On Neutrosophic Crisp Semi Alpha Closed Sets

Neutrosophic Sets and Systems, Vol. 21, 2018

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University of New Mexico

On Neutrosophic Crisp Semi Alpha Closed Sets 1

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Riad K. Al-Hamido , Qays Hatem Imran , Karem A. Alghurabi and Taleb Gharibah

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Department of Mathematics, College of Science, Al-Baath University, Homs, Syria.

E-mail: riad-hamido1983@hotmail.com E-mail: taleb.gharibah@gmail.com 2 Department of Mathematics, College of Education for Pure Science, Al Muthanna University, Samawah, Iraq.

E-mail: qays.imran@mu.edu.iq 3 Department of Mathematics, College of Education for Pure Science, Babylon University, Hilla, Iraq.

E-mail: kareemalghurabi@yahoo.com Abstract. In this paper, we presented another concept of neutrosophic crisp generalized closed sets called neutrosophic crisp semi-ď Ą-closed sets and studied their fundamental properties in neutrosophic crisp topological spaces. We also present neutrosophic crisp semi-ď Ą-closure and neutrosophic crisp semi-ď Ą-interior and study some of their fundamental properties.

Mathematics Subject Classification (2000): 54A40, 03E72.

Keywords: Neutrosophic crisp semi-ď Ą-closed sets, neutrosophic crisp semi-ď Ą-open sets, neutrosophic crisp semi-ď Ą-closure and neutrosophic crisp semi-ď Ą-interior.

1. Introduction The concept of "neutrosophic set" was first given by F. Smarandache [4,5]. A. A. Salama and S. A. Alblowi [1] presented the concept of neutrosophic topological space (briefly NTS). Q. H. Imran, F. Smarandache, R. K. Al-Hamido and R. Dhavaseelan [6] presented the idea of neutrosophic semi- ď Ą -open sets in neutrosophic topological spaces. In 2014, A. A. Salama, F. Smarandache and V. Kroumov [2] presented the concept of neutrosophic crisp topological space (briefly NCTS). The objective of this paper is to present the concept of neutrosophic crisp semi-ď Ą-closed sets and study their fundamental properties in neutrosophic crisp topological spaces. We also present neutrosophic crisp semi-ď Ą-closure and neutrosophic crisp semi-ď Ą-interior and obtain some of its properties.

2. Preliminaries Throughout this paper, (đ?’°, đ?‘‡) (or simply đ?’°) always mean a neutrosophic crisp topological space. The complement of a neutrosophic crisp open set (briefly NC-OS) is called a neutrosophic crisp closed set (briefly NC-CS) in (đ?’°, đ?‘‡). For a neutrosophic crisp set đ?’œ in a neutrosophic crisp topological space (đ?’°, đ?‘‡), đ?‘ đ??śđ?‘?đ?‘™(đ?’œ), đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ) and đ?’œ denote the neutrosophic crisp closure of đ?’œ, the neutrosophic crisp interior of đ?’œ and the neutrosophic crisp complement of đ?’œ, respectively.

Definition 2.1: A neutrosophic crisp subset đ?’œ of a neutrosophic crisp topological space (đ?’°, đ?‘‡) is said to be: (i) A neutrosophic crisp pre-open set (briefly NCP-OS) [3] if đ?’œ ⊆ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?’œ)). The complement of a NCPOS is called a neutrosophic crisp pre-closed set (briefly NCP-CS) in (đ?’°, đ?‘‡). The family of all NCP-OS (resp. NCPCS) of đ?’° is denoted by NCPO(đ?’°) (resp. NCPC(đ?’°)). (ii) A neutrosophic crisp semi-open set (briefly NCS-OS) [3] if đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ )). The complement of a NCS-OS is called a neutrosophic crisp semi-closed set (briefly NCS-CS) in (đ?’°, đ?‘‡). The family of all NCS-OS (resp. NCS-CS) of đ?’° is denoted by NCSO(đ?’°) (resp. NCSC(đ?’°)). (iii) A neutrosophic crisp Îą-open set (briefly NCÎą-OS) [3] if đ?’œ ⊆ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ ))). The complement of a NCÎą-OS is called a neutrosophic crisp Îą-closed set (briefly NCÎą-CS) in (đ?’°, đ?‘‡). The family of all NCÎą-OS (resp. NCÎą-CS) of đ?’° is denoted by NCÎąO(đ?’°) (resp. NCÎąC(đ?’°)).

Definition 2.2: (i) The neutrosophic crisp pre-interior of a neutrosophic crisp set đ?’œ of a neutrosophic crisp topological space (đ?’°, đ?‘‡) is the union of all NCP-OS contained in đ?’œ and is denoted by đ?‘ƒđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)[3].

Riad K. Al-Hamido, Qays Hatem Imran, Karem A. Alghurabi and Taleb Gharibah,, On Neutrosophic Crisp Semi Alpha Closed Sets

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(ii) The neutrosophic crisp semi-interior of a neutrosophic crisp set đ?’œ of a neutrosophic crisp topological space (đ?’°, đ?‘‡) is the union of all NCS-OS contained in đ?’œ and is denoted by đ?‘†đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)[3]. (iii) The neutrosophic crisp Îą-interior of a neutrosophic crisp set đ?’œ of a neutrosophic crisp topological space (đ?’°, đ?‘‡) is the union of all NCÎą-OS contained in đ?’œ and is denoted by đ?›źđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)[3].

Definition 2.3: (i) The neutrosophic crisp pre-closure of a neutrosophic crisp set đ?’œ of a neutrosophic crisp topological space (đ?’°, đ?‘‡) is the intersection of all NCP-CS that contain đ?’œ and is denoted by đ?‘ƒđ?‘ đ??śđ?‘?đ?‘™(đ?’œ)[3]. (ii) The neutrosophic crisp semi-closure of a neutrosophic crisp set đ?’œ of a neutrosophic crisp topological space (đ?’°, đ?‘‡) is the intersection of all NCS-CS that contain đ?’œ and is denoted by đ?‘†đ?‘ đ??śđ?‘?đ?‘™(đ?’œ)[3]. (iii) The neutrosophic crisp Îą-closure of a neutrosophic crisp set đ?’œ of a neutrosophic crisp topological space (đ?’°, đ?‘‡) is the intersection of all NCÎą-CS that contain đ?’œ and is denoted by đ?›źđ?‘ đ??śđ?‘?đ?‘™(đ?’œ)[3].

Proposition 2.4 [7]: In a neutrosophic crisp topological space (�, �) , the following statements hold, and the equality of each statement are not true: (i) Every NC-CS (resp. NC-OS) is a NCι-CS (resp. NCι-OS). (ii) Every NCι-CS (resp. NCι-OS) is a NCS-CS (resp. NCS-OS). (iii) Every NCι-CS (resp. NCι-OS) is a NCP-CS (resp. NCP-OS).

Proposition 2.5 [7]: A neutrosophic crisp subset đ?’œ of a neutrosophic crisp topological space (đ?’°, đ?‘‡) is a NCÎą-CS (resp. NCÎą-OS) iff đ?’œ is a NCS-CS (resp. NCS-OS) and NCP-CS (resp. NCP-OS).

Theorem 2.6 [7]: For any neutrosophic crisp subset đ?’œ of a neutrosophic crisp topological space (đ?’°, đ?‘‡), đ?’œ ∈ NCÎąO(đ?’°) iff there exists a NC-OS â„‹ such that â„‹ ⊆ đ?’œ ⊆ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹)).

Proposition 2.7 [7]: The union of any family of NCÎą-OS is a NCÎą-OS. Proposition 2.8: (i) If đ?’Ś is a NC-OS, then đ?‘†đ?‘ đ??śđ?‘?đ?‘™(đ?’Ś) = đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?’Ś)). (ii) If đ?’œ is a neutrosophic crisp subset of a neutrosophic crisp topological space (đ?’°, đ?‘‡), then đ?‘†đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą đ?‘ đ??śđ?‘?đ?‘™(đ?’œ) = đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?’œ))).

Proof: This follows directly from the definition (2.1) and proposition (2.4).

3. Neutrosophic Crisp Semi-đ?›‚-Closed Sets In this section, we present and study the neutrosophic crisp semi-Îą-closed sets and some of its properties.

Definition 3.1: A neutrosophic crisp subset đ?’œ of a neutrosophic crisp topological space (đ?’°, đ?‘‡) is called neutrosophic crisp semiÎą-closed set (briefly NCSÎą-CS) if there exists a NCÎą-CS â„‹ in đ?’° such that đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„‹) ⊆ đ?’œ ⊆ â„‹ or equivalently if đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?›źđ?‘ đ??śđ?‘?đ?‘™(đ?’œ)) ⊆ đ?’œ. The family of all NCSÎą-CS of đ?’° is denoted by NCSÎąC(đ?’°).

Definition 3.2: A neutrosophic crisp set đ?’œ is called a neutrosophic crisp semi-Îą-open set (briefly NCSÎą-OS) if and only if its complement đ?’œ is a NCSÎą-CS or equivalently if there exists a NCÎą-OS â„‹ in đ?’° such that â„‹ ⊆ đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(â„‹).

The family of all NCSÎą-OS of đ?’° is denoted by NCSÎąO(đ?’°).

Proposition 3.3: It is evident by definitions that in a neutrosophic crisp topological space (�, �), the following hold: (i) Every NC-CS (resp. NC-OS) is a NCSι-CS (resp. NCSι-OS). (ii) Every NCι-CS (resp. NCι-OS) is a NCSι-CS (resp. NCSι-OS).

The converse of Proposition (3.3) need not be true as shown by the following example.

Example 3.4: Let đ?’° = {đ?‘?, đ?‘ž, đ?‘&#x;, đ?‘ }, đ?’œ = ⌊{đ?‘?}, {đ?‘ž, đ?‘ }, {đ?‘&#x;}âŒŞ, â„Ź = ⌊{đ?‘?}, {đ?‘ž}, {đ?‘&#x;}âŒŞ .Then đ?‘‡ = {∅ , đ?’œ, â„Ź, đ?’° } is a neutrosophic crisp topology on đ?’°.

Riad K. Al-Hamido, Qays Hatem Imran, Karem A. Alghurabi and Taleb Gharibah, On Neutrosophic Crisp Semi Alpha Closed Sets

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Neutrosophic Sets and Systems, Vol. 21, 2018

(i) Let â„‹ = ⌊{đ?‘?}, {đ?‘ž, đ?‘&#x;, đ?‘ }, âˆ…âŒŞ, đ?’œ ⊆ â„‹ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?’œ) = đ?’° , the neutrosophic crisp set â„‹ is a NCSÎą-OS but not NC-OS. It is clear that â„‹ = ⌊{đ?‘ž, đ?‘&#x;, đ?‘ }, {đ?‘?}, đ?’°âŒŞ is a NCSÎą-CS but not NC-CS. (ii) Let đ?’Ś = ⌊∅, {đ?‘ž, đ?‘&#x;, đ?‘ }, {đ?‘&#x;, đ?‘ }âŒŞ and so đ?’Ś ⊈ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’Ś))), the neutrosophic crisp set đ?’Ś is a NCSÎąOS but not NCÎą-OS. It is clear that đ?’Ś = ⌊đ?’°, {đ?‘?}, {đ?‘?, đ?‘ž}âŒŞ is a NCSÎą-CS but not NCÎą-CS. Remark 3.5: The concepts of NCSÎą-CS (resp. NCSÎą-OS) and NCP-CS (resp. NCP-OS) are independent, as the following examples show.

Example 3.6: Let đ?’° = {đ?‘?, đ?‘ž, đ?‘&#x;, đ?‘ }, đ?’œ = ⌊{đ?‘?}, {đ?‘ž}, {đ?‘&#x;}âŒŞ, â„Ź = ⌊{đ?‘&#x;}, {đ?‘ž}, {đ?‘ }âŒŞ, đ?’ž = ⌊{đ?‘?, đ?‘&#x;}, {đ?‘ž}, âˆ…âŒŞ, đ?’&#x; = ⌊∅, {đ?‘ž}, {đ?‘&#x;, đ?‘ }âŒŞ.

Then đ?‘‡ = {∅ , đ?’œ, â„Ź, đ?’ž, đ?’&#x;, đ?’° } is a neutrosophic crisp topology on đ?’°. Let â„‹ = ⌊{đ?‘&#x;, đ?‘ }, {đ?‘?, đ?‘ž}, {đ?‘ }âŒŞ, â„Ź ⊆ â„‹ ⊆ đ?‘ đ??śđ?‘?đ?‘™(â„Ź) = ⌊{đ?‘&#x;, đ?‘ }, {đ?‘ž}, âˆ…âŒŞ, the neutrosophic crisp set â„‹ is a NCSÎą-OS but not NCP-OS. It is clear that â„‹ = ⌊{đ?‘ }, {đ?‘?, đ?‘ž}, {đ?‘&#x;, đ?‘ }âŒŞ is a NCSÎą-CS but not NCP-CS. Example 3.7: Let đ?’° = {đ?‘?, đ?‘ž, đ?‘&#x;, đ?‘ }, đ?’œ = ⌊{đ?‘?}, {đ?‘ž}, {đ?‘&#x;}âŒŞ, đ?’œ = ⌊{đ?‘?}, {đ?‘ž, đ?‘ }, {đ?‘&#x;}âŒŞ. Then đ?‘‡ = {∅ , đ?’œ , đ?’œ , đ?’° } is a neutrosophic crisp topology on đ?’°. If đ?’œ = ⌊{đ?‘?, đ?‘ž}, {đ?‘&#x;}, {đ?‘ }âŒŞ, then đ?’œ is a NCP-OS but not NCSÎą-OS. It is clear that đ?’œ = ⌊{đ?‘ }, {đ?‘&#x;}, {đ?‘?, đ?‘ž}âŒŞ is a NCP-CS but not NCSÎą-CS. Remark 3.8: (i) If every NC-OS is a NC-CS and every nowhere neutrosophic crisp dense set is NC-CS in any neutrosophic crisp topological space (đ?’°, đ?‘‡), then every NCSÎą-CS (resp. NCSÎą-OS) is a NC-CS (resp. NC-OS). (ii) If every NC - OS is a NC - CS in any neutrosophic crisp topological space (đ?’°, đ?‘‡) , then every NCSÎą - CS (resp. NCSÎą-OS) is a NCÎą-CS (resp. NCÎą-OS).

Remark 3.9: (i) It is clear that every NCS-CS (resp. NCS-OS) and NCP-CS (resp. NCP-OS) of any neutrosophic crisp topological space (�, �) is a NCSι-CS (resp. NCSι-OS) (by Proposition (2.5) and Proposition (3.3) (ii)). (ii) A NCSι-CS (resp. NCSι-OS) in any neutrosophic crisp topological space (�, �) is a NCP-CS (resp. NCP-OS) if every NC-OS of � is a NC-CS (from Proposition (2.4) (iii) and Remark (3.8) (ii)).

Theorem 3.10: For any neutrosophic crisp subset đ?’œ of a neutrosophic crisp topological space (đ?’°, đ?‘‡). The following properties are equivalent: (i) đ?’œ ∈ NCSÎąO(đ?’°). (ii) There exists a NC-OS, say â„‹, such that â„‹ ⊆ đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹))). (iii) đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)))).

Proof: (đ?‘–) â&#x;š (đ?‘–đ?‘–) Let đ?’œ ∈ NCSÎąO(đ?’°). Then, there exists đ?’Ś ∈ NCÎąO(đ?’°), such that đ?’Ś ⊆ đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?’Ś). Hence there exists â„‹ NC-OS such that â„‹ ⊆ đ?’Ś ⊆ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹))(by Theorem (2.6)). Therefore, đ?‘ đ??śđ?‘?đ?‘™(â„‹) ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?’Ś) ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹))), implies that đ?‘ đ??śđ?‘?đ?‘™(đ?’Ś) ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹))).

Then â„‹ ⊆ đ?’Ś ⊆ đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?’Ś) ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹))). Hence, â„‹ ⊆ đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹))), for some â„‹ NC-OS. (đ?‘–đ?‘–) â&#x;š (đ?‘–đ?‘–đ?‘–) Suppose that there exists a NC-OS â„‹ such that â„‹ ⊆ đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹))). We know that đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ) ⊆ đ?’œ. On the other hand, â„‹ ⊆ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ) (since đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ) is the largest NC-OS contained in đ?’œ).

Hence đ?‘ đ??śđ?‘?đ?‘™(â„‹) ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)), then đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹)) ⊆ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ))), therefore đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹))) ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)))). But đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹))) (by hypothesis). Hence đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹))) ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)))), then đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)))). (đ?‘–đ?‘–đ?‘–) â&#x;š (đ?‘–) Let đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)))) . To prove đ?’œ ∈ NCSÎąO(đ?’°) , let đ?’Ś = đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ) ; we know that đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ) ⊆ đ?’œ. To prove đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)).

Since đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ))) ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)).

Hence, đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)))) ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)))) = đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)).

But đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)))) (by hypothesis). Hence, đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)))) ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)) â&#x;š đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)). Hence, there exists an NC-OS say đ?’Ś , such that đ?’Ś ⊆ đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?’œ). On the other hand, đ?’Ś is a NCÎą-OS (since đ?’Ś is a NC-OS). Hence đ?’œ ∈ NCSÎąO(đ?’°).

Riad K. Al-Hamido, Qays Hatem Imran, Karem A. Alghurabi and Taleb Gharibah, On Neutrosophic Crisp Semi Alpha Closed Sets

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Corollary 3.11: For any neutrosophic crisp subset đ?’œ of a neutrosophic crisp topological space (đ?’°, đ?‘‡), the following properties are equivalent: (i) đ?’œ ∈ NCSÎąC(đ?’°). (ii) There exists a NC-CS â„ą such that đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„ą))) ⊆ đ?’œ ⊆ â„ą. (iii) đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?’œ)))) ⊆ đ?’œ.

Proof: (đ?‘–) â&#x;š (đ?‘–đ?‘–) Let đ?’œ ∈ NCSÎąC(đ?’°), then đ?’œ ∈ NCSÎąO(đ?’°). Hence there is â„‹ NC-OS such that â„‹ ⊆ đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹))) (by Theorem (3.10)). Hence (đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹)))) ⊆ đ?’œ ⊆ â„‹ , i.e., đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„‹ ))) ⊆ đ?’œ ⊆ â„‹ . Let â„‹ = â„ą, where â„ą is a NC-CS in đ?’°.

Then đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„ą))) ⊆ đ?’œ ⊆ â„ą, for some â„ą NC-CS. (đ?‘–đ?‘–) â&#x;š (đ?‘–đ?‘–đ?‘–) Suppose that there exists â„ą NC-CS such that đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą đ?‘ đ??śđ?‘?đ?‘™ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„ą) ⊆ đ?’œ ⊆ â„ą, but đ?‘ đ??śđ?‘?đ?‘™(đ?’œ) is the smallest NC-CS containing đ?’œ. Then đ?‘ đ??śđ?‘?đ?‘™(đ?’œ) ⊆ â„ą, and therefore: đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą đ?‘ đ??śđ?‘?đ?‘™(đ?’œ) ⊆ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„ą) â&#x;š đ?‘ đ??śđ?‘?đ?‘™ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą đ?‘ đ??śđ?‘?đ?‘™(đ?’œ) ⊆ đ?‘ đ??śđ?‘?đ?‘™ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„ą) â&#x;š đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?’œ)))) ⊆ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„ą))) ⊆ đ?’œ â&#x;š đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?’œ)))) ⊆ đ?’œ. (đ?‘–đ?‘–đ?‘–) â&#x;š (đ?‘–) Let đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?’œ)))) ⊆ đ?’œ . To prove đ?’œ ∈ NCSÎąC(đ?’°) , i.e., to prove đ?’œ ∈ NCSÎąO(đ?’°) . Then đ?’œ ⊆ (đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?’œ))))) = đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ )))) , but (đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?’œ))))) = đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ )))).

Hence đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ )))), and therefore đ?’œ ∈ NCSÎąO(đ?’°), i.e., đ?’œ ∈ NCSÎąC(đ?’°).

Theorem 3.12: The union of any family of NCSÎą-OS is a NCSÎą-OS. Proof: Let {đ?’œ } ∈ be a family of NCSÎą-OS. To prove ⋃ ∈ đ?’œ is a NCSÎą-OS. Since đ?’œ ∈ NCSÎąO(đ?’°). Then there is a NCÎą-OS â„Ź such that â„Ź ⊆ đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(â„Ź ), ∀đ?œ† ∈ Λ. Hence ⋃ ∈ â„Ź ⊆ ⋃ ∈ đ?’œ ⊆ ⋃ ∈ đ?‘ đ??śđ?‘?đ?‘™(â„Ź ) ⊆ đ?‘ đ??śđ?‘?đ?‘™(⋃ ∈ â„Ź ). But ⋃ ∈ â„Ź ∈ NCÎąO(đ?’°) (by Proposition (2.7)). Hence ⋃ ∈ đ?’œ ∈ NCSÎąO(đ?’°).

Corollary 3.13: The intersection of any family of NCSÎą-CS is a NCSÎą-CS. Proof: This follows directly from Theorem (3.12).

Remark 3.14: The following diagram shows the relations among the different types of weakly neutrosophic crisp closed sets that were studied in this section: NCP-CS

NCS-CS

NC-CS

+

NCÎą-CS

every nowhere NC-dense set is a NC-CS + every NC-OS is a NC-CS

+ NCSÎą-CS

Diagram (3.1)

Riad K. Al-Hamido, Qays Hatem Imran, Karem A. Alghurabi and Taleb Gharibah, On Neutrosophic Crisp Semi Alpha Closed Sets

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Neutrosophic Sets and Systems, Vol. 21, 2018

4. Neutrosophic Crisp Semi-ď Ą-Closure and Neutrosophic Crisp Semi-ď Ą-Interior We present neutrosophic crisp semi-ď Ą-closure and neutrosophic crisp semi-ď Ą-interior and obtain some of their properties in this section.

Definition 4.1: The intersection of all NCSď Ą - CS in a neutrosophic crisp topological space (đ?’°, đ?‘‡) containing đ?’œ is called neutrosophic crisp semi-ď Ą-closure of đ?’œ and is denoted by đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ), đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ) = â‹‚{â„Ź: đ?’œ ⊆ â„Ź, â„Ź is a NCSď Ą-CS}.

Definition 4.2: The union of all NCSď Ą-OS in a neutrosophic crisp topological space (đ?’°, đ?‘‡) contained in đ?’œ is called neutrosophic crisp semi-ď Ą-interior of đ?’œ and is denoted by đ?‘†ď Ąđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ), đ?‘†ď Ąđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ) = ⋃{â„Ź: â„Ź ⊆ đ?’œ, â„Ź is a NCSď Ą-OS}.

Proposition 4.3: Let đ?’œ be any neutrosophic crisp set in a neutrosophic crisp topological space (đ?’°, đ?‘‡), the following properties are true: (i) đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ) = đ?’œ iff đ?’œ is a NCSď Ą-CS. (ii) đ?‘†ď Ąđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ) = đ?’œ iff đ?’œ is a NCSď Ą-OS. (iii) đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ) is the smallest NCSď Ą-CS containing đ?’œ. (iv) đ?‘†ď Ąđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ) is the largest NCSď Ą-OS contained in đ?’œ.

Proof: (i), (ii), (iii) and (iv) are obvious.

Proposition 4.4: Let đ?’œ be any neutrosophic crisp set in a neutrosophic crisp topological space (đ?’°, đ?‘‡), the following properties hold: (i) đ?‘†ď Ąđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’° − đ?’œ) = đ?’° − (đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ)), (ii) đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’° − đ?’œ) = đ?’° − (đ?‘†ď Ąđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)).

Proof: (i) By definition (2.3), đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ) = â‹‚{â„Ź: đ?’œ ⊆ â„Ź, â„Ź is a NCSď Ą-CS} đ?’° − ( đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ)) = đ?’° − â‹‚{â„Ź: đ?’œ ⊆ â„Ź, â„Ź is a NCSď Ą-CS} = ⋃{đ?’° − â„Ź: đ?’œ ⊆ â„Ź, â„Ź is a NCSď Ą-CS} = ⋃{â„‹: â„‹ ⊆ đ?’° − đ?’œ, â„‹ is a NCSď Ą-OS} = đ?‘†ď Ąđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’° − đ?’œ). (ii) The proof is similar to (i).

Theorem 4.5: Let đ?’œ and â„Ź be two neutrosophic crisp sets in a neutrosophic crisp topological space (đ?’°, đ?‘‡). The following properties hold: (i) đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(∅ ) = ∅ , đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’° ) = đ?’° . (ii) đ?’œ ⊆ đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ). (iii) đ?’œ ⊆ â„Ź â&#x;š đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ) ⊆ đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(â„Ź). (iv) đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œâ‹‚â„Ź) ⊆ đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ)â‹‚đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(â„Ź). (v) đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ)⋃đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(â„Ź) ⊆ đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œâ‹ƒâ„Ź). (vi) đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ)) = đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ).

Proof: (i) and (ii) are evident. (iii) By (ii), â„Ź ⊆ đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(â„Ź). Since đ?’œ ⊆ â„Ź, we have đ?’œ ⊆ đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(â„Ź). But đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(â„Ź) is a NCSď Ą-CS. Thus đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(â„Ź) is a NCSď Ą-CS containing đ?’œ.

Since đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ) is the smallest NCSď Ą-CS containing đ?’œ, we have đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ) ⊆ đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(â„Ź). Hence, đ?’œ ⊆ â„Ź â&#x;š đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ) ⊆ đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(â„Ź). (iv) We know that đ?’œâ‹‚â„Ź ⊆ đ?’œ and đ?’œâ‹‚â„Ź ⊆ â„Ź . Therefore, by (iii), đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œâ‹‚â„Ź) ⊆ đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ) and đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œâ‹‚â„Ź) ⊆ đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(â„Ź). Hence đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œâ‹‚â„Ź) ⊆ đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ)â‹‚đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(â„Ź). (v) Since đ?’œ ⊆ đ?’œâ‹ƒâ„Ź and â„Ź ⊆ đ?’œâ‹ƒâ„Ź, it follows from part (iii) that đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ) ⊆ đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œâ‹ƒâ„Ź) and đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(â„Ź) ⊆ đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œâ‹ƒâ„Ź). Hence đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ)⋃đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(â„Ź) ⊆ đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œâ‹ƒâ„Ź). (vi) Since đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ) is a NCSď Ą-CS, we have by Proposition (4.3)(i), đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ)) = đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ).

Theorem 4.6: Let đ?’œ and â„Ź be two neutrosophic crisp sets in a neutrosophic crisp topological space (đ?’°, đ?‘‡). The following properties hold: (i) đ?‘†ď Ąđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(∅ ) = ∅ , đ?‘†ď Ąđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’° ) = đ?’° . (ii) đ?‘†ď Ąđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ) ⊆ đ?’œ.

Riad K. Al-Hamido, Qays Hatem Imran, Karem A. Alghurabi and Taleb Gharibah, On Neutrosophic Crisp Semi Alpha Closed Sets

Neutrosophic Sets and Systems, Vol. 21, 2018

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(iii) 𝒜 ⊆ ℬ ⟹ 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑆𝑁𝐶𝑖𝑛𝑡(ℬ). (iv) 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜⋂ℬ) ⊆ 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜)⋂𝑆𝑁𝐶𝑖𝑛𝑡(ℬ). (v) 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜)⋃𝑆𝑁𝐶𝑖𝑛𝑡(ℬ) ⊆ 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜⋃ℬ). (vi) 𝑆𝑁𝐶𝑖𝑛𝑡(𝑆𝑁𝐶𝑖𝑛𝑡(𝒜)) = 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜).

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

Proposition 4.7: For any neutrosophic crisp subset 𝒜 of a neutrosophic crisp topological space (𝒰, 𝑇), then: (i) 𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑆𝑁𝐶𝑐𝑙(𝒜) ⊆ 𝑁𝐶𝑐𝑙(𝒜) ⊆ 𝑁𝐶𝑐𝑙(𝒜). (ii) 𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑆𝑁𝐶𝑖𝑛𝑡 𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑁𝐶𝑖𝑛𝑡(𝒜). (iii) 𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑆𝑁𝐶𝑖𝑛𝑡 𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑁𝐶𝑖𝑛𝑡(𝒜). (iv) 𝑁𝐶𝑐𝑙 𝑆𝑁𝐶𝑐𝑙(𝒜) = 𝑆𝑁𝐶𝑐𝑙 𝑁𝐶𝑐𝑙(𝒜) = 𝑁𝐶𝑐𝑙(𝒜). (v) 𝑁𝐶𝑐𝑙 𝑆𝑁𝐶𝑐𝑙(𝒜) = 𝑆𝑁𝐶𝑐𝑙 𝑁𝐶𝑐𝑙(𝒜) = 𝑁𝐶𝑐𝑙(𝒜). (vi) 𝑆𝑁𝐶𝑐𝑙(𝒜) = 𝒜⋃𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝒜)))). (vii) 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝒜⋂𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)))). (viii) 𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝒜)) ⊆ 𝑆𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑐𝑙(𝒜) .

Proof: We shall prove only (ii), (iii), (iv), (vii) and (viii). (ii) To prove 𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑆𝑁𝐶𝑖𝑛𝑡 𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑁𝐶𝑖𝑛𝑡(𝒜), we know that 𝑁𝐶𝑖𝑛𝑡(𝒜) is a NCOS. It follows that 𝑁𝐶𝑖𝑛𝑡(𝒜) is a NCS-OS. Hence 𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑆𝑁𝐶𝑖𝑛𝑡 𝑁𝐶𝑖𝑛𝑡(𝒜) (by Proposition (4.3)).

Therefore: 𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑆𝑁𝐶𝑖𝑛𝑡 𝑁𝐶𝑖𝑛𝑡(𝒜) …...….................................................................(1) Since 𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ⟹ 𝑁𝐶𝑖𝑛𝑡 𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ⟹ 𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) . Also, 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝒜 ⟹ 𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑁𝐶𝑖𝑛𝑡(𝒜).

Hence: 𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) …………………...................................................(2) Therefore by (1) and (2), we get 𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑆𝑁𝐶𝑖𝑛𝑡 𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑁𝐶𝑖𝑛𝑡(𝒜). (iii) Now we prove 𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑆𝑁𝐶𝑖𝑛𝑡 𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑁𝐶𝑖𝑛𝑡(𝒜).

Since 𝑁𝐶𝑖𝑛𝑡(𝒜) is NC-OS, therefore 𝑁𝐶𝑖𝑛𝑡(𝒜) is NCS-OS. Therefore by Proposition (4.3): 𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑆𝑁𝐶𝑖𝑛𝑡 𝑁𝐶𝑖𝑛𝑡(𝒜) ………............................................................................(1) Now, to prove 𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) , we have 𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ⟹ 𝑁𝐶𝑖𝑛𝑡 𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ⟹ 𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) .

Also, 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝒜 ⟹ 𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑁𝐶𝑖𝑛𝑡(𝒜).

Hence: 𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ………………............................................(2) Therefore by (1) and (2), we get 𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑆𝑁𝐶𝑖𝑛𝑡 𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝑁𝐶𝑖𝑛𝑡(𝒜). (iv) To prove 𝑁𝐶𝑐𝑙 𝑆𝑁𝐶𝑐𝑙(𝒜) = 𝑆𝑁𝐶𝑐𝑙 𝑁𝐶𝑐𝑙(𝒜) = 𝑁𝐶𝑐𝑙(𝒜). We know that 𝑁𝐶𝑐𝑙(𝒜) is a NC-CS, so it is NCS-CS. Hence by proposition (4.3), we have: 𝑁𝐶𝑐𝑙(𝒜) = 𝑆𝑁𝐶𝑐𝑙 𝑁𝐶𝑐𝑙(𝒜) ….......(1) To prove 𝑵𝑪𝒄𝒍(𝓐) = 𝑵𝑪𝒄𝒍 𝑺𝑵𝑪𝒄𝒍(𝓐) , we have 𝑺𝑵𝑪𝒄𝒍(𝓐) ⊆ 𝑵𝑪𝒄𝒍(𝓐) (by part (i)).

Then 𝑵𝑪𝒄𝒍 𝑺𝑵𝑪𝒄𝒍(𝓐) ⊆ 𝑵𝑪𝒄𝒍 𝑵𝑪𝒄𝒍(𝓐) = 𝑵𝑪𝒄𝒍(𝓐) ⟹ 𝑵𝑪𝒄𝒍 𝑺𝑵𝑪𝒄𝒍(𝓐) ⊆ 𝑵𝑪𝒄𝒍(𝓐).

Since 𝓐 ⊆ 𝑺𝑵𝑪𝒄𝒍(𝓐) ⊆ 𝑵𝑪𝒄𝒍 𝑺𝑵𝑪𝒄𝒍(𝓐) , then 𝓐 ⊆ 𝑵𝑪𝒄𝒍 𝑺𝑵𝑪𝒄𝒍(𝓐) .

Hence, 𝑁𝐶𝑐𝑙(𝒜) ⊆ 𝑁𝐶𝑐𝑙 𝑁𝐶𝑐𝑙 𝑆𝑁𝐶𝑐𝑙(𝒜) = 𝑁𝐶𝑐𝑙 𝑆𝑁𝐶𝑐𝑙(𝒜) ⟹ 𝑁𝐶𝑐𝑙(𝒜) ⊆ 𝑁𝐶𝑐𝑙 𝑆𝑁𝐶𝑐𝑙(𝒜) and therefore: 𝑁𝐶𝑐𝑙(𝒜) = 𝑁𝐶𝑐𝑙 𝑆𝑁𝐶𝑐𝑙(𝒜) ……...........................................................(2) Now, by (1) and (2), we get that 𝑁𝐶𝑐𝑙 𝑆𝑁𝐶𝑐𝑙(𝒜) = 𝑆𝑁𝐶𝑐𝑙 𝑁𝐶𝑐𝑙(𝒜) . Hence 𝑁𝐶𝑐𝑙 𝑆𝑁𝐶𝑐𝑙(𝒜) = 𝑆𝑁𝐶𝑐𝑙 𝑁𝐶𝑐𝑙(𝒜) = 𝑁𝐶𝑐𝑙(𝒜). (vii) To prove 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝒜⋂𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)))), since 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ∈ NCSO(𝒰) ⟹ 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙 𝑁𝐶𝑖𝑛𝑡(𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ))) = 𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)))) (by part (ii)). Hence, 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)))), also 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝒜. Then: 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝒜⋂𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜))))......................................................(1) To prove 𝒜⋂𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)))) is a NCS-OS contained in 𝒜.

It is clear that 𝒜⋂𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)))) ⊆ 𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)))) and also it is clear that 𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)) ⟹ 𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑖𝑛𝑡(𝒜)) ⊆ 𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜))) ⟹ 𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜))) ⟹ 𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)) ⊆ 𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜))) and 𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)) ⟹ 𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)))) and 𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝒜 ⟹ 𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝒜⋂𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)))) . We get 𝑁𝐶𝑖𝑛𝑡(𝒜) ⊆ 𝒜⋂𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)))) ⊆ 𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)))). Hence 𝒜⋂𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)))) is a NCS-OS (by Proposition (4.3)). Also, 𝒜⋂𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)))) is contained in 𝒜.

Then 𝒜⋂𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)))) ⊆ 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) (since 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) is the largest NCS - OS contained in 𝒜). Hence: 𝒜⋂𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)))) ⊆ 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜)...............(2) By (1) and (2), we get that 𝑆𝑁𝐶𝑖𝑛𝑡(𝒜) = 𝒜⋂𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝒜)))). (viii) To prove that 𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝒜)) ⊆ 𝑆𝑁𝐶𝑖𝑛𝑡 𝑆𝑁𝐶𝑐𝑙(𝒜) , we know that 𝑆𝑁𝐶𝑐𝑙(𝒜) is a NCS-CS, therefore 𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡 𝑁𝐶𝑐𝑙(𝑆𝑁𝐶𝑐𝑙(𝒜) ))) ⊆ 𝑆𝑁𝐶𝑐𝑙(𝒜) (by Corollary (3.11)). Hence 𝑁𝐶𝑖𝑛𝑡 𝑁𝐶𝑐𝑙(𝒜) ⊆ 𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝑁𝐶𝑖𝑛𝑡(𝑁𝐶𝑐𝑙(𝒜))) ⊆ 𝑆𝑁𝐶𝑐𝑙(𝒜) (by part (iv)).

Therefore, Riad K. Al-Hamido, Qays Hatem Imran, Karem A. Alghurabi and Taleb Gharibah, On Neutrosophic Crisp Semi Alpha Closed Sets

34 đ?‘†ď Ąđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą đ?‘ đ??śđ?‘?đ?‘™(đ?’œ) (ii)).

Neutrosophic Sets and Systems, Vol. 21, 2018

⊆ đ?‘†ď Ąđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ) â&#x;š đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą đ?‘ đ??śđ?‘?đ?‘™(đ?’œ) ⊆ đ?‘†ď Ąđ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘†ď Ąđ?‘ đ??śđ?‘?đ?‘™(đ?’œ)) (by

Theorem 4.8: For any neutrosophic crisp subset đ?’œ of a neutrosophic crisp topological space (đ?’°, đ?‘‡). The following properties are equivalent: (i) đ?’œ ∈ NCSď ĄO(đ?’°). (ii) â„‹ ⊆ đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹))), for some NC-OS â„‹. (iii) â„‹ ⊆ đ?’œ ⊆ đ?‘†đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹)), for some NC-OS â„‹. (iv) đ?’œ ⊆ đ?‘†đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ))).

Proof: (đ?‘–) â&#x;š (đ?‘–đ?‘–) Let đ?’œ ∈ NCSď ĄO(đ?’°), then đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)))) and đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ) ⊆ đ?’œ. Hence â„‹ ⊆ đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹))), where â„‹ = đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ). (đ?‘–đ?‘–) â&#x;š (đ?‘–đ?‘–đ?‘–) Suppose â„‹ ⊆ đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹))), for some NC-OS â„‹. But đ?‘†đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą đ?‘ đ??śđ?‘?đ?‘™(â„‹) = đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹))) (by Proposition (2.8)). Then â„‹ ⊆ đ?’œ ⊆ đ?‘†đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹)) , for some NC-OS â„‹. (đ?‘–đ?‘–đ?‘–) â&#x;š (đ?‘–đ?‘Ł) Suppose that â„‹ ⊆ đ?’œ ⊆ đ?‘†đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹)), for some NC-OS â„‹. Since â„‹ is a NC-OS contained in đ?’œ. Then â„‹ ⊆ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ) â&#x;š đ?‘ đ??śđ?‘?đ?‘™(â„‹) ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)) â&#x;š đ?‘†đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹)) ⊆ đ?‘†đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ))). But đ?’œ ⊆ đ?‘†đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„‹)) (by hypothesis), then đ?’œ ⊆ đ?‘†đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ))). (đ?‘–đ?‘Ł) â&#x;š (đ?‘–) Let đ?’œ ⊆ đ?‘†đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ))).

But đ?‘†đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ))) = đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)))) (by Proposition (2.8)).

Hence, đ?’œ ⊆ đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?’œ)))) â&#x;š đ?’œ ∈ NCSď ĄO(đ?’°).

Corollary 4.9: For any neutrosophic crisp subset â„Ź of a neutrosophic crisp topological space (đ?’°, đ?‘‡), the following properties are equivalent: (i) â„Ź ∈ NCSď ĄC(đ?’°). (ii) đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„ą))) ⊆ â„Ź ⊆ â„ą, for some â„ą NC-CS. (iii) đ?‘†đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„ą)) ⊆ â„Ź ⊆ â„ą, for some â„ą NC-CS. (iv) đ?‘†đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„Ź))) ⊆ â„Ź.

Proof: (đ?‘–) â&#x;š (đ?‘–đ?‘–) Let â„Ź ∈ NCSď ĄC(đ?’°) â&#x;š đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„Ź)))) ⊆ â„Ź (by Corollary(3.11)) and â„Ź ⊆ đ?‘ đ??śđ?‘?đ?‘™(â„Ź). Hence we obtain đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„Ź)))) ⊆ â„Ź ⊆ đ?‘ đ??śđ?‘?đ?‘™(â„Ź).

Therefore, đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„ą))) ⊆ â„Ź ⊆ â„ą, where â„ą = đ?‘ đ??śđ?‘?đ?‘™(â„Ź). (đ?‘–đ?‘–) â&#x;š (đ?‘–đ?‘–đ?‘–) Let đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„ą))) ⊆ â„Ź ⊆ â„ą, for some â„ą NC-CS. But đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„ą))) = đ?‘†đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„ą)) (by Proposition (2.8)). Hence đ?‘†đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„ą)) ⊆ â„Ź ⊆ â„ą, for some â„ą NC-CS. (đ?‘–đ?‘–đ?‘–) â&#x;š (đ?‘–đ?‘Ł) Let đ?‘†đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„ą)) ⊆ â„Ź ⊆ â„ą, for some â„ą NC-CS. Since â„Ź ⊆ â„ą (by hypothesis), then we have đ?‘ đ??śđ?‘?đ?‘™(â„Ź) ⊆ â„ą â&#x;š đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„Ź) ⊆ đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„ą) â&#x;š đ?‘†đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„Ź))) ⊆ đ?‘†đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(â„ą)) ⊆ â„Ź â&#x;š đ?‘†đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„Ź))) ⊆ â„Ź. (đ?‘–đ?‘Ł) â&#x;š (đ?‘–) Let đ?‘†đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„Ź))) ⊆ â„Ź.

But đ?‘†đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„Ź))) = đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„Ź)))) (by Proposition (2.8)).

Hence, đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(đ?‘ đ??śđ?‘–đ?‘›đ?‘Ą(đ?‘ đ??śđ?‘?đ?‘™(â„Ź)))) ⊆ â„Ź â&#x;š â„Ź ∈ NCSď ĄC(đ?’°).

5. Conclusion In this work, we have the new concept of neutrosophic crisp closed sets called neutrosophic crisp semi-ď Ąclosed sets and studied their fundamental properties in neutrosophic crisp topological spaces. The neutrosophic crisp semi-ď Ą-closed sets can obtain to derive a new decomposition of neutrosophic crisp continuity.

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Received: June 18, 2018. Accepted: July 9, 2018.

Riad K. Al-Hamido, Qays Hatem Imran, Karem A. Alghurabi and Taleb Gharibah, On Neutrosophic Crisp Semi Alpha Closed Sets