �� ⊑1������ − ����������(��). Then �� ⊑ ���������� �� ⨅(1������ − ���������� �� ) = 0������ . Thus, ��= 0������ . Hence ���������� �� − �� does not include non-empty ���������� in (��,��).

Proposition 3.16:

A neutrosophic set �� is ������ℊ��ℊ���� in a ����������(��,��) iff ���������� �� − �� does not include non-empty ��������ℊ���� in (��,��).

Proof:

Let �� be a ������ℊ��ℊ���� in a ����������(��,��) and let ℜ be any ��������ℊ���� in (��,��) such that ℜ ⊑ ���������� �� − ��. Since �� is a ������ℊ��ℊ����, we have ���������� �� ⊑1������ − ℜ. This implies ℜ ⊑1������ − ����������(��). Then ℜ ⊑ ���������� �� ⨅(1������ − ���������� �� ) = 0������ . Thus, ℜ is empty. Conversely, suppose that ���������� �� − �� does not include non-empty ��������ℊ���� in (��,��). Let �� ⊑ �� and �� is ��������ℊ����. If ���������� �� ⊑ �� then ���������� �� ⨅(1������ − ��) is non-empty. Since ���������� �� is ���������� and 1������ − �� is ��������ℊ����, we have ���������� �� ⨅(1������ − ��) is not empty ��������ℊ���� of ���������� �� − ��, which is a contradiction. Therefore ���������� �� ⋢ ��. Hence �� is a ������ℊ��ℊ����.

Proposition 3.17:

If �� is a ������ℊ��ℊ���� in a ����������(��,��) and �� ⊑ �� ⊑ ����������(��), then �� is a ������ℊ��ℊ���� in (��,��).

Proof:

Assume the set �� is a ������ℊ��ℊ���� in a ����������(��,��). Suppose the set �� is a ��������ℊ���� in (��,��) where �� ⊑ ��. So, �� ⊑ ��. Because �� is a ������ℊ��ℊ����, it observes that ����������(��)⊑ ��. Currently, �� ⊑ ����������(��) suggests ����������(��)⊑ ����������(����������(��)) =����������(��). Thus, ����������(��)⊑ ��. Hence �� is a ������ℊ��ℊ����.

Proposition 3.18:

Let �� ⊑ �� ⊑ �� and if �� is a ������ℊ��ℊ���� in �� then �� is a ������ℊ��ℊ���� relative to ��.

Proof:

�� ⊑ ��⨅�� where �� is a ��������ℊ���� in ��. Then �� ⊑ �� and hence ���������� �� ⊑ ��. This implies that ��⨅���������� �� ⊑ ��⨅��. Thus �� is a ������ℊ��ℊ���� relative to ��.

Proposition 3.19:

If �� is a ��������ℊ���� and a ������ℊ��ℊ���� in a ����������(��,��), then �� is a ���������� in (��,��).

Proof:

Suppose that �� is a ��������ℊ���� and a ������ℊ��ℊ���� in a ����������(��,��), then ����������(��)⊑ �� and since �� ⊑ ����������(��). Thus, ���������� �� =��. Hence �� is a ����������.

Theorem 3.20:

For each �� ∈ �� either ��, 0.1,0.1 is a ��������ℊ���� or 1������ − ��, 0.1,0.1 is a ������ℊ��ℊ���� in ��.

Proof:

If ��, 0.1,0.1 is not a ��������ℊ���� in �� then 1������ − ��, 0.1,0.1 is not a ��������ℊ���� and the only ��������ℊ���� containing 1������ − ��, 0.1,0.1 is the space �� itself. Therefore ���������� 1������ − ��, 0.1,0.1 ⊑1������ and so 1������ − ��, 0.1,0.1 is a ������ℊ��ℊ���� in ��.

Proposition 3.21:

If �� and �� are ������ℊ��ℊ������ in a ����������(��,��), then ��⨅�� is a ������ℊ��ℊ����.

Proof:

Let �� and �� be ������ℊ��ℊ������ in a ����������(��,��). Then 1������ − �� and 1������ − �� are ������ℊ��ℊ������. By proposition (3.14), 1������ − �� ⨆(1������ − ��) is a ������ℊ��ℊ����. Since 1������ − �� ⨆(1������ − ��) = 1������ −(��⨅��). Hence ��⨅�� is a ������ℊ��ℊ����.

Theorem 3.22:

A neutrosophic set �� is ������ℊ��ℊ���� iff �� ⊑ ������������(��) where �� is a ������ℊ��ℊ���� and �� ⊑ ��.

Proof:

Suppose that �� ⊑ ������������(��) where �� is a ������ℊ��ℊ���� and �� ⊑ ��. Then 1������ − �� ⊑1������ − �� and 1������ − �� is a ��������ℊ���� by theorem (3.13) part (ii). Now, ����������(1������ − ��) = 1������ − ������������(��)⊑1������ − ��. Then 1������ − �� is a ������ℊ��ℊ����. Hence �� is a ������ℊ��ℊ����. Conversely, let �� be a ������ℊ��ℊ���� and �� be a ������ℊ��ℊ���� and �� ⊑ ��. Then 1������ − �� ⊑1������ − ��. Since 1������ − �� is a ������ℊ��ℊ���� and 1������ − �� is a ��������ℊ����, we have ����������(1������ − ��)⊑1������ − ��. Then �� ⊑ ������������(��).

Theorem 3.23:

If �� ⊑ �� ⊑ �� where �� is a ������ℊ��ℊ���� relative to �� and �� is a ������ℊ��ℊ���� in ��, then �� is a ������ℊ��ℊ���� in ��.

Proof:

Let �� be a ��������ℊ���� in �� and suppose that �� ⊑ ��. Then ��=��⨅�� is a ��������ℊ���� in ��. But �� is a ������ℊ��ℊ���� relative to ��. Therefore �� ⊑ ��������������(��). Since ��������������(��) is a ���������� relative to ��. We have �� ⊑ ��⨅�� ⊑ ��, for some ���������� �� in ��. Since �� is a ������ℊ��ℊ���� in ��, we have �� ⊑ ������������(��)⊑ ��. Therefore �� ⊑ ������������(��)⨅�� ⊑ ��⨅�� ⊑ ��. It follows that �� ⊑ ������������(��). Thus, �� is a ������ℊ��ℊ���� in ��.

Theorem 3.24:

If �� is a ������ℊ��ℊ���� in a ����������(��,��) and ������������(��)⊑ �� ⊑ ��, then �� is a ������ℊ��ℊ���� in (��,��).

Proof:

Suppose that �� is a ������ℊ��ℊ���� in a ����������(��,��) and ������������(��)⊑ �� ⊑ ��. Then 1������ − �� is a ������ℊ��ℊ���� and 1������ − �� ⊑1������ − �� ⊑ ����������(1������ − ��). Then 1������ − �� is a ������ℊ��ℊ���� by proposition (3.17). Hence, �� is a ������ℊ��ℊ����.

Theorem 3.25:

For a neutrosophic subset �� of a ����������(��,��), the following statements are equivalent: (i) �� is a ������ℊ��ℊ����. (ii) ���������� �� − �� contains no non-empty ��������ℊ����. (iii) ���������� �� − �� is a ������ℊ��ℊ����.

Proof:

Follows from proposition (3.16) and proposition (3.18). 

Remark 3.26:

The subsequent illustration reveals the relative among the diverse kinds of ����������:

������������ ������ℊ������ ��������ℊ����

���������� ������ℊ��ℊ����

��������ℊ���� ������ℊ����

������������ ��������ℊ���� ������ℊ������

Fig. 3.1

4. Neutrosophic ������-Closure and Neutrosophic ������-Interior

We present neutrosophic ℊ��ℊ-closure and neutrosophic ℊ��ℊ-interior and obtain some of its properties in this section.

Definition 4.1:

The intersection of all ������ℊ��ℊ������ in a ����������(��,��) containing �� is called neutrosophic ℊ��ℊ-closure of �� and is denoted by ������ℊ��ℊ����(��).

Definition 4.2:

The union of all ������ℊ��ℊ������ in a ����������(��,��) contained in �� is called neutrosophic ℊ��ℊ-interior of �� and is denoted by ������ℊ��ℊ������ �� .

Proposition 4.3:

Let �� be any neutrosophic set in a ����������(��,��). Then the following properties hold: (i) ������ℊ��ℊ������ �� =�� iff �� is a ������ℊ��ℊ����. (ii) ������ℊ��ℊ����(��) =�� iff �� is a ������ℊ��ℊ����. (iii) ������ℊ��ℊ������ �� is the largest ������ℊ��ℊ���� contained in ��. (iv) ������ℊ��ℊ����(��) is the smallest ������ℊ��ℊ���� containing ��.

Proof:

(i), (ii), (iii) and (iv) are obvious. 

Proposition 4.4:

Let �� be any neutrosophic set in a ����������(��,��). Then the following properties hold: (i) ������ℊ��ℊ������ 1������ − �� = 1������ −(������ℊ��ℊ���� �� ), (ii) ������ℊ��ℊ���� 1������ − �� = 1������ −(������ℊ��ℊ������ �� ).

Proof:

(i) By definition, ������ℊ��ℊ���� �� =⨅{��:�� ⊑ ��,�� is a ������ℊ��ℊ����} 1������ −(������ℊ��ℊ���� �� ) = 1������ − ⨅{��:�� ⊑ ��,�� is a ������ℊ��ℊ����} =⨆{1������ − ��:�� ⊑ ��,�� is a ������ℊ��ℊ����} =⨆{��:�� ⊑1������ − ��,�� is a ������ℊ��ℊ����} =������ℊ��ℊ������ 1������ − �� . (ii) The evidence is analogous to (i). 

Theorem 4.5:

Let �� and �� be two neutrosophic sets in a ����������(��,��). Then the following properties hold: (i) ������ℊ��ℊ����(0������ ) = 0������ , ������ℊ��ℊ����(1������ ) = 1������ . (ii) �� ⊑ ������ℊ��ℊ���� �� . (iii) �� ⊑ �� ⟹������ℊ��ℊ���� �� ⊑ ������ℊ��ℊ���� �� . (iv) ������ℊ��ℊ���� ��⨅�� ⊑ ������ℊ��ℊ���� �� ⨅������ℊ��ℊ���� �� . (v) ������ℊ��ℊ���� ��⨆�� =������ℊ��ℊ���� �� ⨆������ℊ��ℊ���� �� . (vi) ������ℊ��ℊ����(������ℊ��ℊ���� �� ) =������ℊ��ℊ���� �� .

Proof:

(i) and (ii) are obvious. (iii) By part (ii), �� ⊑ ������ℊ��ℊ���� �� . Since �� ⊑ ��, we have �� ⊑ ������ℊ��ℊ���� �� . But ������ℊ��ℊ���� �� is a ������ℊ��ℊ����. Thus ������ℊ��ℊ���� �� is a ������ℊ��ℊ���� containing ��. Since ������ℊ��ℊ����(��) is the smallest ������ℊ��ℊ���� containing ��, we have ������ℊ��ℊ���� �� ⊑ ������ℊ��ℊ���� �� . (iv) We know that ��⨅�� ⊑ �� and ��⨅�� ⊑ ��. Therefore, by part (iii), ������ℊ��ℊ���� ��⨅�� ⊑ ������ℊ��ℊ���� �� and ������ℊ��ℊ���� ��⨅�� ⊑ ������ℊ��ℊ���� �� . Hence ������ℊ��ℊ���� ��⨅�� ⊑ ������ℊ��ℊ���� �� ⨅������ℊ��ℊ���� �� . (v) Since �� ⊑ ��⨆�� and �� ⊑ ��⨆��, it follows from part (iii) that ������ℊ��ℊ���� �� ⊑ ������ℊ��ℊ���� ��⨆�� and ������ℊ��ℊ���� �� ⊑ ������ℊ��ℊ���� ��⨆�� . Hence ������ℊ��ℊ���� �� ⨆������ℊ��ℊ���� �� ⊑ ������ℊ��ℊ���� ��⨆�� ………… (1)

Since ������ℊ��ℊ���� �� and ������ℊ��ℊC�� �� are ������ℊ��ℊ������, ������ℊ��ℊ���� �� ⨆������ℊ��ℊ���� �� is also ������ℊ��ℊ���� by proposition (3.14). Also �� ⊑ ������ℊ��ℊ���� �� and �� ⊑ ������ℊ��ℊ���� �� implies that ��⨆�� ⊑ ������ℊ��ℊ���� �� ⨆������ℊ��ℊ���� �� . Thus ������ℊ��ℊ���� �� ⨆������ℊ��ℊ���� �� is a ������ℊ��ℊ���� containing ��⨆��. Since ������ℊ��ℊ���� ��⨆�� is the smallest ������ℊ��ℊ���� containing ��⨆��, we have ������ℊ��ℊ���� ��⨆�� ⊑ ������ℊ��ℊ���� �� ⨆������ℊ��ℊ���� �� …………. (2) From (1) and (2), we have ������ℊ��ℊ���� ��⨆�� =������ℊ��ℊ���� �� ⨆������ℊ��ℊ���� �� . (vi) Since ������ℊ��ℊ���� �� is a ������ℊ��ℊ����, we have by proposition (4.3) part (ii), ������ℊ��ℊ����(������ℊ��ℊ���� �� ) =������ℊ��ℊ���� �� .

Theorem 4.6:

Let �� and �� be two neutrosophic sets in a ����������(��,��). Then the following properties hold: (i) ������ℊ��ℊ������(0������ ) = 0������ , ������ℊ��ℊ������(1������ ) = 1������ . (ii) ������ℊ��ℊ������ �� ⊑ ��. (iii) �� ⊑ �� ⟹������ℊ��ℊ������ �� ⊑ ������ℊ��ℊ������ �� . (iv) ������ℊ��ℊ������ ��⨅�� =������ℊ��ℊ������ �� ⨅������ℊ��ℊ������ �� . (v) ������ℊ��ℊ������ ��⨆�� ⊒ ������ℊ��ℊ������ �� ⨆������ℊ��ℊ������ �� . (vi) ������ℊ��ℊ������(������ℊ��ℊ������ �� ) =������ℊ��ℊ������ �� .

Proof:

(i), (ii), (iii), (iv), (v) and (vi) are obvious. 

Definition 4.7:

A ����������(��,��) is called a neutrosophic ��1 2 -space (in short, ��������1 2 -space) if each ������ℊ���� in this space is a

Definition 4.8:

A ����������(��,��) is called a neutrosophic ��ℊ��ℊ -space (in short, ��������ℊ��ℊ -space) if each ������ℊ��ℊ���� in this space

is a ����������.

Proposition 4.9:

Every ��������1 2 -space is a ��������ℊ��ℊ -space.

Proof:

Let (��,��) be a ��������1

2

-space and let �� be a ������ℊ��ℊ���� in ��. Then �� is a ������ℊ����, by theorem (3.2) part (ii). Since (��,��) is a ��������1

2 -space, then �� is a ���������� in ��. Hence (��,��) is a ��������ℊ��ℊ -space.

Theorem 4.10:

For a ����������(��,��), the following statements are equivalent: (i) (��,��) is a ��������ℊ��ℊ -space. (ii) Every singleton of a ����������(��,��) is either ��������ℊ���� or ����������.

Proof:

(i) ⟹(ii) Assume that for some �� ∈ �� the neutrosophic set ��, 0.1,0.1 is not a ��������ℊ���� in a ����������(��,��). Then the only ��������ℊ���� containing 1������ − ��, 0.1,0.1 is the space �� itself and 1������ − ��, 0.1,0.1 is a ������ℊ��ℊ���� in (��,��). By assumption 1������ − ��, 0.1,0.1 is a ���������� in (��,��) or equivalently ��, 0.1,0.1 is a ����������.

(ii) ⟹(i) Let �� be a ������ℊ��ℊ���� in (��,��) and let �� ∈ ����������(��). By assumption ��, 0.1,0.1 is either ��������ℊ���� or ����������. Case(1). Suppose ��, 0.1,0.1 is a ��������ℊ����. If �� ∉ �� then ���������� �� − �� contains a non-empty ��������ℊ���� ��, 0.1,0.1 which is a contradiction to proposition (3.18). Therefore �� ∈ ��. Case(2). Suppose ��, 0.1,0.1 is a ����������. Since �� ∈ ���������� �� , ��, 0.1,0.1 ⨅�� ≠0������ and therefore ���������� �� ⊑ �� or equivalently �� is a ���������� in a ����������(��,��).

5. Conclusion

The concept of ������ℊ��ℊ���� identified utilizing ��������ℊ���� constructs a neutrosophic topology and sits between the concept of ���������� and the concept of ������ℊ����. The ������ℊ��ℊ���� can be used to derive a new decomposition of ������ℊ��ℊ-continuity and new ������ℊ��ℊ-separation axioms.

Funding: This research received no external funding.

Acknowledgements: The authors are highly grateful to the Referees for their constructive suggestions.

Conflicts of Interest: The authors declare no conflict of interest.

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