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ISSN 2229 – 5046

ON INFRA TOPOLOGICAL SPACES

ADEL. M. AL-ODHARI*

Department of Mathematics, Faculty of Education,

Arts and Sciences (Khwlan), Sana'a University, P.O. Box: 13509, Yemen.

(Received On: 27-10-15; Revised & Accepted On: 19-11-15)

ABSRACT

I

n this paper we introduced and investigate infra-topological spaces which deduced from topological spaces and

studies the properties of subsets of infra topological spaces such as infra topological space, infra derived set, infra-interior set , infra-closure set, infra-exterior set and infra-boundary set.

Keywords: infra - topological space, infra -derived set, infra- interior point, infra- closure, infra-exterior point and

infra-boundary set.

1. INTRODUCTION

In 1983, A.S.Mashhour et al. [1] introduced the supra topological space and studied 𝒔𝒔-ontinous functions and

π’”π’”βˆ—-continuous functions.In this paper we introduced Infra -Topological Space (ITS) and analogue concepts associated

with topological space. Such as, derived set (resp.closure, interior, exterior and infra-boundary) of subset 𝑨𝑨 of infra-space 𝑿𝑿. we will be denoted by π’Šπ’Šπ’Šπ’Šπ’”π’”(𝑨𝑨) (resp. π’Šπ’Šπ’Šπ’Šπ’Šπ’Š(𝑨𝑨),π’Šπ’Šπ’Šπ’Šπ’Šπ’Š(𝑨𝑨),π’Šπ’Šπ’Šπ’Šπ’Šπ’Š(𝑨𝑨) and π’Šπ’Šπ’Šπ’Šπ’Šπ’Š(𝑨𝑨)). Many results of topologic space remain valid in infra- topological space, Whereas some become invalid in infra- topological space.

2. INFRA -TOPOLOGICAL SPACES

Definition 2.1: Let 𝑋𝑋 be any arbitrary set. An Infra-topological space on 𝑋𝑋 is a collection πœπœπ‘–π‘–π‘‹π‘‹ of subsets of 𝑋𝑋 such that the following axioms are satisfying:

Ax-1 βˆ…,𝑋𝑋 ∈ πœπœπ‘–π‘–π‘‹π‘‹,.

Ax-2 The intersection of the elements of any subcollection of πœπœπ‘–π‘–π‘‹π‘‹ in 𝑋𝑋. i.e,𝐼𝐼𝐼𝐼𝑂𝑂𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹, 1≀ 𝑖𝑖 ≀ 𝑛𝑛 β†’ β‹‚ 𝑂𝑂𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹.

Terminology, the ordered pair (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) is called Infra-Topological Space. we simply say 𝑋𝑋 is aInfra -space.

Definition 2.2: Let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) be an (ITS) and 𝐴𝐴 βŠ‚ 𝑋𝑋. 𝐴𝐴 is called infra -open set (IOS) if 𝐴𝐴 ∈ πœπœπ‘–π‘–π‘‹π‘‹.

Definition 2.3: Let 𝑋𝑋 be any arbitrary set and 𝜏𝜏= {βˆ…,𝑋𝑋}, then (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) is called indiscrete infra-topology space or is called trivial infra- topological space

Definition 2.4: Let 𝑋𝑋 be any countable arbitrary set and 𝜏𝜏=𝛲𝛲(𝑋𝑋) the set of all subsets of 𝑋𝑋, then (𝑋𝑋,𝜏𝜏) is called

discrete infra-topology space or is called maximal infra- topological space.

Theorem 2.1: Let (𝑋𝑋,𝜏𝜏) be a topological -space (TS), then (𝑋𝑋,𝜏𝜏) is an infra-topological space (IITS).

Proof: Suppose that (𝑋𝑋,𝜏𝜏) is a topological space, then by axioms it is clear that (𝑋𝑋,𝜏𝜏) is infra topological space. The converse of above theorem is not true.

Example 2.1: If 𝑋𝑋= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐} and πœπœπ‘–π‘–π‘‹π‘‹ =οΏ½βˆ…,𝑋𝑋, {π‘Žπ‘Ž}, {𝑏𝑏}οΏ½, then (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) is infra- topological space, but not topological space.

Corresponding Author: Adel. M. Al-Odhari*

Department of Mathematics, Faculty of Education,

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Theorem 2.2: Let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) be infra-topological space. Then: 1. βˆ…,𝑋𝑋 are infra -open set.

2. Any arbitrary intersections of infra- open sets are infra- open sets. 3. Finite union of infra-open sets may not be infra-open sets.

Proof

1. It is clear that βˆ…,𝑋𝑋 are infra open set by Ax-1 and definition 2.1. 2. Let πΆπΆπ‘–π‘–βˆˆπ½π½ ∈ πœπœπ‘–π‘–π‘‹π‘‹ , by Ax-2 and definition 2.1.β‹‚ 𝑂𝑂𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹are infra open set. 3. By counter example 2.1:{π‘Žπ‘Ž}, {𝑏𝑏}∈ πœπœπ‘–π‘–π‘‹π‘‹, but {π‘Žπ‘Ž}βˆͺ{𝑏𝑏} = {π‘Žπ‘Ž,𝑏𝑏}βˆ‰ πœπœπ‘–π‘–π‘‹π‘‹.

Theorem 2.3: let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) and (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹βˆ— ) be two infra topological Spaces on set 𝑋𝑋. Then the intersection πœπœπ‘–π‘–π‘‹π‘‹ and πœπœπ‘–π‘–π‘‹π‘‹βˆ— is aninfra topological space.

Proof: let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) and (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹βˆ— ) be two infra topological Spaces on set 𝑋𝑋.

By Ax-1βˆ…,𝑋𝑋 ∈ πœπœπ‘–π‘–π‘‹π‘‹ and βˆ…,𝑋𝑋 ∈ πœπœπ‘–π‘–π‘‹π‘‹βˆ— , soβˆ…,𝑋𝑋 ∈ πœπœπ‘–π‘–π‘‹π‘‹ ∩ πœπœπ‘–π‘–π‘‹π‘‹βˆ— .Suppose that 𝑂𝑂𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹ ∩ πœπœπ‘–π‘–π‘‹π‘‹βˆ— .

1≀ 𝑖𝑖 ≀ 𝑛𝑛, implies that 𝑂𝑂𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹ and 𝑂𝑂𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹βˆ— Consequently, ⋂𝑖𝑖𝑛𝑛=1𝑂𝑂𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹ and ⋂𝑛𝑛𝑖𝑖=1 𝑂𝑂𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹βˆ— and hence

⋂𝑛𝑛𝑖𝑖=1 𝑂𝑂𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹ ∩ πœπœπ‘–π‘–π‘‹π‘‹βˆ— .

Theorem 2.4: Let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) and (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹βˆ— ) be two infra topological Spaces on set 𝑋𝑋. then the union πœπœπ‘–π‘–π‘‹π‘‹ and πœπœπ‘–π‘–π‘‹π‘‹βˆ— is an infra topological space.

Proof: Let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) and (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹βˆ— ) be tow infra topological infra spaces on set 𝑋𝑋. By Ax-1,βˆ…,𝑋𝑋 ∈ πœπœπ‘–π‘–π‘‹π‘‹ and βˆ…,𝑋𝑋 ∈ πœπœπ‘–π‘–π‘‹π‘‹βˆ— , so βˆ…,𝑋𝑋 ∈ πœπœπ‘–π‘–π‘‹π‘‹ βˆͺ πœπœπ‘–π‘–π‘‹π‘‹βˆ— .

Suppose that 𝑂𝑂𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹βˆͺ πœπœπ‘–π‘–π‘‹π‘‹βˆ— ; 1≀ 𝑖𝑖 ≀ 𝑛𝑛, implies that 𝑂𝑂𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹ or 𝑂𝑂𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹βˆ— for some 𝑖𝑖, 1≀ 𝑖𝑖 ≀ 𝑛𝑛. So ⋂𝑛𝑛𝑖𝑖=1 𝑂𝑂𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹ or ⋂𝑛𝑛𝑖𝑖=1 𝑂𝑂𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹βˆ— and hence ⋂𝑛𝑛𝑖𝑖=1 𝑂𝑂𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹βˆͺ πœπœπ‘–π‘–π‘‹π‘‹βˆ— .

Remark: The union of infra- topological spaces may not be infra-topological space, in general, by the following example.

Example 2.2: Let 𝑋𝑋 be a set 𝑋𝑋= {π‘Žπ‘Ž,𝑏𝑏,𝑐𝑐,𝑑𝑑}, πœπœπ‘–π‘–π‘‹π‘‹ =οΏ½βˆ…,𝑋𝑋, {π‘Žπ‘Ž}, {𝑐𝑐}, {π‘Žπ‘Ž,𝑏𝑏}, {π‘Žπ‘Ž,𝑐𝑐}οΏ½ and πœπœπ‘–π‘–π‘‹π‘‹βˆ—=οΏ½βˆ…,𝑋𝑋, {𝑐𝑐}, {𝑑𝑑}, {𝑏𝑏,𝑐𝑐}, {𝑐𝑐,𝑑𝑑}οΏ½.

Now πœπœπ‘–π‘–π‘‹π‘‹βˆͺ πœπœπ‘–π‘–π‘‹π‘‹βˆ—=οΏ½βˆ…,𝑋𝑋, {π‘Žπ‘Ž}, {𝑏𝑏}, {𝑐𝑐}, {π‘Žπ‘Ž,𝑏𝑏}, {π‘Žπ‘Ž,𝑐𝑐}, {𝑏𝑏,𝑐𝑐}, {𝑐𝑐,𝑑𝑑}οΏ½. is not infra-topological space.

3. PROPERTIES OF SUBSETS ON INFRA TOPOLOGICAL SPACES

Definition 3.1: Let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) be an (ITS) and 𝐴𝐴 βŠ‚ 𝑋𝑋. A point π‘₯π‘₯ ∈ 𝑋𝑋is called Infra-Cluster Point (ICP) of 𝐴𝐴, if for all

Infra-open set 𝑂𝑂containingπ‘₯π‘₯, then 𝐴𝐴 ∩(𝑂𝑂\{π‘₯π‘₯})β‰  βˆ….

Definition 3.2: Let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) be an (ITS) and 𝐴𝐴 βŠ‚ 𝑋𝑋. The set of all Infra-Cluster Point (ICP) of A is called the Infra-Derived Set (IDS) of 𝐴𝐴 and is denoted by 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴).

Definition 3.3: Let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) be infra-topological space. A subset 𝐢𝐢 βŠ‚ 𝑋𝑋 is called infra-closed set in 𝑋𝑋 if 𝑋𝑋 βˆ’ 𝐢𝐢 is infra-open set in 𝑋𝑋. That is 𝐢𝐢 is infra-closed set (ICS) iff 𝑋𝑋 βˆ’ 𝐢𝐢 ∈ πœπœπ‘–π‘–π‘‹π‘‹.

Theorem 3.1: Let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) be infra-topological space. Then: i. βˆ…,𝑋𝑋 ∈ πœπœπ‘–π‘–π‘‹π‘‹ are infra-closed set.

ii. Any arbitrary finite intersections of infra-closed sets is an infra-closed sets.

Proof:

i. Since 𝑋𝑋 βˆ’ βˆ…=𝑋𝑋 ∈ πœπœπ‘–π‘–π‘‹π‘‹ π‘Žπ‘Žπ‘›π‘›π‘‘π‘‘ 𝑋𝑋 βˆ’ 𝑋𝑋=βˆ… ∈ πœπœπ‘–π‘–π‘‹π‘‹ are infra-closed sets.

ii. Let {𝐢𝐢𝑖𝑖 ∢ 𝑖𝑖 ∈ 𝐼𝐼}be an arbitrary family of infra closed sets such that 𝐢𝐢𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹ for all 𝑖𝑖 ∈ 𝐼𝐼.Now, 𝑋𝑋 βˆ’ πΆπΆπ‘–π‘–βˆˆ πœπœπ‘–π‘–π‘‹π‘‹ is infra-open set for all 𝑖𝑖 ∈ 𝐼𝐼 .

But 𝑋𝑋 βˆ’ 𝐢𝐢𝑖𝑖 =𝐢𝐢𝑖𝑖𝑐𝑐 ∈ πœπœπΌπΌπ‘‹π‘‹, then ∩ 𝐢𝐢𝑖𝑖𝑐𝑐= ∩(𝑋𝑋 βˆ’ 𝐢𝐢𝑖𝑖) =𝑋𝑋 βˆ’βˆ© 𝐢𝐢𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹,βˆ€π‘–π‘– ∈ 𝐼𝐼. Hence ∩ 𝐢𝐢𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹,βˆ€π‘–π‘– ∈ 𝐼𝐼 is infra-closed set.

Remark: Finite union of infra-closed sets may not be infra-closed sets, in general.

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Remark: Since 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴) is the intersection of all infra closed sets containing in 𝐴𝐴, then 𝐴𝐴 βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴) and 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴) is the smallest infra closed sets .

Definition 3.5: Let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) be an (ITS) and 𝐴𝐴 βŠ‚ 𝑋𝑋. The Infra-Interior Points (IIP) of 𝐴𝐴 is a set denoted by 𝑖𝑖𝑖𝑖𝑖𝑖 (𝐴𝐴) and given by: 𝑖𝑖𝑖𝑖𝑖𝑖 (𝐴𝐴) =βˆͺ{𝑂𝑂𝑖𝑖: 𝑂𝑂𝑖𝑖 βŠ‚ 𝐴𝐴, 𝑂𝑂𝑖𝑖 ∈ πœπœπ‘–π‘–π‘‹π‘‹}.That is, 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴) is the union of all infra-open set contained in the set 𝐴𝐴.

Remark: Since 𝑖𝑖𝑖𝑖𝑖𝑖 (𝐴𝐴) is the union of all infra-open sets contained in 𝐴𝐴, then𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)βŠ‚ 𝐴𝐴 and 𝑖𝑖𝑐𝑐𝑖𝑖 (𝐴𝐴) is the smallest infra-open sets. Also if 𝑂𝑂is infra-open setcontained in 𝐴𝐴, then 𝑂𝑂 βŠ‚ 𝑖𝑖𝑖𝑖𝑖𝑖 (𝐴𝐴).

Definition 3.6 Let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) be an (ITS) and 𝐴𝐴 βŠ‚ 𝑋𝑋. The Infra-Exterior Points (IEP) of 𝐴𝐴 is a set denoted by 𝑖𝑖𝑖𝑖𝑖𝑖 (𝐴𝐴) and given by: 𝑖𝑖𝑖𝑖𝑖𝑖 (𝐴𝐴) =𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴𝑐𝑐).That is, Set of all infra-interior point of complement of 𝐴𝐴 .

Definition 3.7: Let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) be an (ITS) and 𝐴𝐴 βŠ‚ 𝑋𝑋. The Infra-Boundary Points (IBP) of 𝐴𝐴 is a set denoted by 𝑖𝑖𝑏𝑏𝑖𝑖 (𝐴𝐴) and given by:𝑖𝑖𝑏𝑏𝑖𝑖 (𝐴𝐴) =𝑋𝑋\𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)βˆͺ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)

Theorem 3.2: Let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) be an (ITS) and 𝐴𝐴,𝐡𝐡 βŠ‚ 𝑋𝑋.The Infra-Derived Set Axioms (IDSA) satisfies the followings:

(IDSA )1 : 𝑖𝑖𝑑𝑑𝑖𝑖( βˆ…) =βˆ….

(IDSA )2 : If 𝐴𝐴 βŠ‚ 𝐡𝐡, then 𝑖𝑖𝑑𝑑𝑖𝑖( 𝐴𝐴)βŠ‚ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐡𝐡).

(IDSA )3 : if π‘₯π‘₯ ∈ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴), then π‘₯π‘₯ ∈ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴\{π‘₯π‘₯}).

(IDSA )4 : 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴 ∩ 𝐡𝐡)βŠ‚ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)∩ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐡𝐡).

(IDSA )5 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴 βˆͺ 𝐡𝐡) =𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)βˆͺ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐡𝐡).

Proof:

(IDSA )1 : Suppose that 𝑖𝑖𝑑𝑑𝑖𝑖( βˆ…)β‰  βˆ… β†’ βˆƒ π‘₯π‘₯ ∈ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)βˆ‹ βˆ… ∩(𝑂𝑂\{π‘₯π‘₯})β‰  βˆ…

β†’ π‘₯π‘₯ ∈ βˆ…π‘Žπ‘Žπ‘›π‘›π‘‘π‘‘ π‘₯π‘₯ βˆ‰ βˆ….That is contradiction.

β†’ 𝑖𝑖𝑑𝑑𝑖𝑖( βˆ…) =βˆ….

(IDSA)2 : Suppose that 𝐴𝐴 βŠ‚ 𝐡𝐡. Let π‘₯π‘₯ ∈ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴) β†’ βˆ€π‘‚π‘‚ βˆ‹ π‘₯π‘₯,𝐴𝐴 ∩(𝑂𝑂\{π‘₯π‘₯})β‰  βˆ….

β†’ βˆ€π‘‚π‘‚ βˆ‹ π‘₯π‘₯,𝐡𝐡 ∩(𝑂𝑂\{π‘₯π‘₯})β‰  βˆ…. β†’ π‘₯π‘₯ ∈ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐡𝐡)

β†’ 𝑖𝑖𝑑𝑑𝑖𝑖( 𝐴𝐴)βŠ‚ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐡𝐡).

(IDSA)3 : Assume that π‘₯π‘₯ ∈ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)β†’ βˆ€π‘‚π‘‚ βˆ‹ π‘₯π‘₯,𝐴𝐴 ∩(𝑂𝑂\{π‘₯π‘₯})β‰  βˆ….

β†’ βˆ€π‘‚π‘‚ βˆ‹ π‘₯π‘₯,𝐴𝐴 ∩(𝑂𝑂 ∩{π‘₯π‘₯}𝑐𝑐)β‰  βˆ…. β†’ βˆ€π‘‚π‘‚ βˆ‹ π‘₯π‘₯,𝐴𝐴 ∩(𝑂𝑂 ∩{π‘₯π‘₯}π‘π‘βˆ©{π‘₯π‘₯}𝑐𝑐)β‰  βˆ…. β†’ βˆ€π‘‚π‘‚ βˆ‹ π‘₯π‘₯,𝐴𝐴 ∩({π‘₯π‘₯}π‘π‘βˆ© 𝑂𝑂 ∩{π‘₯π‘₯}𝑐𝑐)β‰  βˆ…. β†’ βˆ€π‘‚π‘‚ βˆ‹ π‘₯π‘₯, (𝐴𝐴 ∩{π‘₯π‘₯}𝑐𝑐)∩(𝑂𝑂 ∩{π‘₯π‘₯}𝑐𝑐)β‰  βˆ…. β†’ βˆ€π‘‚π‘‚ βˆ‹ π‘₯π‘₯, (𝐴𝐴\{π‘₯π‘₯} )∩(𝑂𝑂 \{π‘₯π‘₯} )β‰  βˆ…. β†’ π‘₯π‘₯ ∈ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴\{π‘₯π‘₯}).

(IDSA )4: Since 𝐴𝐴 ∩ 𝐡𝐡 βŠ‚ 𝐴𝐴 ∧ 𝐴𝐴 ∩ 𝐡𝐡 βŠ‚ 𝐡𝐡

β†’ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴 ∩ 𝐡𝐡)βŠ‚ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)∧ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴 ∩ 𝐡𝐡)βŠ‚ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐡𝐡)) .

β†’ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴 ∩ 𝐡𝐡)βŠ‚ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)∩ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐡𝐡)) .

It can be easily shown by example the equality is not hold like topological space.

(IDSA )5: Since 𝐴𝐴 βŠ‚ 𝐴𝐴 βˆͺ 𝐡𝐡 and 𝐡𝐡 βŠ‚ 𝐴𝐴 βˆͺ 𝐡𝐡 , then 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)βŠ‚ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴 βˆͺ 𝐡𝐡) and 𝑖𝑖𝑑𝑑𝑖𝑖(𝐡𝐡)βŠ‚ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴 βˆͺ 𝐡𝐡), hence

𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)βˆͺ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐡𝐡)βŠ‚ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴 βˆͺ 𝐡𝐡). Conversely,

Suppose that π‘₯π‘₯ ∈ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴 βˆͺ 𝐡𝐡)β†’ βˆ€π‘‚π‘‚ βˆ‹ π‘₯π‘₯, (𝐴𝐴 βˆͺ 𝐡𝐡)∩(𝑂𝑂\{π‘₯π‘₯})β‰  βˆ….

β†’ βˆ€π‘‚π‘‚ βˆ‹ π‘₯π‘₯,𝐴𝐴 ∩(𝑂𝑂\{π‘₯π‘₯})β‰  βˆ… βˆͺ 𝐡𝐡 ∩(𝑂𝑂\{π‘₯π‘₯})β‰  βˆ….

β†’ π‘₯π‘₯ ∈ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)βˆͺ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐡𝐡).Hence,

𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴 βˆͺ 𝐡𝐡) =𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)βˆͺ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐡𝐡).

Theorem 3.3: Let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) be an (ITS) and 𝐴𝐴,𝐡𝐡 βŠ‚ 𝑋𝑋.The Infra Closure Point Axioms (ICPA) satisfying the following conditions:

(ICPA)1 : 𝐴𝐴 is infra-closed iff𝐴𝐴=𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴).

(ICPA)2 : 𝑖𝑖𝑐𝑐𝑖𝑖(βˆ…) =βˆ…π‘Žπ‘Žπ‘›π‘›π‘‘π‘‘π‘–π‘–π‘π‘π‘–π‘–( 𝑋𝑋) =𝑋𝑋.

(ICPA)3 : 𝑖𝑖𝑐𝑐𝑖𝑖�𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)οΏ½=𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴).

(ICPA)4 : If 𝐴𝐴 βŠ‚ 𝐡𝐡, then 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐡𝐡).

(4)

Proof:

(ICPA)1 : Suppose that 𝐴𝐴 is infra-closed set. Since 𝐴𝐴 βŠ‚ 𝐴𝐴 and𝐴𝐴 ∩ 𝐴𝐴=𝐴𝐴 β†’ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)βŠ‚ 𝐴𝐴, Also 𝐴𝐴 βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)β†’ 𝐴𝐴=

𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴). Conversely, Let 𝐴𝐴=𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴), obviously,

𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)is the smallest infra-closed set. Hence 𝐴𝐴 is infra-closed set.

(ICPA)2 : Since π‘‹π‘‹π‘Žπ‘Žπ‘›π‘›π‘‘π‘‘βˆ… are infra- closed sets, so by(ICPA)1𝑖𝑖𝑐𝑐𝑖𝑖(βˆ…) =βˆ…π‘Žπ‘Žπ‘›π‘›π‘‘π‘‘π‘–π‘–π‘π‘π‘–π‘–(𝑋𝑋) =𝑋𝑋. (ICPA)3: Since𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴) is the

intersection of all infra-closed sets are closed sets, then 𝑖𝑖𝑐𝑐𝑖𝑖�𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)οΏ½=𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴).

(ICPA)4: Consider 𝐴𝐴 βŠ‚ 𝐡𝐡. Since 𝐴𝐴 βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴) and 𝐡𝐡 βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐡𝐡), so 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐡𝐡).

(ICPA)5: Since (𝐴𝐴 ∩ 𝐡𝐡 βŠ‚ 𝐴𝐴 ∧ 𝐴𝐴 ∩ 𝐡𝐡 βŠ‚ 𝐡𝐡 ), then 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴 ∩ 𝐡𝐡)βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)and

𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴 ∩ 𝐡𝐡)βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐡𝐡)β†’ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴 ∩ 𝐡𝐡)βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)∩ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐡𝐡).

Remark: In (ICPA)5 the equality does not work like topological space.

Theorem 3.4: Let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) be an (ITS) and 𝐴𝐴,𝐡𝐡 βŠ‚ 𝑋𝑋.The Infra-Interior Points Axioms (IIPA) given by:

(IIPA)1: 𝐴𝐴 is infra-open set iff𝐴𝐴=𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴).

(IIPA)2: 𝑖𝑖𝑖𝑖𝑖𝑖( 𝑋𝑋) =𝑋𝑋and𝑖𝑖𝑖𝑖𝑖𝑖( βˆ…) =βˆ…

(IIPA)3: 𝑖𝑖𝑖𝑖𝑖𝑖�𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)οΏ½=𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴).

(IIPA)4 : If 𝐴𝐴 βŠ‚ 𝐡𝐡, then 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)βŠ‚ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡).

(IIPA)5 : 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴 ∩ 𝐡𝐡) =𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)∩ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡).

Proof:

(IIPA)1 : Suppose that 𝐴𝐴 is infra-open set. Since 𝐴𝐴 βŠ‚ 𝐴𝐴, then 𝐴𝐴 is infra-open set containing itself, so 𝐴𝐴 βŠ‚ 𝑖𝑖𝑖𝑖𝑖𝑖 (𝐴𝐴) and

𝑖𝑖𝑖𝑖𝑖𝑖 (𝐴𝐴)βŠ‚ 𝐴𝐴, that implies𝐴𝐴=𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴). Conversely, Let 𝐴𝐴=𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴), suppose that 𝐴𝐴=𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴). Since 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴) is infra-open set, then 𝐴𝐴 is infra- open set.

(IIPA)2: Since 𝑋𝑋,βˆ… are infra-open sets, by (IIPA)1 , we have 𝑖𝑖𝑖𝑖𝑖𝑖( 𝑋𝑋) =𝑋𝑋 and 𝑖𝑖𝑖𝑖𝑖𝑖( βˆ…) =βˆ….

(IIPA)3: Since 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴) is infra-open set. so by (IIPA)1𝑖𝑖𝑖𝑖𝑖𝑖�𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)οΏ½=𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴).

(IIPA)4∢ Suppose that If 𝐴𝐴 βŠ‚ 𝐡𝐡. Let𝑂𝑂𝑖𝑖 ∈ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)β†’ 𝑂𝑂𝑖𝑖 βŠ‚ 𝐴𝐴 β†’ 𝑂𝑂𝑖𝑖 βŠ‚ 𝐡𝐡 β†’ π‘‚π‘‚π‘–π‘–βˆˆ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡).

Therefore 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)βŠ‚ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡).

(IIPA)5: Let 𝑂𝑂𝑖𝑖 ∈ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)∩ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡) ↔ 𝑂𝑂𝑖𝑖 ∈ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴) ∧ 𝑂𝑂𝑖𝑖 ∈ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡).

↔βˆͺ 𝑂𝑂𝑖𝑖,𝑂𝑂𝑖𝑖 βŠ‚ 𝐴𝐴,βˆ€π‘–π‘– ∧βˆͺ 𝑂𝑂𝑖𝑖,𝑂𝑂𝑖𝑖 βŠ‚ 𝐡𝐡,βˆ€π‘–π‘–. ↔βˆͺ 𝑂𝑂𝑖𝑖,𝑂𝑂𝑖𝑖 βŠ‚ 𝐴𝐴 ∩ 𝐡𝐡,βˆ€π‘–π‘–.

↔ 𝑂𝑂𝑖𝑖 ∈ 𝑖𝑖𝑖𝑖𝑖𝑖 (𝐴𝐴 ∩ 𝐡𝐡),βˆ€π‘–π‘–.

Theorem 3.5: Let (π‘‹π‘‹πœπœπ‘–π‘–π‘‹π‘‹) be an (ITS) and 𝐴𝐴,𝐡𝐡 βŠ‚ 𝑋𝑋.The Infra -Exterior points Axioms (IEPA) given by:

(IEPA)1 : 𝑖𝑖𝑖𝑖𝑖𝑖( 𝑋𝑋) =βˆ…and𝑖𝑖𝑖𝑖𝑖𝑖( βˆ…) =𝑋𝑋.

(IEPA)2 : 𝑖𝑖𝑖𝑖𝑖𝑖( 𝐴𝐴)βŠ‚ 𝐴𝐴𝐢𝐢.

(IEPA)3 : 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴 βˆͺ 𝐡𝐡) =𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)∩ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡).

(IEPA)4 : If 𝐴𝐴 βŠ‚ 𝐡𝐡, then 𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡)βŠ‚ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴).

(IEPA)5 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴 ∩ 𝐡𝐡)βŠ‚ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)βˆͺ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡).

Proof:

(IEPA)1: 𝑖𝑖𝑖𝑖𝑖𝑖(𝑋𝑋) =𝑖𝑖𝑖𝑖𝑖𝑖(𝑋𝑋𝑐𝑐) =𝑖𝑖𝑖𝑖𝑖𝑖(βˆ…) =βˆ… and 𝑖𝑖𝑖𝑖𝑖𝑖( βˆ…) =𝑖𝑖𝑖𝑖𝑖𝑖(βˆ…π‘π‘) =𝑖𝑖𝑖𝑖𝑖𝑖(𝑋𝑋) =𝑋𝑋.

(IEPA)2:𝑖𝑖𝑖𝑖𝑖𝑖( 𝐴𝐴) =𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴𝑐𝑐)βŠ‚ 𝐴𝐴𝐢𝐢.

(IEPA)3:𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴 βˆͺ 𝐡𝐡) =𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴 βˆͺ 𝐡𝐡)𝑐𝑐 =𝑖𝑖𝑖𝑖𝑖𝑖(π΄π΄π‘π‘βˆ© 𝐡𝐡𝑐𝑐) =𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴𝑐𝑐)∩ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡𝑐𝑐)

=𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)∩ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡).

(IEPA)4: let 𝐴𝐴 βŠ‚ 𝐡𝐡 β†’ π΅π΅π‘π‘βŠ‚ 𝐴𝐴𝑐𝑐 ⟢ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡𝑐𝑐)βŠ‚ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴𝑐𝑐)β†’ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡)βŠ‚ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴).

(5)

Theorem 3.6: Let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) be an (ITS) and 𝐴𝐴 βŠ‚ 𝑋𝑋.The Infra-Boundary Points Axioms (IBPA) given by:

(IBPA)1 : 𝑖𝑖𝑏𝑏𝑖𝑖( 𝑋𝑋) =𝑖𝑖𝑏𝑏𝑖𝑖(βˆ…) =βˆ….

(IBPA)2 : 𝑖𝑖𝑏𝑏𝑖𝑖( 𝐴𝐴 ∩ 𝐡𝐡) =𝑖𝑖𝑏𝑏𝑖𝑖(𝐴𝐴)βˆͺ 𝑖𝑖𝑏𝑏𝑖𝑖(𝐡𝐡).

Proof:

(IBPA)1: 𝑖𝑖𝑏𝑏𝑖𝑖(𝑋𝑋) =𝑋𝑋\𝑖𝑖𝑖𝑖𝑖𝑖(𝑋𝑋)βˆͺ 𝑖𝑖𝑖𝑖𝑖𝑖(𝑋𝑋) =𝑋𝑋\𝑋𝑋 βˆͺ βˆ…=𝑋𝑋\𝑋𝑋=βˆ….

𝑖𝑖𝑏𝑏𝑖𝑖(βˆ…) =𝑋𝑋\𝑖𝑖𝑖𝑖𝑖𝑖(βˆ…)βˆͺ 𝑖𝑖𝑖𝑖𝑖𝑖(βˆ…) =𝑋𝑋\βˆ… βˆͺ 𝑋𝑋=𝑋𝑋\𝑋𝑋=βˆ…

(IBPA)2:𝑖𝑖𝑏𝑏𝑖𝑖( 𝐴𝐴 ∩ 𝐡𝐡) =𝑋𝑋\𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴 ∩ 𝐡𝐡)βˆͺ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴 ∩ 𝐡𝐡 )

= 𝑋𝑋\𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)∩ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡)βˆͺ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴 ∩ 𝐡𝐡 ) =𝑋𝑋\𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)βˆͺ 𝑋𝑋\𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡)βˆͺ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴 ∩ 𝐡𝐡 ) =𝑋𝑋\𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)βˆͺ 𝑋𝑋\𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡) βˆͺ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)βˆͺ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐡𝐡)) =𝑖𝑖𝑏𝑏𝑖𝑖(𝐴𝐴)βˆͺ 𝑖𝑖𝑏𝑏𝑖𝑖(𝐡𝐡).

The following theorem illustrate the relations between 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴),𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴),𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴),𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)and 𝑖𝑖𝑏𝑏𝑖𝑖(𝐴𝐴).

Theorem 3.7: Let (𝑋𝑋,πœπœπ‘–π‘–π‘‹π‘‹) be an (ITS) and 𝐴𝐴 βŠ‚ 𝑋𝑋.then: 1. 𝐴𝐴 βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)β†’ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)βŠ‚ 𝑖𝑖𝑑𝑑𝑖𝑖�𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)οΏ½.

2. 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)βŠ‚ 𝐴𝐴 β†’ 𝑖𝑖𝑑𝑑𝑖𝑖�𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)οΏ½ βŠ‚ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴). 3. If 𝐴𝐴 is infra-closed , then 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)βŠ‚ 𝐴𝐴. 4. 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴) =𝐴𝐴 βˆͺ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴).

5. 𝑖𝑖𝑏𝑏𝑖𝑖(𝐴𝐴) =𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)\𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴). 6. 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴) =𝑖𝑖𝑏𝑏𝑖𝑖(𝐴𝐴)βˆͺ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴). 7. 𝑖𝑖𝑏𝑏𝑖𝑖(𝐴𝐴)βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴).

8. 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)∩ 𝑖𝑖𝑏𝑏𝑖𝑖(𝐴𝐴) =βˆ….

Proof:

(1) Let𝐴𝐴 βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴). By(IDSA )2𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)βŠ‚ 𝑖𝑖𝑑𝑑𝑖𝑖�𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)οΏ½. (2) Let 𝑖𝑖𝑖𝑖𝑖𝑖( 𝐴𝐴)βŠ‚ 𝐴𝐴. By(IDSA )2𝑖𝑖𝑑𝑑𝑖𝑖�𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)οΏ½ βŠ‚ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴).

(3) Let𝐴𝐴 be a infra closed set and π‘₯π‘₯ ∈ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴), then βˆ€π‘‚π‘‚ βˆ‹ π‘₯π‘₯,𝐴𝐴 ∩(𝑂𝑂 βˆ’(π‘₯π‘₯))β‰  βˆ… Hence π‘₯π‘₯ ∈ 𝐴𝐴and 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)βŠ‚ 𝐴𝐴. (4) Since 𝐴𝐴 βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴) and 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)βŠ‚ 𝑖𝑖𝑑𝑑𝑖𝑖�𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)οΏ½ βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴). we have 𝐴𝐴 βˆͺ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴). Another direction,

To show that 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)βŠ‚ 𝐴𝐴 βˆͺ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴).

Let π‘₯π‘₯ ∈ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴), but 𝐴𝐴 βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴), then π‘₯π‘₯ ∈ 𝐴𝐴 or π‘₯π‘₯ βˆ‰ 𝐴𝐴.

Probability 1) Ifπ‘₯π‘₯ ∈ 𝐴𝐴 , then π‘₯π‘₯ ∈ 𝐴𝐴 βˆͺ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴).

Probability 2) if π‘₯π‘₯ βˆ‰ 𝐴𝐴,Let π‘₯π‘₯ βˆ‰ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴)β†’ βˆƒπ‘‚π‘‚ βˆ‹ π‘₯π‘₯,𝐴𝐴 ∩(𝑂𝑂\{π‘₯π‘₯} =βˆ…, but π‘₯π‘₯ βˆ‰ 𝐴𝐴, that is contradiction, therefore

π‘₯π‘₯ ∈ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴) and π‘₯π‘₯ ∈ 𝐴𝐴 βˆͺ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴).

So 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴) =𝐴𝐴 βˆͺ 𝑖𝑖𝑑𝑑𝑖𝑖(𝐴𝐴).

(5) By defn:𝑖𝑖𝑏𝑏𝑖𝑖 (𝐴𝐴) =𝑋𝑋\𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)βˆͺ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴) =𝑋𝑋\𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)∩ 𝑋𝑋\𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴) =𝑋𝑋\𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)∩ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)

Since 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)βŠ‚ 𝑋𝑋 β†’ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)∩(𝑋𝑋\𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴) =𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)\𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴). Then we have

𝑖𝑖𝑏𝑏𝑖𝑖(𝐴𝐴) =𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)\𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴). (6) By (1) 𝑖𝑖𝑏𝑏𝑖𝑖(𝐴𝐴) =𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)\𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)

β†’ 𝑖𝑖𝑏𝑏𝑖𝑖(𝐴𝐴)βˆͺ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴) =𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)\𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)βˆͺ 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴) =𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴). (7) By (2) it is clear that 𝑖𝑖𝑏𝑏𝑖𝑖(𝐴𝐴)βŠ‚ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴).

(8) 𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)∩ 𝑖𝑖𝑏𝑏𝑖𝑖(𝐴𝐴) =𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴)∩ 𝑖𝑖𝑐𝑐𝑖𝑖(𝐴𝐴)\𝑖𝑖𝑖𝑖𝑖𝑖(𝐴𝐴) =βˆ….

Theorem 3.8: Let 𝑋𝑋 be any finite set and 𝐴𝐴 βŠ‚ 𝑋𝑋 and order (π‘œπ‘œ (𝐴𝐴) = 1). The collection πœπœπ‘–π‘–π‘‹π‘‹ = {βˆ…,𝑋𝑋}βˆͺ{𝐴𝐴 βŠ‚

𝑋𝑋 π‘–π‘–π‘’π‘’π‘π‘β„Ž π‘‘π‘‘β„Žπ‘Žπ‘Žπ‘‘π‘‘ π‘œπ‘œ 𝐴𝐴=1 is infra topological space.

Proof: Since βˆ…,𝑋𝑋 ∈{βˆ…,𝑋𝑋}, so βˆ…,𝑋𝑋 ∈ πœπœπ‘–π‘–π‘‹π‘‹ and Ax-1 is hold. Now, Assume that π΄π΄π‘–π‘–βˆˆ πœπœπ‘–π‘–π‘‹π‘‹, 1≀ 𝑖𝑖 ≀ 𝑛𝑛. And π‘œπ‘œ (𝐴𝐴 ) = 1) Then π΄π΄π‘–π‘–βˆ© βˆ…=βˆ…,βˆ€π‘–π‘–π‘Žπ‘Žπ‘›π‘›π‘‘π‘‘ π΄π΄π‘–π‘–βˆ© 𝑋𝑋= 𝐴𝐴𝑖𝑖,βˆ€π‘–π‘–, so that Ax-2 is hold .

(6)

REFERENCES

1. S. Mashhour,A. A. Allam, F. S. Mahmoud and F. H. Khedr, On Supra Topological spaces, Indian J. Pure and Appl.Math,14(1983),502-510.

2. J. Dugundji, Topology, Universal Book Stall, New Delhi, (1990).

3. J. R. Munkres, Topology A First Course, Prentice-Hall, Inc. Englewood Cliffis, New Jersey,(1975). 4. O. Njastad, On Some Classes of Nearly Open Sets, Pacific Journal of mathematics, Vol.15, No, 31965. 5. R. Brown, Topology: A geometric Account of General Topology, Homotopy Types and The Fundamental

Groupiod, John Wiley & Sons, New York. Chichester. Brisbanc. Toronto.

Source of support: Nil, Conflict of interest: None Declared

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