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WR0 SPACE IN MINIMAL gCLOSED SETS OF TYPE1 G. Venkateswarlu1, V. Amarendra Babu2, K. Bhagya Lakshmi3 and V.B.V.N. Prasad4 1Research
scholar, Dravidian University, Kuppam, Chittore(District), A.P., India. of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar, Guntur Dist., A.P., India. 3Department of Mathematics, K.I.T.S. Engineering College, Guntur (District) Andhra Pradesh, India. 4 Department of Mathematics, Koneru Lakshmaiah Education Foundation, Vaddeswaram, A.P., India ***ABSTRACT  In this article we introduce and study 3. MAIN RESULT: 2Department
Weakly minimal closes sets of Type1R0 spaces, Weakly minimal gclosed sets of Type1R0 spaces, Weakly minimal closed sets of Type1iR0 spaces, Weakly minimal closed sets of Type1dR0 spaces, Weakly minimal closed sets of Type1bR0 spaces, and also discuss the interrelationships among separation properties along with several counter examples.
Now we state and prove our first main result. Before that we first introduce the following definitions and notations. Note: The collections of all increasing Weaklyminimal closed sets of Type1, increasing Weaklyminimal gclosed sets Type1 is denoted by Wimicl(Z, T1), Wimigcl(Z, T1). [resp. decreasing and balanced is denoted by Wdmicl (Z, T1), Wdmigcl(Z, T1), Wbmicl(Z, T1), Wbmigcl (Z, T1)]. Topological ordered space is denoted by TOS, WeaklyMinimal closed sets of Type 1, WeaklyMinimal gclosed sets of Type 1, set is denoted by W micl(Z, T1), Wmigcl (Z, Τ1) and αclosed, βclosed, Ψclosed is denoted by αcl,βcl, Ψcl.
Key Wards: minimal open sets, minimal gopen sets, minimal open sets of Type1, minimal gopen set of Type1.
1. INTRODUCTION. L.Nachbin[6] Topology and order, D.Van Nostrand Inc., Princeton, New Jersy studied increasing [resp. decreasing, balanced] open sets in 1965. K. Bhagya Lakshmi, J. Venkateswara Rao[13] studied WR0 Type spaces in topological ordered spaces in 2014. G.Venkareswarlu, V.Amarendra Babu, and M.K.R.S Veera kumar [11] introduced and studied minimal open sets of Type1 sets, minimal gopen sets of Type1 sets in 2016. G.Venkateswarlu, V.Amarendra Babu, K. Bhagya Lakshmi and V.B.V.N. Prasad [14] studied WC0 Spaces in 2019.
WE INTRODUCE THE FOLLOWING DEFINITIONS: DEFINITION 3.1: In a topological space (Z, Ƭ), a non empty closed subset F of Z is called a minimal closed sets of Type1 if ∃ at least one nomempty open set U such that F⊆U or U=Z. DEFINITION 3.2: In a topological space (Z, Ƭ), a non empty g closed subset F of Z is called a minimal g closed sets of Type1 if ∃ at least one nomempty g open set U such that F⊆U or U=Z.
In this article we introduce new separation axioms of type Weakly minimal closed sets Type1R0 spaces, Weakly minimal gclosed sets of Type1R0 spaces, Weakly minimal g closed sets of Type1iR0 spaces, Weakly minimal g closed sets of Type1dR0 spaces, Weakly minimal g closed sets of Type1bR0 spaces, and discuss the interrelationships among separation properties along with several counter examples.
DEFINITION 3.3: A space (Z, Ƭ) is called a minimal closed sets of Type1 R0space if micl{x} contained in G where G is Minimal closed sets of Type1( brifly micl{x}) and xϵGϵƬ DEFINITION 3.4: A space (Z, Ƭ) is called a minimal gclosed sets of Type1 R0space if migcl{x} contained in G where G is Minimal gclosed sets of Type1( brifly migcl{x}) and xϵGϵƬ
2. PRELIMINARIES. DEFINITION 2.1[11]: In a topological space (X, Ƭ), an open sub set U of X is called a minimal open sets of Type! If ∃ at least one nonempty closed set F such that F⊆U or U = Ф.
DEFINITION 3.5: A space (Z, Ƭ) is called a minimal gclosed sets of Type1 αR0space if for xϵGϵmigO(Z, Ƭ) αmigcl{x} contained in G where G is Minimal gclosed sets of Type1( brifly migcl{x}) and xϵGϵƬ
DEFINITION 2.2[11]: In a topological space (X, Ƭ), an open sub set U of X is called a minimal g open sets of Type! If ∃ at least one nonempty g closed set F such that F⊆U or U = Ф.
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DEFINITION 3.6: A space (Z, Ƭ) is called a minimal gclosed sets of Type1 ΨR0space if for xϵGϵmigO(Z, Ƭ) Ψmigcl{x} contained in G where G is Minimal gclosed sets of Type1( brifly migcl{x}) and xϵGϵƬ

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DEFINITION 3.7: A space (Z, Ƭ) Called Weakly minimal closed sets of Type1R0 if the intersection of micl{x} is nonempty set xϵZ
THEOREM 3.17: In a TOS (Z, Ƭ, ≤), every Wi migcl(Z, Τ1) – R0 space is a WiαR0 space but not converse. Proof: Suppose (Z, Ƭ ) be a Wi migcl(Z, Τ1) – R0 space. Then the intersection of i migcl{x} is empty xϵZ by fact, everyi migcl(Z, Τ1)is a imicl(Z, T1) and then every imicl(Z, T1) is iαcl set Then iαcl{x} contained in imigcl{x} x∈Z. That implies the intersection of iαcl{x} contained ini migcl{x}. But the intersection of imigcl{x} empty set x∈Z. we get the intersection of iαcl{x} is empty set x∈Z Hence (Z, Τ) is WiαR0 space.
DEFINITION 3.8: A space (Z, Ƭ) Called Weakly minimal gclosed sets of Type1R0 if the intersection of migcl{x} is nonempty set xϵZ DEFINITION 3.9: A space (Z, Ƭ) Called Weakly minimal closed sets of Type1iR0 if the intersection of imicl{x} ia nonempty set xϵZ DEFINITION 3.10: A space (Z, Ƭ) Called Weakly minimal gclosed sets of Type1iR0 if the intersection of imigcl{x} is nonempty set xϵZ
DEFINITION 3.14: A space (Z, Ƭ) Called Weakly minimal gclosed sets of Type1bR0 if the intersection of bmigcl{x} is nonempty xϵZ
EXAMPLE 3.18: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {ζ₁, Ω₃} }. ≤4 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, δ₂) }, (Ω₃, ζ₁), (Ω₃, δ₂) }. migcl(Z, Τ1) are Ф, Z, {ζ₁}, {δ₂, Ω₃} imᵢgcl(Z, Τ1) are Ф, Z αcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}. iαcl sets are Ф, Z, {δ₂}. imigcl{ζ₁} is Z imigcl{δ₂} is Z imigcl {Ω₃} is Z The intersection of i mᵢgcl{x} is Z x∈Z iαcl{ζ₁}is Z iαcl{δ₂} is {δ₂} iαcl{Ω₃} is Z The intersection of iαcl{x} is {δ₂} not equal to Z x∈Z
THEOREM 3.15: In a TOS (Z, Ƭ, ≤), every W migcl(Z, Τ1) – R0 space is a WαR0 space but not converse.
THEOREM 3.19: In a TOS (Z, Ƭ, ≤), every Wd migcl(Z, Τ1) – R0 space is a WdαR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a W migcl(Z, Τ1) – R0 space. Then the intersection of migcl{x} is empty set xϵZ by fact, every migcl(Z, Τ1)is a micl(Z, T1) and then every micl(Z, T1) is αcl set Then αcl{x} contained in migcl{x} x∈Z. That implies the intersection of αcl{x} contained in migcl{x}. But the intersection of migcl{x} empty set xϵZ we get the intersection of αcl{x} is empty set xϵZ. Hence (Z, Τ) is W αR0 space.
Proof: Suppose (Z, Ƭ ) be a Wd migcl(Z, Τ1) – R0 space. Then the intersection of d migcl{x} is empty xϵZ by fact, every d migcl(Z, Τ1)is a dmicl(Z, T1) and then every dmicl(Z, T1) is dαcl set Then dαcl{x} contained in dmigcl{x} x∈Z. That implies the intersection of dαcl{x} contained in d migcl{x}. But the intersection of dmigcl{x} empty set x∈Z. we get the intersection of dαcl{x} is empty set x∈Z Hence (Z, Τ) is WdαR0 space.
EXAMPLE 3.16: Let Z = {ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁ } }. migcl(Z, Τ1) are Ф, Z αcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃} migcl{ζ₁} is Z migcl{δ₂} is Z migcl{Ω₃} is Z The intersection of mᵢgcl{x} is Z x∈Z αcl{ζ₁} is Z αcl{δ₂} is {δ₂} αcl{Ω₃} is {Ω₃} The intersection of αcl{x} is empty set not equal to Z x∈Z
EXAMPLE 3.20: Let Z={ζ₁, δ₂, Ω₃} and T= { Ф, Z, {ζ₁}, {ζ₁, Ω₃} }. ≤2 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, δ₂) }, (Ω₃, δ₂) }. migcl(Z, Τ1) are Ф, Z dmigcl(Z, Τ1) are Ф, Z αcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}. dαcl sets are Ф, Z, {Ω₃}. dmigcl{ζ₁} is Z dmigcl{δ₂} is Z dmigcl{Ω₃} is Z The intersection of dmigcl{x} is Z x∈Z dαcl{ζ₁} is Z dαcl{δ₂} is Z dαcl{Ω₃} is {Ω₃} The intersection of dαcl{x} is {Ω₃} not equal to Z x∈Z
DEFINITION 3.11: A space (Z, Ƭ) Called Weakly minimal closed sets of Type1dR0 if the intersection of dmicl{x} nonempty set xϵZ DEFINITION 3.12: A space (Z, Ƭ) Called Weakly minimal gclosed sets of Type1dR0 if the intersection of dmigcl{x} is nonempty set xϵZ DEFINITION 3.13: A space (Z, Ƭ) Called Weakly minimal closed sets of Type1bR0 if the intersection of bmicl{x} nonempty set xϵZ
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THEOREM 3.21: In a TOS (Z, Ƭ, ≤), every Wb migcl(Z, Τ1) – R0 space is a WbαR0 space but not converse.
THEOREM 3.25: In a TOS (Z, Ƭ, ≤), every Wi migcl(Z, Τ1) – R0 space is a WdαR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a Wb migcl(Z, Τ1) – R0 space. Then the intersection of b migcl{x} is empty xϵZ by fact, every b migcl(Z, Τ1)is a bmicl(Z, T1) and then every bmicl(Z, T1) is bαcl set Then bαcl{x} contained in bmigcl{x} x∈Z. That implies the intersection of bαcl{x} contained in b migcl{x}. But the intersection of bmigcl{x} empty set x∈Z. we get the intersection of bαcl{x} is empty set x∈Z Hence (Z, Τ) is WbαR0 space.
Proof: Suppose (Z, Ƭ ) be a Wi migcl(Z, Τ1) – R0 space. Then the intersection of i migcl{x} is empty xϵZ by fact, every i migcl(Z, Τ1)is a dmicl(Z, T1) and then every dmicl(Z, T1) is dαcl set Then dαcl{x} contained in imigcl{x} x∈Z. That implies the intersection of dαcl{x} contained in i migcl{x}. But the intersection of imigcl{x} empty set x∈Z. we get the intersection of dαcl{x} is empty set x∈Z Hence (Z, Τ) is WdαR0 space.
EXAMPLE 3.22: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {δ₂}, {ζ₁, δ₂}, {δ₂, Ω₃} }. ≤9 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, Ω₃) } }. migcl(Z, Τ1) are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃} bmigcl(Z, Τ1) are Ф, Z αcl sets are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃} bαcl sets are Ф, Z, {ζ₁, Ω₃}. bmigcl{ζ₁} is Z bmigcl{δ₂} is Z bmigcl{Ω₃} is Z The intersection of bmigcl{x} is Z x∈Z bαcl{ζ₁} is {ζ₁, Ω₃} bαcl{δ₂} is Z bαcl{Ω₃} is {ζ₁, Ω₃} The intersection of bαcl{x} is {ζ₁, Ω₃} not equal to Z x∈Z
EXAMPLE3.26: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {δ₂}, {ζ₁, δ₂}, {ζ₁, Ω₃} }. ≤1 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, δ₂), (ζ₁, Ω₃), (δ₂, Ω₃) } migcl(Z, Τ1) are Ф, Z, {δ₂}, {Ω₃}, {ζ₁, Ω₃} imigcl(Z, Τ1) are Ф, Z, {Ω₃} αcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃} d αcl sets are Ф, Z imigcl{ζ₁} is Z imigcl{δ₂} is Z imigcl{Ω₃} is {Ω₃} The intersection of imigcl{x} is {Ω₃} x∈Z. dαc;{ζ₁} is Z dαcl{δ₂} is Z dαcl{Ω₃}is Z The intersection of dαcl{x} is Z not equal to {Ω₃} x∈Z.
THEOREM 3.23: In a TOS (Z, Ƭ, ≤), every Wi migcl(Z, Τ1) – R0 space is a WbαR0 space but not converse.
THEOREM 3.27: In a TOS (Z, Ƭ, ≤), every Wb migcl(Z, Τ1) – R0 space is a WiαR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a Wi migcl(Z, Τ1) – R0 space. Then the intersection of i migcl{x} is empty xϵZ by fact, every i migcl(Z, Τ1)is a bmicl(Z, T1) and then every bmicl(Z, T1) is bαcl set Then bαcl{x} contained in imigcl{x} x∈Z. That implies the intersection of bαcl{x} contained in i migcl{x}. But the intersection of imigcl{x} empty set x∈Z. we get the intersection of bαcl{x} is empty set x∈Z Hence (Z, Τ) is WbαR0 space.
Proof: Suppose (Z, Ƭ ) be a Wb migcl(Z, Τ1) – R0 space. Then the intersection of b migcl{x} is empty xϵZ by fact, every b migcl(Z, Τ1)is a imicl(Z, T1) and then every imicl(Z, T1) is iαcl set Then iαcl{x} contained in bmigcl{x} x∈Z. That implies the intersection of iαcl{x} contained in b migcl{x}. But the intersection of bmigcl{x} empty set x∈Z. we get the intersection of iαcl{x} is empty set x∈Z Hence (Z, Τ) is WiαR0 space.
EXAMPLE 3.24: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁} }. ≤9 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, Ω₃) } }. migcl(Z, Τ1) are Ф, Z imigcl(Z, Τ1) are Ф, Z αcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}. b αcl sets are Ф, Z, {δ₂}. imigcl{ζ₁} is Z imigcl{δ₂} is Z imigcl{Ω₃} is Z The intersection of imigcl{Zx is Z x∈Z. bαcl{ζ₁} is Z bαcl{δ₂} is {δ₂} bαcl{Ω₃} is Z The intersection of bαcl{x} is {δ₂} not equal to Z x∈Z.
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EXAMPLE 3.28: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {δ₂}, {ζ₁, δ₂}, {δ₂, Ω₃} }. ≤1 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, δ₂) }, (ζ₁, Ω₃), (δ₂, Ω₃) }. migcl(Z, Τ1) are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃} bmigcl(Z, Τ1) are Ф, Z αcl sets are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃} iαcl sets are Ф, Z, {Ω₃}, {δ₂, Ω₃} bmigcl{ζ₁} is Z bmigcl{δ₂} is Z bmigcl{Ω₃} is Z The intersection of bmigcl{x} is Z x∈Z. iαcl{ζ₁} is {ζ₁} iαcl{δ₂} is {δ₂, Ω₃} iαcl{Ω₃} is {Ω₃}

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The intersection of iαcl{x} is empty set not equal to Z x∈Z.
iαcl{ζ₁} is Z iαcl{δ₂} is {δ₂} iαl{Ω₃} is {Ω₃} The intersection of iαcl{x} is empty set not equal to Z x∈Z
THEOREM 3.29: In a TOS (Z, Ƭ, ≤), every Wb migcl(Z, Τ1) – R0 space is a WdαR0 space but not converse. Proof. Suppose (Z, Ƭ ) be a Wb migcl(Z, Τ1) – R0 space. Then the intersection of b migcl{x} is empty xϵZ by fact, every b migcl(Z, Τ1)is a dmicl(Z, T1) and then every dmicl(Z, T1) is dαcl set Then dαcl{x} contained in bmigcl{x} x∈Z. That implies the intersection of dαcl{x} contained in b migcl{x}. But the intersection of bmigcl{x} empty set x∈Z. we get the intersection of dαcl{x} is empty set x∈Z Hence (Z, Τ) is WdαR0 space.
THEOREM 3.33: In a TOS (Z, Ƭ, ≤), every Wd migcl(Z, Τ1) – R0 space is a WbαR0 space but not converse. Proof: Suppose (Z, Ƭ ) be a Wd migcl(Z, Τ1) – R0 space. Then the intersection of d migcl{x} is empty xϵZ by fact, every d migcl(Z, Τ1)is a bmicl(Z, T1) and then every bmicl(Z, T1) is bαcl set Then bαcl{x} contained in dmigcl{x} x∈Z. That implies the intersection of bαcl{x} contained in d migcl{x}. But the intersection of dmigcl{x} empty set x∈Z. we get the intersection of bαcl{x} is empty set x∈Z Hence (Z, Τ) is WbαR0 space.
EXAMPLE 3.30: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {δ₂}, {ζ₁, δ₂}, {δ₂, Ω₃} }. ≤2 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, δ₂) }, (Ω₃, δ₂) } migcl(Z, Τ1) are Ф, Z, {δ₂}, {Ω₃}, {ζ₁, Ω₃} bmigcl(Z, Τ1) are Ф,Z αcl sets are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃} dαcl sets are Ф, Z, {ζ₁, Ω₃} bmigcl{ζ₁} is Z bmigcl{δ₂} is Z bmigcl{Ω₃} is Z The intersection of bmigcl{x} is Z x∈Z. dαcl{ζ₁} is {ζ₁} dαcl{δ₂} is Z dαcl{Ω₃} is {Ω₃} The intersection of dαcl{x} is empty set not equal to Z x∈Z.
EXAMPLE 3.34: Let Z = { ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {δ₂}, {ζ₁, δ₂}, {ζ₁, Ω₃} }. ≤4 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (Ω₃, ζ₁), (Ω₃, δ₂) }. migcl(Z, Τ1) are Ф, Z, {δ₂}, {Ω₃}, {ζ₁, Ω₃} dmigcl(Z, Τ1) are Ф,Z, {ζ₁, Ω₃] αci sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃} bαcl sets are Ф, Z dmigcl{ζ₁} is {ζ₁, Ω₃} dmigcl{δ₂} is Z dmigcl{Ω₃} is {ζ₁, Ω₃} The intersection of dmigcl{x} is {ζ₁, Ω₃} x∈Z. bαcl{ζ₁} is Z bαcl{δ₂} is Z bαcl{Ω₃} is Z The intersection of bαcl{x} is Z not equal to ζ₁, Ω₃} x∈Z.
THEOREM 3.31: In a TOS (Z, Ƭ, ≤), every Wd migcl(Z, Τ1) – R0 space is a WiαR0 space but not converse.
THEOREM 3.35: In a TOS (Z, Ƭ, ≤), every W migcl(Z, Τ1) – R0 space is a WβR0 space but not converse.
migcl(Z,
Proof: Suppose (Z, Ƭ ) be a WdΤ1) – R0 space. Then the intersection of d migcl{x} is empty xϵZ by fact, every d migcl(Z, Τ1)is a imicl(Z, T1) and then every imicl(Z, T1) is iαcl set Then iαcl{x} contained in dmigcl{x} x∈Z. That implies the intersection of iαcl{x} contained in d migcl{x}. But the intersection of dmigcl{x} empty set x∈Z. we get the intersection of iαcl{x} is empty set x∈Z Hence (Z, Τ) is WiαR0 space.
Proof: Suppose (Z, Ƭ ) be a W migcl(Z, Τ1) – R0 space. Then the intersection of migcl{x} is empty set xϵZ by fact, every migcl(Z, Τ1)is a micl(Z, T1) and then every micl(Z, T1) is βcl set Then βcl{x} contained in migcl{x} x∈Z. That implies the intersection of βcl{x} contained in migcl{x}. But the intersection of migcl{x} empty set xϵZ we get the intersection of βcl{x} is empty set xϵZ. Hence (Z, Τ) is W βR0 space.
EXAMPLE 3.32: Let Z= { ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {ζ₁, Ω₃} }. ≤3 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, δ₂), (ζ₁, Ω₃) }. migcl(Z, Τ1) are Ф, Z dmigci(Z, Τ1) are Ф,Z αcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃} iαcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}. dmigcl{ζ₁} is Z dmigcl{δ₂} is Z dmigcl{Ω₃} is Z The intersection of dmigcl{Z} is Z x∈Z
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EXAMPLE 3.36: Let Z = { ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {δ₂}, {ζ₁, δ₂} }. migcl(Z, Τ1) are Ф, Z βcl sets are Ф, Z, {ζ₁}, {δ₂}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃} migcl{ζ₁} is Z migcl {δ₂} is Z migcl {Ω₃} is Z The intersection of migcl{x} is Z x∈Z. βcl{ζ₁} is {ζ₁} βcl{δ₂} is {δ₂}

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The intersection of dβck{x} is {δ₂} not equal to Z x∈Z.
βcl{Ω₃} is {Ω₃} The intersection of βcl{x} is empty set not equal to Z x∈Z. THEOREM 3.37: In a TOS (Z, Ƭ, ≤), every Wi– R0 space is a WiβR0 space but not converse.
migcl(Z,
THEOREM 3.41: In a TOS (Z, Ƭ, ≤), every Wb migcl(Z, Τ1) – R0 space is a WbβR0 space but not converse. Proof. Suppose (Z, Ƭ ) be a Wb migcl(Z, Τ1) – R0 space. Then the intersection of b migcl{x} is empty xϵZ by fact, every b migcl(Z, Τ1)is a bmicl(Z, T1) and then every bmicl(Z, T1) is bβcl set Then bβcl{x} contained in bmigcl{x} x∈Z. That implies the intersection of bβcl{x} contained in b migcl{x}. But the intersection of bmigcl{x} empty set x∈Z. we get the intersection of bβcl{x} is empty set x∈Z Hence (Z, Τ) is WbβR0 space.
Τ1)
Proof: Suppose (Z, Ƭ ) be a Wi migcl(Z, Τ1) – R0 space. Then the intersection of i migcl{x} is empty xϵZ by fact, every i migcl(Z, Τ1)is a imicl(Z, T1) and then every imicl(Z, T1) is iβcl set Then iβcl{x} contained in imigcl{x} x∈Z. That implies the intersection of iβcl{x} contained in i migcl{x}. But the intersection of imigcl{x} empty set x∈Z. we get the intersection of iβcl{x} is empty set x∈Z Hence (Z, Τ) is WiβR0 space.
EXAMPLE 3.42: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {δ₂}, {ζ₁, δ₂}, {δ₂, Ω₃} }. ≤9 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, Ω₃) } }. migcl(Z, Τ1) are Ф, Z, {δ₂}, {Ω₃}, {ζ₁, Ω₃} bmigcl(Z, Τ1) are Ф, Z, {δ₂} βclsets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃} bβcl sets are Ф, Z, {δ₂}, {ζ₁, Ω₃}. bmigcl{ζ₁} is Z bmigcl{δ₂} = {δ₂} bmigcl{Ω₃} = Z The intersection of bmigcl{x} is {δ₂} x∈Z. bβcl{ζ₁} is {ζ₁, Ω₃} bβcl{δ₂} is {δ₂} bαcl{Ω₃} is {ζ₁, Ω₃} The intersection of bβcl{x} is Ф not equal to{δ₂} x∈Z.
EXAMPLE 3.38: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {ζ₁, Ω₃} }. ≤4 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, δ₂) }, (Ω₃, ζ₁), (Ω₃, δ₂) }. migcl(Z, Τ1) a re Ф, Z imiclg(Z, Τ1) are Ф, Z βcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}. Iβcl sets are Ф, Z, {δ₂}. imigcl{ζ₁} is Z imigcl{δ₂} is Z imigcl{Ω₃} is Z The intersection of imigcl{x} is Z x∈Z. iβcl{ζ₁} is Z iβcl{δ₂} is {δ₂} iβcl{Ω₃} is Z The intersection of iβcl{x} is {δ₂} not equal to Z x∈Z.
THEOREM 3.43: In a TOS (Z, Ƭ, ≤), every Wi migcl(Z, Τ1) – R0 space is a WbβR0 space but not converse.
THEOREM 3.39: In a TOS (Z, Ƭ, ≤), every Wd migcl(Z, Τ1) – R0 space is a WdβR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a Wi migcl(Z, Τ1) – R0 space. Then the intersection of i migcl{x} is empty xϵZ by fact, every i migcl(Z, Τ1)is a bmicl(Z, T1) and then every bmicl(Z, T1) is bβcl set Then bβcl{x} contained in imigcl{x} x∈Z. That implies the intersection of bβcl{x} contained in i migcl{x}. But the intersection of imigcl{x} empty set x∈Z. we get the intersection of bβcl{x} is empty set x∈Z Hence (Z, Τ) is WbβR0 space.
Proof: Suppose (Z, Ƭ ) be a Wd migcl(Z, Τ1) – R0 space. Then the intersection of d migcl{x} is empty xϵZ by fact, every d migcl(Z, Τ1)is a dmicl(Z, T1) and then every dmicl(Z, T1) is dβcl set Then dβcl{x} contained in dmigcl{x} x∈Z. That implies the intersection of dβcl{x} contained in d migcl{x}. But the intersection of dmigcl{x} empty set x∈Z. we get the intersection of dβcl{x} is empty set x∈Z Hence (Z, Τ) is WdβR0 space.
EXAMPLE 3.44: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {δ₂, Ω₃} }. ≤5 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, Ω₃), (δ₂, Ω₃) } migcl(Z, Τ1) are Ф, Z, {ζ₁}, {δ₂, Ω₃} imigcl(Z, Τ1) are Ф,Z,{δ₂, Ω₃} βcl sets are Ф, Z, {ζ₁}, {δ₂}, {Ω₃}, {ζ₁, δ₂}, {δ₂, Ω₃}, {ζ₁, Ω₃} bβcl sets are Ф, Z imigcl{ζ₁} is Z imigcl{δ₂} is {δ₂, Ω₃} imigcl{Ω₃} is {δ₂, Ω₃} The intersection of imigcl{Z} is {δ₂, Ω₃} x∈Z. bβcl{ζ₁} is Z bβcl{δ₂} is Z bβcl{Ω₃} is Z The intersection of bβcl{x} is Z not equal to {δ₂, Ω₃} x∈Z.
EXAMPLE 3.40: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁} }, ≤4 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, δ₂) }, (Ω₃, ζ₁), (Ω₃, δ₂) }. migcl(Z, Τ1) are Ф, Z dmigcl(Z, Τ1) are Ф, Z βcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}. dβcl sets are Ф, Z, {Ω₃}. dmigcl{ζ₁} is Z dmigcl{δ₂} is Z dmigcl{Ω₃} is Z The intersection of dmigcl{x} is Z x∈Z. dβcl{ζ₁} is Z dβcl{δ₂} is {δ₂} dβcl{Ω₃} is Z
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THEOREM 3.45: In a TOS (Z, Ƭ, ≤), every Wi migcl(Z, Τ1) – R0 space is a WdβR0 space but not converse.
THEOREM 3.49: In a TOS (Z, Ƭ, ≤), every Wb migcl(Z, Τ1) – R0 space is a WdβR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a Wi migcl(Z, Τ1) – R0 space. Then the intersection of i migcl{x} is empty xϵZ by fact, every i migcl(Z, Τ1)is a dmicl(Z, T1) and then every dmicl(Z, T1) is dβcl set Then dβcl{x} contained in imigcl{x} x∈Z. That implies the intersection of dβcl{x} contained in i migcl{x}. But the intersection of imigcl{x} empty set x∈Z. we get the intersection of dβcl{x} is empty set x∈Z Hence (Z, Τ) is WdβR0 space.
Proof: Suppose (Z, Ƭ ) be a Wb migcl(Z, Τ1) – R0 space. Then the intersection of b migcl{x} is empty xϵZ by fact, every b migcl(Z, Τ1)is a dmicl(Z, T1) and then every dmicl(Z, T1) is dβcl set Then dβcl{x} contained in bmigcl{x} x∈Z. That implies the intersection of dβcl{x} contained in b migcl{x}. But the intersection of bmigcl{x} empty set x∈Z. we get the intersection of dβcl{x} is empty set x∈Z Hence (Z, Τ) is WdβR0 space.
EXAMPLE 3.46: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {ζ₁, δ₂}, {ζ₁, Ω₃} }. ≤1 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, δ₂), (ζ₁, Ω₃), (δ₂, Ω₃)} migcl(Z, Τ1) are Ф, Z, {δ₂}, {Ω₃} imigcl(Z, Τ1) are Ф, Z, {Ω₃} βcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃} dβcl sets are Ф, Z imigcl{ζ₁} is Z imigcl{δ₂} is Z imigcl{Ω₃} is {Ω₃} The intersection of imigcl{x} is {Ω₃} x∈Z. dβcl{ζ₁} is Z dβcl{δ₂} is Z dβcl{Ω₃} isZ The intersection of dβcl{x} is Z not equal to{Ω₃} x∈Z.
EXAMPLE 3.50: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {ζ₁, δ₂} }. ≤10 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (Ω₃, ζ₁) }, (δ₂, Ω₃), (δ₂, ζ₁) }. migcl(Z, Τ1) are Ф, Z bmigcl(Z, Τ1) are Ф,Z βcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃} } dβcl sets are Ф, Z, {ζ₁, Ω₃} bmigcl{ζ₁} is Z bmigcl{δ₂} is Z bmigcl{Ω₃} is Z The intersection of bmigcl{x} is Z x∈Z. dβcl{ζ₁} is Z dβcl{δ₂} is {δ₂} dβcl{Ω₃} are {δ₂, Ω₃} The intersection of dβcl{x} is {δ₂} not equal to Z x∈Z.
THEOREM 3.47: In a TOS (Z, Ƭ, ≤), every Wb migcl(Z, Τ1) – R0 space is a WiβR0 space but not converse.
THEOREM 3.51: In a TOS (Z, Ƭ, ≤), every Wd migcl(Z, Τ1) – R0 space is a WiβR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a Wb migcl(Z, Τ1) – R0 space. Then the intersection of b migcl{x} is empty xϵZ by fact, every b migcl(Z, Τ1)is a imicl(Z, T1) and then every imicl(Z, T1) is iβcl set Then iβcl{x} contained in bmigcl{x} x∈Z. That implies the intersection of iβcl{x} contained in b migcl{x}. But the intersection of bmigcl{x} empty set x∈Z.
Proof: Suppose (Z, Ƭ ) be a Wd migcl(Z, Τ1) – R0 space. Then the intersection of d migcl{x} is empty xϵZ by fact, every d migcl(Z, Τ1)is a imicl(Z, T1) and then every imicl(Z, T1) is iβcl set Then iαcl{x} contained in dmigcl{x} x∈Z. That implies the intersection of iβcl{x} contained in d migcl{x}. But the intersection of dmigcl{x} empty set x∈Z. we get the intersection of iβcl{x} is empty set x∈Z Hence (Z, Τ) is WiβR0 space.
EXAMPLE 3.48: Let Z = { ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃} }. ≤1 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (δ₂, ζ₁) }, (ζ₁, Ω₃), (δ₂, Ω₃) }. migcl(Z, Τ1) are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃} bmigl(Z, Τ1) are Ф, Z βcl sets are Ф, Z, {ζ₁}, {δ₂}, {Ω₃}, {ζ₁, δ₂}, {ζ₁, Ω₃} iβcl sets are Ф, Z, {Ω₃}, {ζ₁, Ω₃} bmigcl{ζ₁} is Z bmigcl{δ₂} is Z bmigcl{Ω₃} is Z The intersection of bmigcl{Z} is Z x∈Z. iβcl{ζ₁} is {ζ₁, Ω₃} iβcl{δ₂} is Z iβcl{Ω₃} is {Ω₃} The intersection of iβcl{Z} is {Ω₃} not equal to Z x∈Z.
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EXAMPLE 3.52: Let Z = { ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {Ω₃}, {δ₂, Ω₃} }, ≤7 = {(ζ₁, ζ₁), (δ₂, Ω₃), (Ω₃, Ω₃), (δ₂, ζ₁) } }. migcl(Z, Τ1) are Ф, Z dmigcl(Z, Τ1) are Ф,Z βclsets are Ф, Z, {ζ₁}, {δ₂}, {ζ₁, δ₂} iβcl sets are Ф, Z, {ζ₁}, {ζ₁, δ₂}. dmigcl{ζ₁} is Z dmigcl{δ₂} is Z dmigcl{Ω₃} is Z The intersection of dmigcl{x} is Z x∈Z. iβcl{ζ₁} is {ζ₁} iβcl{δ₂} is {ζ₁, δ₂} iβcl{Ω₃} is Z The intersection of iβcl{x} is {ζ₁} not equal to Z x∈Z.

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THEOREM 3.53: In a TOS (Z, Ƭ, ≤), every Wd migcl(Z, Τ1) – R0 space is a WbβR0 space but not converse.
THEOREM 3.57: In a TOS (Z, Ƭ, ≤), every Wi migcl(Z, Τ1) – R0 space is a WiΨR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a Wd migcl(Z, Τ1) – R0 space. Then the intersection of d migcl{x} is empty xϵZ by fact, every d migcl(Z, Τ1)is a bmicl(Z, T1) and then every bmicl(Z, T1) is bβcl set Then bβcl{x} contained in dmigcl{x} x∈Z. That implies the intersection of bβcl{x} contained in d migcl{x}. But the intersection of dmigcl{x} empty set x∈Z. we get the intersection of bβcl{x} is empty set x∈Z Hence (Z, Τ) is WbβR0 space.
Proof: Suppose (Z, Ƭ ) be a Wi migcl(Z, Τ1) – R0 space. Then the intersection of i migcl{x} is empty xϵZ by fact, every i migcl(Z, Τ1)is a imicl(Z, T1) and then every imicl(Z, T1) is iΨcl set Then iΨcl{x} contained in imigcl{x} x∈Z. That implies the intersection of iΨcl{x} contained in i migcl{x}. But the intersection of imigcl{x} empty set x∈Z. we get the intersection of iΨcl{x} is empty set x∈Z Hence (Z, Τ) is WiΨR0 space.
EXAMPLE 3.54: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {δ₂}, {ζ₁, δ₂}, {δ₂, Ω₃} }, ≤3 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, δ₂), (ζ₁, Ω₃) }. migcl(Z, Τ1) are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃} dmigcl(Z, Τ1) are Ф,Z, {ζ₁} βcl sets are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃} bβcl sets are Ф, Z dmigcl{ζ₁} is {ζ₁} dmigcl{δ₂} is Z dmigcl{Ω₃} is Z The intersection of dmigcl{x} is {ζ₁} x∈Z. bβcl{ζ₁} is Z bβcl{δ₂} is Z bβcl{Ω₃} is Z The intersection of bβcl{x} is Z not equal to {ζ₁} x∈Z.
EXAMPLE 3.58: Let Z = { ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃} }. ≤7 = {(ζ₁, ζ₁), (δ₂, Ω₃), (Ω₃, Ω₃), (δ₂, ζ₁) } }. migcl(Z, Τ1) are Ф, Z imigcl(Z, Τ1) are Ф, Z Ψcl sets are Ф, Z, {ζ₁}, {δ₂}, {Ω₃}, {ζ₁, δ₂}, {ζ₁, Ω₃}. iΨcl sets are Ф, Z, {ζ₁}, {Ω₃}, {ζ₁, δ₂}, {ζ₁, Ω₃} imigcl{ζ₁} is Z imigcl{δ₂} is Z imigcl{Ω₃} is Z The intersection of imigcl{x} is Z x∈Z. iΨcl{ζ₁} is {ζ₁} iΨcl{δ₂} is {ζ₁, δ₂} iΨcl{Ω₃} is {Ω₃} The intersection of iΨcl{x} is empty but not equal to Z x∈Z.
THEOREM 3.55: In a TOS (Z, Ƭ, ≤), every W migcl(Z, Τ1) – R0 space is a WΨR0 space but not converse.
THEOREM 3.59: In a TOS (Z, Ƭ, ≤), every Wd migcl(Z, Τ1) – R0 space is a WdΨR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a W migcl(Z, Τ1) – R0 space. Then the intersection of migcl{x} is empty set xϵZ by fact, every migcl(Z, Τ1)is a micl(Z, T1) and then every micl(Z, T1) is Ψcl set Then βcl{x} contained in migcl{x} x∈Z. That implies the intersection of Ψcl{x} contained in migcl{x}. But the intersection of migcl{x} is empty set x∈Z. we get the intersection of Ψcl{x} is empty set x∈Z. Hence (Z, Τ) is W ΨR0 space.
Proof: Suppose (Z, Ƭ ) be a Wd migcl(Z, Τ1) – R0 space. Then the intersection of d migcl{x} is empty xϵZ by fact, every d migcl(Z, Τ1)is a dmicl(Z, T1) and then every dmicl(Z, T1) is dΨcl set Then dΨcl{x} contained in dmigcl{x} x∈Z. That implies the intersection of dΨcl{x} contained in d migcl{x}. But the intersection of dmigcl{x} empty set x∈Z. we get the intersection of dΨcl{x} is empty set x∈Z Hence (Z, Τ) is WdΨR0 space.
EXAMPLE 3.56: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {δ₂ }, {Ω₃], {δ₂, Ω₃} } migcl(Z, Τ1) are Ф, Z Ψcl sets are Ф, Z, {ζ₁}, {δ₂}, {Ω₃}, {ζ₁, δ₂} {ζ₁, Ω₃} migcl{ζ₁} is Z migcl{δ₂} is Z migcl{Ω₃} is Z The intersection of migcl{x} is Z x∈Z. Ψcl{ζ₁} is {ζ₁} Ψcl{δ₂} is {δ₂} Ψcl{Ω₃} is {Ω₃} The intersection of Ψcl{x} ia empty set but not equal to Z x∈Z.
EXAMPLE 3.60: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {ζ₁, Ω₃} }. ≤6 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (δ₂, ζ₁) }, (ζ₁, Ω₃), {δ₂, Ω₃} }. migcl(Z, Τ1) are Ф, Z dmigcl(Z, Τ1) are Ф, Z Ψcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}. dΨcl sets are Ф, Z, {δ₂}. dmigcl{ζ₁} is Z dmigcl{δ₂} is Z dmigcl{Ω₃} is Z The intersection of dmigcl{x} is Z x∈Z. dΨcl{ζ₁} is Z dΨcl{δ₂} is {δ₂} dΨcl{Ω₃} is Z The intersection of dαcl{x} is {δ₂}not equal to Z x∈Z.
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THEOREM 3.61: In a TOS (Z, Ƭ, ≤), every Wb migcl(Z, Τ1) – R0 space is a WbΨR0 space but not converse.
THEOREM 3.65: In a TOS (Z, Ƭ, ≤), every Wi migcl(Z, Τ1) – R0 space is a WdΨR0 space but not converse. Proof. Suppose (Z, Ƭ ) be a Wi migcl(Z, Τ1) – R0 space. Then the intersection of i migcl{x} is empty xϵZ by fact, every i migcl(Z, Τ1)is a dmicl(Z, T1) and then every dmicl(Z, T1) is dΨcl set Then dΨcl{x} contained in imigcl{x} x∈Z. That implies the intersection of dΨcl{x} contained in i migcl{x}. But the intersection of imigcl{x} empty set x∈Z. we get the intersection of dΨcl{x} is empty set x∈Z Hence (Z, Τ) is WdΨR0 space.
Proof: Suppose (Z, Ƭ ) be a Wb migcl(Z, Τ1) – R0 space. Then the intersection of b migcl{x} is empty xϵZ by fact, every b migcl(Z, Τ1)is a bmicl(Z, T1) and then every bmicl(Z, T1) is bΨcl set Then bΨcl{x} contained in bmigcl{x} x∈Z. That implies the intersection of bΨcl{x} contained in b migcl{x}. But the intersection of bmigcl{x} empty set x∈Z. we get the intersection of bΨcl{x} is empty set x∈Z Hence (Z, Τ) is WbΨR0 space. EXAMPLE 3.62: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {ζ₁, Ω₃} }. ≤9 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, Ω₃) } }. migcl(Z, Τ1) are Ф, Z bmigcl(Z, Τ1) are Ф, Z Ψcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃} bΨcl sets are Ф, Z, {δ₂}. bmigcl{ζ₁} is Z bmigcl{δ₂} is Z bmigcl{Ω₃} is Z The intersection of bmigcl{x} is Z x∈Z. bΨcl{ζ₁} is Z bΨcl{δ₂} is {δ₂} bΨcl{Ω₃} is Z The intersection of bΨcl{x} is {δ₂} is not equal to Z x∈Z.
EXAMPLE 3.66: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {δ₂}, {ζ₁, δ₂} }. ≤4 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, δ₂), (Ω₃, ζ₁), (Ω₃, δ₂) } migcl(Z, Τ1) are Ф, Z imigcl(Z, Τ1) are Ф, Z Ψcl sets are Ф, Z, {ζ₁}, {δ₂} {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃} dΨcl sets are Ф, Z, {Ω₃}, {ζ₁, Ω₃} imigcl{ζ₁} is Z imigcl{δ₂} is Z imigcl{Ω₃} is Z The intersection of imigcl{x} is Z x∈Z. dΨcl{ζ₁} is {ζ₁, Ω₃} dΨcl{δ₂} is Z dΨcl{Ω₃} is {Ω₃} The intersection of dΨcl{x} is{Ω₃] not equal to Z x∈Z.
THEOREM 3.63: In a TOS (Z, Ƭ, ≤), every Wi migcl(Z, Τ1) – R0 space is a WbΨR0 space but not converse.
THEOREM 3.67: In a TOS (Z, Ƭ, ≤), every Wb migcl(Z, Τ1) – R0 space is a WiΨR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a Wi migcl(Z, Τ1) – R0 space. Then the intersection of i migcl{x} is empty xϵZ by fact, every i migcl(Z, Τ1)is a bmicl(Z, T1) and then every bmicl(Z, T1) is bΨcl set Then bΨcl{x} contained in imigcl{x} x∈Z. That implies the intersection of bΨcl{x} contained in i migcl{x}. But the intersection of imigcl{x} empty set x∈Z. we get the intersection of bΨcl{x} is empty set x∈Z Hence (Z, Τ) is WbΨR0 space.
Proof: Suppose (Z, Ƭ ) be a Wb migcl(Z, Τ1) – R0 space. Then the intersection of b migcl{x} is empty xϵZ by fact, every b migcl(Z, Τ1)is a imicl(Z, T1) and then every imicl(Z, T1) is iΨcl set Then iΨcl{x} contained in bmigcl{x} x∈Z. That implies the intersection of iΨcl{x} contained in b migcl{x}. But the intersection of bmigcl{x} empty set x∈Z. we get the intersection of iΨcl{x} is empty set x∈Z Hence (Z, Τ) is WiΨR0 space.
EXAMPLE 3.64: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {δ₂], {ζ₁, δ₂], {ζ₁, Ω₃} }. ≤1 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, δ₂), (ζ₁, Ω₃), (δ₂, Ω₃) }. migcl(Z, Τ1) are Ф, Z, {δ₂], {Ω₃], {ζ₁, Ω₃} imigcl(Z, Τ1) are Ф, Z, {Ω₃} Ψcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃} bΨcl sets are Ф, Z imigcl{ζ₁} is Z imigcl{δ₂} is Z imigcl{Ω₃} is {Ω₃} The intersection of imigcl{x} is {Ω₃} x∈Z. bΨcl{ζ₁} is Z Ψcl{δ₂} is Z bΨcl{Ω₃} is Z The intersection of bΨcl{x} is Z but not equal to {Ω₃} x∈Z.
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EXAMPLE 3.68: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {δ₂, Ω₃} }. ≤2 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, δ₂) }, (Ω₃, δ₂) }. migcl(Z, Τ1) are Ф, Z, {ζ₁}, {δ₂, Ω₃} bmigcl(Z, Τ1) are Ф, Z Ψcl sets are Ф, Z, {ζ₁}, {δ₂, Ω₃} iΨcl sets are Ф, Z, {δ₂, Ω₃} bmiggcl{ζ₁} is Z bmigcl{δ₂} is Z bmigcl{Ω₃} is Z The intersection of bmigcl{x} is Z x∈Z. iΨcl{ζ₁} is Z iΨcl{δ₂} is {δ₂, Ω₃} iΨcl{Ω₃} is {δ₂, Ω₃} The intersection of iΨcl{x} is {δ₂, Ω₃} not equal to Z x∈Z.

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THEOREM 3.69: In a TOS (Z, Ƭ, ≤), every Wb migcl(Z, Τ1) – R0 space is a WdΨR0 space but not converse.
THEOREM 3.73: In a TOS (Z, Ƭ, ≤), every Wd migcl(Z, Τ1) – R0 space is a WbβR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a Wb migcl(Z, Τ1) – R0 space. Then the intersection of b migcl{x} is empty xϵZ by fact, every b migcl(Z, Τ1)is a dmicl(Z, T1) and then every dmicl(Z, T1) is dΨcl set Then dΨcl{x} contained in bmigcl{x} x∈Z. That implies the intersection of dΨcl{x} contained in b migcl{x}. But the intersection of bmigcl{x} empty set x∈Z. we get the intersection of dΨcl{x} is empty set x∈Z Hence (Z, Τ) is WdΨR0 space.
Proof: Suppose (Z, Ƭ ) be a Wd migcl(Z, Τ1) – R0 space. Then the intersection of d migcl{x} is empty xϵZ by fact, every d migcl(Z, Τ1)is a bmicl(Z, T1) and then every bmicl(Z, T1) is bΨcl set Then bΨcl{x} contained in dmigcl{x} x∈Z. That implies the intersection of bΨcl{x} contained in d migcl{x}. But the intersection of dmigcl{x} empty set x∈Z. we get the intersection of bΨcl{x} is empty set x∈Z Hence (Z, Τ) is WbΨR0 space.
EXAMPLE 3.70: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {δ₂}, {ζ₁, δ₂}, {δ₂, Ω₃} }. ≤5 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, Ω₃) }, (δ₂, Ω₃) } migcl(Z, Τ1) are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃} bmigcl(Z, Τ1) are Ф,Z Ψcl sets are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃} dΨcl sets are Ф, Z, {ζ₁} bmigcl{ζ₁} is Z bmigcl{δ₂} is Z bmigcl{Ω₃} is Z The intersection of bmigcl{x} is Z x∈Z. dΨcl{ζ₁} is {ζ₁} dΨcl{δ₂} is Z dΨcl{Ω₃} is Z The intersection of Ψcl{x} is {ζ₁} not equal to Z x∈Z.
EXAMPLE 3.74: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁}, {ζ₁, Ω₃} }. ≤7 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (δ₂, ζ₁) }. migcl(Z, Τ1) are Ф, Z, {ζ₁}, {δ₂, Ω₃} dmigcl(Z, Τ1) are Ф,Z, {δ₂, Ω₃] Ψcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃} bΨcl sets are Ф, Z, {Ω₃} dmigcl{ζ₁} is Z bΨcl{ζ₁} is Z cl dmig {δ₂} is {δ₂, Ω₃} bΨcl{δ₂} is Z cl dmig {Ω₃} is {δ₂, Ω₃} bΨcl{Ω₃} is {Ω₃} The intersection of dmigcl{x} is {δ₂, Ω₃} The intersection of bΨcl{x} is [Ω₃} not equal to {δ₂, Ω₃} bΨcl{ζ₁} is Z bΨcl{δ₂} is Z bΨcl{Ω₃} is {Ω₃} The intersection of bΨcl{x} is [Ω₃} not equal to {δ₂, Ω₃}
THEOREM 3.71: In a TOS (Z, Ƭ, ≤), every Wd migcl(Z, Τ1) – R0 space is a WiΨR0 space but not converse. Proof: Suppose (Z, Ƭ ) be a Wd migcl(Z, Τ1) – R0 space. Then the intersection of d migcl{x} is empty xϵZ by fact, every d migcl(Z, Τ1)is a imicl(Z, T1) and then every imicl(Z, T1) is iΨcl set Then iΨcl{x} contained in dmigcl{x} x∈Z. That implies the intersection of iΨcl{x} contained in d migcl{x}. But the intersection of dmigcl{x} empty set x∈Z. we get the intersection of iΨcl{x} is empty set x∈Z Hence (Z, Τ) is WiΨR0 space.
The following diagrams shows the above results.
EXAMPLE 3.72: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃} }. ≤3 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, δ₂), (ζ₁, Ω₃) }. migcl(Z, Τ1) are Ф, Z dmigcl(Z, Τ1) are Ф, Z Ψcl are Ф, Z, {ζ₁}, {δ₂}, {Ω₃}, {ζ₁, δ₂}, {ζ₁, Ω₃} iΨcl sets are Ф, Z, {δ₂}, {Ω₃} dmigcl{ζ₁} is Z dmigcl{δ₂} is Z dmigcl{Ω₃} is Z The intersections of dmigcl{x} is Z x∈Z. iΨcl{ζ₁} is Z iΨcl{δ₂} is {δ₂} iΨcl{Ω₃} is {Ω₃} The intersection of iΨcl{x} is empty not equal to Z x∈Z.
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some classes of nearly open sets, Pacific J. Math., 15(1965), 961970.
REFERENCES: [1].
K. Krishna Rao, some concepts in topological ordered spaces using semiopen sets, preopen sets; αopen sets and βopen sets, Ph.D Thesis, Archarya Nagarjuna University, October, 2012.
[2].
K. Krishna Rao and R.Chudamani, αhomomorphisms in topological ordered spaces, International Journal of Mathematics, Tcchnology and Humanities, 52 (2012), 541560.
[3].
N. Levine, Generalized closed sets in topology, Rend. Circ. Math. Palermo, 19(2)(1970), 8996.
[4].
A.S.Mashhour, I.A.Hasanein and S.N.El.Deeb, αcontinuous and αopen mappings, Acta Math. Hung. 41(34) (1983), 213218.
[5].
H. Maki, R. Devi and K. Balachandran, Generalized αclosed sets in topology, Bull. Fukuoka Univ. Ed. Part III, 42(1993), 1321.
[6].
L Nachbin, Topology and order, D.Van Nostrand Inc., Princeton, New Jersy (1965)[8]. O. Njastad, On
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[7].
V.V.S. Ramachnadram, B. Sankara Rao and M.K.R.S. Veera kumar, gclosed type, g*closed type and sgclosed type sets in topological ordered spaces, Diophantus j. Math., 4(1)(2015), 19.
[8].
M.K.R.S Veera kumar, Homeomorphisms in topological ordered spaces, Acta Ciencia Indica,ZZVIII (M)(1)2002, 6776.
[9].
M.K.R.S. Veera kumar, Between closed sets and gclosed sets Mem. Fac. Sci.Kochi Univ. Ser.A, Math., 21(2000), 119.
[10]. K. Bhagya Lakshmi, V. Amarendra babu and M.K.R.S. Veera kumar, Generalizwd αclosed type sets in topological ordered spaces, Archimedes J. Maths., 5(1)(2015),18. [11] G.venkateswarlu, V. Amarendra babu and M.K.R.S. Veera kumar, International J. Math. And Nature., 2(1)(2016), 114. [12]. M.K.R.S. Veera kumar, Between semiclosed sets and semipreclosed sets, Rend. Lstit. Mat. Univ. Trieste, XXXII(2000), 2541. [13]. K. Bhagya Lakshmi, j. Venkareswara Rao, Archimedes J. Math., 4(3)(2014), 111128. [14]. G.Venkateswarlu, V.Amarendra Babu, K. Bhagya Lakshmi and V.B.V.N. Prasad, IJMTE Journal, Volume IX, issue III, March2019.

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