International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 06 Issue: 03 | Mar 2019
p-ISSN: 2395-0072
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W-R0 SPACE IN MINIMAL g-CLOSED 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 Type1-R0 spaces, Weakly
minimal g-closed sets of Type1-R0 spaces, Weakly minimal
closed sets of Type1-i-R0 spaces, Weakly minimal closed sets
of Type1-d-R0 spaces, Weakly minimal closed sets of Type1b-R0 spaces, and also discuss the inter-relationships 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 Weakly-minimal closed
sets of Type1, increasing Weakly-minimal g-closed sets
Type1 is denoted by W-i-micl(Z, T1), W-i-mi-gcl(Z, T1).
[resp. decreasing and balanced is denoted by W-d-micl
(Z, T1), W-d-mi-gcl(Z, T1), W-b-micl(Z, T1), W-b-mi-gcl
(Z, T1)]. Topological ordered space is denoted by TOS,
Weakly-Minimal closed sets of Type 1, Weakly-Minimal
g-closed sets of Type 1, set is denoted by W- micl(Z, T1),
W-mi-gcl (Z, Τ1) and α-closed, β-closed, Ψ-closed is
denoted by αcl,βcl, Ψcl.
Key Wards: minimal open sets, minimal g-open sets,
minimal open sets of Type1, minimal g-open 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 W-R0 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 g-open sets of Type1 sets in 2016.
G.Venkateswarlu, V.Amarendra Babu, K. Bhagya Lakshmi
and V.B.V.N. Prasad [14] studied W-C0 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 nom-empty 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 nom-empty 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 Type1-R0 spaces,
Weakly minimal g-closed sets of Type1-R0 spaces, Weakly
minimal g- closed sets of Type1-i-R0 spaces, Weakly
minimal g- closed sets of Type1-d-R0 spaces, Weakly
minimal g- closed sets of Type1-b-R0 spaces, and discuss
the inter-relationships among separation properties along
with several counter examples.
DEFINITION 3.3: A space (Z, Ƭ) is called a minimal closed
sets of Type1 R0-space 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 R0-space if mi-gcl{x} contained in G
where G is Minimal g-closed sets of Type1( brifly mi-gcl{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 non-empty 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 α-R0-space if for xϵGϵmi-g-O(Z, Ƭ)
αmi-gcl{x} contained in G where G is Minimal g-closed sets
of Type1( brifly mi-gcl{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 non-empty 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 Ψ-R0-space if for xϵGϵmi-g-O(Z, Ƭ)
Ψmi-gcl{x} contained in G where G is Minimal g-closed sets
of Type1( brifly mi-gcl{x}) and xϵGϵƬ
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International Research Journal of Engineering and Technology (IRJET)
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Volume: 06 Issue: 03 | Mar 2019
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DEFINITION 3.7: A space (Z, Ƭ) Called Weakly minimal
closed sets of Type1--R0 if the intersection of micl{x} is
non-empty set xϵZ
THEOREM 3.17: In a TOS (Z, Ƭ, ≤), every W-i- mi-gcl(Z, Τ1)
– R0 space is a W-iαR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a W-i- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of i- mi-gcl{x} is empty
xϵZ by fact, everyi- mi-gcl(Z, Τ1)is a i-micl(Z, T1) and then
every i-micl(Z, T1) is
iαcl set Then iαcl{x} contained in i-mi-gcl{x} x∈Z.
That implies the intersection of iαcl{x} contained ini- migcl{x}. But the intersection of i-mi-gcl{x} empty set x∈Z.
we get the intersection of iαcl{x} is empty set x∈Z Hence
(Z, Τ) is W-iαR0 space.
DEFINITION 3.8: A space (Z, Ƭ) Called Weakly minimal gclosed sets of Type1--R0 if the intersection of mi-gcl{x} is
non-empty set xϵZ
DEFINITION 3.9: A space (Z, Ƭ) Called Weakly minimal
closed sets of Type1-i-R0 if the intersection of i-micl{x} ia
non-empty set xϵZ
DEFINITION 3.10: A space (Z, Ƭ) Called Weakly minimal
g-closed sets of Type1-i-R0 if the intersection of i-mi-gcl{x}
is non-empty set xϵZ
DEFINITION 3.14: A space (Z, Ƭ) Called Weakly minimal
g-closed sets of Type1-b-R0 if the intersection of b-mi-gcl{x}
is non-empty 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) }.
mi-gcl(Z, Τ1) are Ф, Z, {ζ₁}, {δ₂, Ω₃}
i-mᵢ-gcl(Z, Τ1) are Ф, Z
αcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}.
iαcl sets are Ф, Z, {δ₂}.
i-mi-gcl{ζ₁} is Z
i-mi-gcl{δ₂} is Z
i-mi-gcl {Ω₃} is Z
The intersection of i- mᵢ-gcl{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- mi-gcl(Z, Τ1) –
R0 space is a W-αR0 space but not converse.
THEOREM 3.19: In a TOS (Z, Ƭ, ≤), every W-d- mi-gcl(Z, Τ1)
– R0 space is a W-dαR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a W- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of mi-gcl{x} is empty set
xϵZ by fact, every mi-gcl(Z, Τ1)is a micl(Z, T1) and then
every micl(Z, T1) is
αcl set Then αcl{x} contained in mi-gcl{x} x∈Z.
That implies the intersection of αcl{x} contained in migcl{x}. But the intersection of mi-gcl{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 W-d- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of d- mi-gcl{x} is empty
xϵZ by fact, every d- mi-gcl(Z, Τ1)is a d-micl(Z, T1) and
then every d-micl(Z, T1) is
dαcl set Then dαcl{x} contained in d-mi-gcl{x} x∈Z.
That implies the intersection of dαcl{x} contained in d- migcl{x}. But the intersection of d-mi-gcl{x} empty set x∈Z.
we get the intersection of dαcl{x} is empty set x∈Z Hence
(Z, Τ) is W-dαR0 space.
EXAMPLE 3.16: Let Z = {ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁ } }.
mi-gcl(Z, Τ1) are Ф, Z
αcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}
mi-gcl{ζ₁} is Z
mi-gcl{δ₂} is Z
mi-gcl{Ω₃} is Z
The intersection of mᵢ-gcl{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) }.
mi-gcl(Z, Τ1) are Ф, Z
d-mi-gcl(Z, Τ1) are Ф, Z
αcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}.
dαcl sets are Ф, Z, {Ω₃}.
d-mi-gcl{ζ₁} is Z
d-mi-gcl{δ₂} is Z
d-mi-g-cl{Ω₃} is Z
The intersection of d-mi-gcl{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 Type1-d--R0 if the intersection of d-micl{x}
non-empty set xϵZ
DEFINITION 3.12: A space (Z, Ƭ) Called Weakly minimal
g-closed sets of Type1-d--R0 if the intersection of d-migcl{x} is non-empty set xϵZ
DEFINITION 3.13: A space (Z, Ƭ) Called Weakly minimal
closed sets of Type1-b--R0 if the intersection of b-micl{x}
non-empty set xϵZ
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Volume: 06 Issue: 03 | Mar 2019
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THEOREM 3.21: In a TOS (Z, Ƭ, ≤), every W-b- mi-gcl(Z, Τ1)
– R0 space is a W-bαR0 space but not converse.
THEOREM 3.25: In a TOS (Z, Ƭ, ≤), every W-i- mi-gcl(Z, Τ1)
– R0 space is a W-dαR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a W-b- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of b- mi-gcl{x} is empty
xϵZ by fact, every b- mi-gcl(Z, Τ1)is a b-micl(Z, T1) and
then every b-micl(Z, T1) is
bαcl set Then bαcl{x} contained in b-mi-gcl{x} x∈Z.
That implies the intersection of bαcl{x} contained in b- migcl{x}. But the intersection of b-mi-gcl{x} empty set x∈Z.
we get the intersection of bαcl{x} is empty set x∈Z Hence
(Z, Τ) is W-bαR0 space.
Proof: Suppose (Z, Ƭ ) be a W-i- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of i- mi-gcl{x} is empty
xϵZ by fact, every i- mi-gcl(Z, Τ1)is a d-micl(Z, T1) and
then every d-micl(Z, T1) is
dαcl set Then dαcl{x} contained in i-mi-gcl{x} x∈Z.
That implies the intersection of dαcl{x} contained in i- migcl{x}. But the intersection of i-mi-gcl{x} empty set x∈Z.
we get the intersection of dαcl{x} is empty set x∈Z Hence
(Z, Τ) is W-dαR0 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) } }.
mi-gcl(Z, Τ1) are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃}
b-mi-gcl(Z, Τ1) are Ф, Z
αcl sets are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃}
bαcl sets are Ф, Z, {ζ₁, Ω₃}.
b-mi-gcl{ζ₁} is Z
b-mi-gcl{δ₂} is Z
b-mi-gcl{Ω₃} is Z
The intersection of b-mi-gcl{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) }
mi-gcl(Z, Τ1) are Ф, Z, {δ₂}, {Ω₃}, {ζ₁, Ω₃}
i-mi-gcl(Z, Τ1) are Ф, Z, {Ω₃}
αcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃}
d αcl sets are Ф, Z
i-mi-gcl{ζ₁} is Z
i-mi-gcl{δ₂} is Z
i-mi-gcl{Ω₃} is {Ω₃}
The intersection of i-mi-gcl{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 W-i- mi-gcl(Z, Τ1)
– R0 space is a W-bαR0 space but not converse.
THEOREM 3.27: In a TOS (Z, Ƭ, ≤), every W-b- mi-gcl(Z, Τ1)
– R0 space is a W-iαR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a W-i- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of i- mi-gcl{x} is empty
xϵZ by fact, every i- mi-gcl(Z, Τ1)is a b-micl(Z, T1) and
then every b-micl(Z, T1) is
bαcl set Then bαcl{x} contained in i-mi-gcl{x} x∈Z.
That implies the intersection of bαcl{x} contained in i- migcl{x}. But the intersection of i-mi-gcl{x} empty set x∈Z.
we get the intersection of bαcl{x} is empty set x∈Z Hence
(Z, Τ) is W-bαR0 space.
Proof: Suppose (Z, Ƭ ) be a W-b- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of b- mi-gcl{x} is empty
xϵZ by fact, every b- mi-gcl(Z, Τ1)is a i-micl(Z, T1) and
then every i-micl(Z, T1) is
iαcl set Then iαcl{x} contained in b-mi-gcl{x} x∈Z.
That implies the intersection of iαcl{x} contained in b- migcl{x}. But the intersection of b-mi-gcl{x}
empty set x∈Z.
we get the intersection of iαcl{x} is empty set x∈Z Hence
(Z, Τ) is W-iαR0 space.
EXAMPLE 3.24: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {ζ₁} }.
≤9 = {(ζ₁, ζ₁), (δ₂, δ₂), (Ω₃, Ω₃), (ζ₁, Ω₃) } }.
mi-gcl(Z, Τ1) are Ф, Z
i-mi-gcl(Z, Τ1) are Ф, Z
αcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}.
b αcl sets are Ф, Z, {δ₂}.
i-mi-gcl{ζ₁} is Z
i-mi-gcl{δ₂} is Z
i-mi-gcl{Ω₃} is Z
The intersection of i-mi-gcl{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) }.
mi-gcl(Z, Τ1) are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃}
b-mi-gcl(Z, Τ1) are Ф, Z
αcl sets are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃}
iαcl sets are Ф, Z, {Ω₃}, {δ₂, Ω₃}
b-mi-gcl{ζ₁} is Z
b-migcl{δ₂} is Z
b-migcl{Ω₃} is Z
The intersection of b-mi-gcl{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 W-b- mi-gcl(Z, Τ1)
– R0 space is a W-dαR0 space but not converse.
Proof. Suppose (Z, Ƭ ) be a W-b- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of b- mi-gcl{x} is empty
xϵZ by fact, every b- mi-gcl(Z, Τ1)is a d-micl(Z, T1) and
then every d-micl(Z, T1) is
dαcl set Then dαcl{x} contained in b-mi-gcl{x} x∈Z.
That implies the intersection of dαcl{x} contained in b- migcl{x}. But the intersection of b-mi-gcl{x}
empty set x∈Z.
we get the intersection of dαcl{x} is empty set x∈Z Hence
(Z, Τ) is W-dαR0 space.
THEOREM 3.33: In a TOS (Z, Ƭ, ≤), every W-d- mi-gcl(Z, Τ1)
– R0 space is a W-bαR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a W-d- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of d- mi-gcl{x} is empty
xϵZ by fact, every d- mi-gcl(Z, Τ1)is a b-micl(Z, T1) and
then every b-micl(Z, T1) is
bαcl set Then bαcl{x} contained in d-mi-gcl{x} x∈Z.
That implies the intersection of bαcl{x} contained in d- migcl{x}. But the intersection of d-mi-gcl{x}
empty set x∈Z.
we get the intersection of bαcl{x} is empty set x∈Z Hence
(Z, Τ) is W-bαR0 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) }
mi-gcl(Z, Τ1) are Ф, Z, {δ₂}, {Ω₃}, {ζ₁, Ω₃}
b-mi-gcl(Z, Τ1) are Ф,Z
αcl sets are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃}
dαcl sets are Ф, Z, {ζ₁, Ω₃}
b-mi-gcl{ζ₁} is Z
b-mi-gcl{δ₂} is Z
b-mi-gcl{Ω₃} is Z
The intersection of b-mi-gcl{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) }.
mi-gcl(Z, Τ1) are Ф, Z, {δ₂}, {Ω₃}, {ζ₁, Ω₃}
d-mi-gcl(Z, Τ1) are Ф,Z, {ζ₁, Ω₃]
αci sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃}
bαcl sets are Ф, Z
d-mi-gcl{ζ₁} is {ζ₁, Ω₃}
d-mi-gcl{δ₂} is Z
d-mi-gcl{Ω₃} is {ζ₁, Ω₃}
The intersection of d-migcl{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 W-d- mi-gcl(Z, Τ1)
– R0 space is a W-iαR0 space but not converse.
THEOREM 3.35: In a TOS (Z, Ƭ, ≤), every W- mi-gcl(Z, Τ1) –
R0 space is a W-βR0 space but not converse.
mi-gcl(Z,
Proof: Suppose (Z, Ƭ ) be a W-dΤ1) – R0 space.
Then the intersection of d- mi-gcl{x} is empty
xϵZ by fact, every d- mi-gcl(Z, Τ1)is a i-micl(Z, T1) and
then every i-micl(Z, T1) is
iαcl set Then iαcl{x} contained in d-mi-gcl{x} x∈Z.
That implies the intersection of iαcl{x} contained in d- migcl{x}. But the intersection of d-mi-gcl{x}
empty set x∈Z.
we get the intersection of iαcl{x} is empty set x∈Z Hence
(Z, Τ) is W-iαR0 space.
Proof: Suppose (Z, Ƭ ) be a W- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of mi-gcl{x} is empty set
xϵZ by fact, every mi-gcl(Z, Τ1)is a micl(Z, T1) and then
every micl(Z, T1) is
βcl set Then βcl{x} contained in mi-gcl{x} x∈Z.
That implies the intersection of βcl{x} contained in migcl{x}. But the intersection of mi-gcl{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) }.
mi-gcl(Z, Τ1) are Ф, Z
d-mi-gci(Z, Τ1) are Ф,Z
αcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}
iαcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}.
d-mi-gcl{ζ₁} is Z
d-mi-gcl{δ₂} is Z
d-mi-gcl{Ω₃} is Z
The intersection of d-mi-gcl{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} }.
mi-gcl(Z, Τ1) are Ф, Z
βcl sets are Ф, Z, {ζ₁}, {δ₂}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃}
mi-gcl{ζ₁} is Z
mi-gcl {δ₂} is Z
mi-gcl {Ω₃} is Z
The intersection of mi-gcl{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 W-i– R0 space is a W-iβR0 space but not converse.
mi-gcl(Z,
THEOREM 3.41: In a TOS (Z, Ƭ, ≤), every W-b- mi-gcl(Z, Τ1)
– R0 space is a W-bβR0 space but not converse.
Proof. Suppose (Z, Ƭ ) be a W-b- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of b- mi-gcl{x} is empty
xϵZ by fact, every b- mi-gcl(Z, Τ1)is a b-micl(Z, T1) and
then every b-micl(Z, T1) is
bβcl set Then bβcl{x} contained in b-mi-gcl{x} x∈Z.
That implies the intersection of bβcl{x} contained in b- migcl{x}. But the intersection of b-mi-gcl{x} empty set x∈Z.
we get the intersection of bβcl{x} is empty set x∈Z Hence
(Z, Τ) is W-bβR0 space.
Τ1)
Proof: Suppose (Z, Ƭ ) be a W-i- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of i- mi-gcl{x} is empty
xϵZ by fact, every i- mi-gcl(Z, Τ1)is a i-micl(Z, T1) and
then every i-micl(Z, T1) is
iβcl set Then iβcl{x} contained in i-mi-gcl{x} x∈Z.
That implies the intersection of iβcl{x} contained in i- migcl{x}. But the intersection of i-mi-gcl{x} empty set x∈Z.
we get the intersection of iβcl{x} is empty set x∈Z Hence
(Z, Τ) is W-iβR0 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) } }.
mi-gcl(Z, Τ1) are Ф, Z, {δ₂}, {Ω₃}, {ζ₁, Ω₃}
b-mi-gcl(Z, Τ1) are Ф, Z, {δ₂}
βclsets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃}
bβcl sets are Ф, Z, {δ₂}, {ζ₁, Ω₃}.
b-mi-gcl{ζ₁} is Z
b-mi-gcl{δ₂} = {δ₂}
b-mi-gcl{Ω₃} = Z
The intersection of b-mi-gcl{x} is {δ₂} x∈Z.
bβcl{ζ₁} is {ζ₁, Ω₃}
bβcl{δ₂} is {δ₂}
bαcl{Ω₃} is {ζ₁, Ω₃}
The intersection of b-βcl{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) }.
mi-gcl(Z, Τ1) a re Ф, Z
i-mi-clg(Z, Τ1) are Ф, Z
βcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}.
Iβcl sets are Ф, Z, {δ₂}.
i-mi-gcl{ζ₁} is Z
i-mi-gcl{δ₂} is Z
i-mi-gcl{Ω₃} is Z
The intersection of i-mi-gcl{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 W-i- mi-gcl(Z, Τ1)
– R0 space is a W-bβR0 space but not converse.
THEOREM 3.39: In a TOS (Z, Ƭ, ≤), every W-d- mi-gcl(Z, Τ1)
– R0 space is a W-dβR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a W-i- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of i- mi-gcl{x} is empty
xϵZ by fact, every i- mi-gcl(Z, Τ1)is a b-micl(Z, T1) and
then every b-micl(Z, T1) is
bβcl set Then bβcl{x} contained in i-mi-gcl{x} x∈Z.
That implies the intersection of bβcl{x} contained in i- migcl{x}. But the intersection of i-mi-gcl{x} empty set x∈Z.
we get the intersection of bβcl{x} is empty set x∈Z Hence
(Z, Τ) is W-bβR0 space.
Proof: Suppose (Z, Ƭ ) be a W-d- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of d- mi-gcl{x} is empty
xϵZ by fact, every d- mi-gcl(Z, Τ1)is a d-micl(Z, T1) and
then every d-micl(Z, T1) is
dβcl set Then dβcl{x} contained in d-mi-gcl{x} x∈Z.
That implies the intersection of dβcl{x} contained in d- migcl{x}. But the intersection of d-mi-gcl{x} empty set x∈Z.
we get the intersection of dβcl{x} is empty set x∈Z Hence
(Z, Τ) is W-dβR0 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) }
mi-gcl(Z, Τ1) are Ф, Z, {ζ₁}, {δ₂, Ω₃}
i-mi-gcl(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
i-mi-gcl{ζ₁} is Z
i-mi-gcl{δ₂} is {δ₂, Ω₃}
i-mi-gcl{Ω₃} is {δ₂, Ω₃}
The intersection of i-mi-gcl{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) }.
mi-gcl(Z, Τ1) are Ф, Z
d-mi-gcl(Z, Τ1) are Ф, Z
βcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}.
dβcl sets are Ф, Z, {Ω₃}.
d-mi-gcl{ζ₁} is Z
d-mi-gcl{δ₂} is Z
d-mi-gcl{Ω₃} is Z
The intersection of d-mi-gcl{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 W-i- mi-gcl(Z, Τ1)
– R0 space is a W-dβR0 space but not converse.
THEOREM 3.49: In a TOS (Z, Ƭ, ≤), every W-b- mi-gcl(Z, Τ1)
– R0 space is a W-dβR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a W-i- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of i- mi-gcl{x} is empty
xϵZ by fact, every i- mi-gcl(Z, Τ1)is a d-micl(Z, T1) and
then every d-micl(Z, T1) is
dβcl set Then dβcl{x} contained in i-mi-gcl{x} x∈Z.
That implies the intersection of dβcl{x} contained in i- migcl{x}. But the intersection of i-mi-gcl{x} empty set x∈Z.
we get the intersection of dβcl{x} is empty set x∈Z Hence
(Z, Τ) is W-dβR0 space.
Proof: Suppose (Z, Ƭ ) be a W-b- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of b- mi-gcl{x} is empty
xϵZ by fact, every b- mi-gcl(Z, Τ1)is a d-micl(Z, T1) and
then every d-micl(Z, T1) is
dβcl set Then dβcl{x} contained in b-mi-gcl{x} x∈Z.
That implies the intersection of dβcl{x} contained in b- migcl{x}. But the intersection of b-mi-gcl{x}
empty set x∈Z.
we get the intersection of dβcl{x} is empty set x∈Z Hence
(Z, Τ) is W-dβR0 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)}
mi-gcl(Z, Τ1) are Ф, Z, {δ₂}, {Ω₃}
i-mi-gcl(Z, Τ1) are Ф, Z, {Ω₃}
βcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}
dβcl sets are Ф, Z
i-mi-gcl{ζ₁} is Z
i-mi-gcl{δ₂} is Z
i-mi-gcl{Ω₃} is {Ω₃}
The intersection of i-mi-gcl{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) }.
mi-gcl(Z, Τ1) are Ф, Z
b-mi-gcl(Z, Τ1) are Ф,Z
βcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃} }
dβcl sets are Ф, Z, {ζ₁, Ω₃}
b-mi-gcl{ζ₁} is Z
b-mi-gcl{δ₂} is Z
b-mi-gcl{Ω₃} is Z
The intersection of b-mi-gcl{x} is Z x∈Z.
dβcl{ζ₁} is Z
dβcl{δ₂} is {δ₂}
dβcl{Ω₃} are {δ₂, Ω₃}
The intersection of d-βcl{x} is {δ₂} not equal to Z x∈Z.
THEOREM 3.47: In a TOS (Z, Ƭ, ≤), every W-b- mi-gcl(Z, Τ1)
– R0 space is a W-iβR0 space but not converse.
THEOREM 3.51: In a TOS (Z, Ƭ, ≤), every W-d- mi-gcl(Z, Τ1)
– R0 space is a W-iβR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a W-b- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of b- mi-gcl{x} is empty
xϵZ by fact, every b- mi-gcl(Z, Τ1)is a i-micl(Z, T1) and
then every i-micl(Z, T1) is
iβcl set Then iβcl{x} contained in b-mi-gcl{x} x∈Z.
That implies the intersection of iβcl{x} contained in b- migcl{x}. But the intersection of b-mi-gcl{x}
empty set x∈Z.
Proof: Suppose (Z, Ƭ ) be a W-d- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of d- mi-gcl{x} is empty
xϵZ by fact, every d- mi-gcl(Z, Τ1)is a i-micl(Z, T1) and
then every i-micl(Z, T1) is
iβcl set Then iαcl{x} contained in d-mi-gcl{x} x∈Z.
That implies the intersection of iβcl{x} contained in d- migcl{x}. But the intersection of d-mi-gcl{x}
empty set x∈Z.
we get the intersection of iβcl{x} is empty set x∈Z Hence
(Z, Τ) is W-iβR0 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) }.
mi-gcl(Z, Τ1) are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃}
b-mi-gl(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}
b-mi-gcl{ζ₁} is Z
b-mi-gcl{δ₂} is Z
b-mi-gcl{Ω₃} is Z
The intersection of b-mi-gcl{Z} is Z x∈Z.
iβcl{ζ₁} is {ζ₁, Ω₃}
iβcl{δ₂} is Z
iβcl{Ω₃} is {Ω₃}
The intersection of i-βcl{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) } }.
mi-gcl(Z, Τ1) are Ф, Z
d-mi-gcl(Z, Τ1) are Ф,Z
βclsets are Ф, Z, {ζ₁}, {δ₂}, {ζ₁, δ₂}
iβcl sets are Ф, Z, {ζ₁}, {ζ₁, δ₂}.
d-mi-gcl{ζ₁} is Z
d-mi-gcl{δ₂} is Z
d-mi-gcl{Ω₃} is Z
The intersection of d-mi-gcl{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 W-d- mi-gcl(Z, Τ1)
– R0 space is a W-bβR0 space but not converse.
THEOREM 3.57: In a TOS (Z, Ƭ, ≤), every W-i- mi-gcl(Z, Τ1)
– R0 space is a W-iΨR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a W-d- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of d- mi-gcl{x} is empty
xϵZ by fact, every d- mi-gcl(Z, Τ1)is a b-micl(Z, T1) and
then every b-micl(Z, T1) is
bβcl set Then bβcl{x} contained in d-mi-gcl{x} x∈Z.
That implies the intersection of bβcl{x} contained in d- migcl{x}. But the intersection of d-mi-gcl{x}
empty set x∈Z.
we get the intersection of bβcl{x} is empty set x∈Z Hence
(Z, Τ) is W-bβR0 space.
Proof: Suppose (Z, Ƭ ) be a W-i- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of i- mi-gcl{x} is empty
xϵZ by fact, every i- mi-gcl(Z, Τ1)is a i-micl(Z, T1) and
then every i-micl(Z, T1) is
iΨcl set Then iΨcl{x} contained in i-mi-gcl{x} x∈Z.
That implies the intersection of iΨcl{x} contained in i- migcl{x}. But the intersection of i-mi-gcl{x} empty set x∈Z.
we get the intersection of iΨcl{x} is empty set x∈Z Hence
(Z, Τ) is W-iΨR0 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) }.
mi-gcl(Z, Τ1) are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃}
d-migcl(Z, Τ1) are Ф,Z, {ζ₁}
βcl sets are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃}
bβcl sets are Ф, Z
d-mi-gcl{ζ₁} is {ζ₁}
d-mi-gcl{δ₂} is Z
d-mi-gcl{Ω₃} is Z
The intersection of d-mi-gcl{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) } }.
mi-gcl(Z, Τ1) are Ф, Z
i-mi-gcl(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}
i-mi-gcl{ζ₁} is Z
i-mi-gcl{δ₂} is Z
i-mi-gcl{Ω₃} is Z
The intersection of i-mi-gcl{x} is Z x∈Z.
iΨcl{ζ₁} is {ζ₁}
iΨcl{δ₂} is {ζ₁, δ₂}
iΨcl{Ω₃} is {Ω₃}
The intersection of i-Ψcl{x} is empty but not equal to Z
x∈Z.
THEOREM 3.55: In a TOS (Z, Ƭ, ≤), every W- mi-gcl(Z, Τ1)
– R0 space is a W-ΨR0 space but not converse.
THEOREM 3.59: In a TOS (Z, Ƭ, ≤), every W-d- mi-gcl(Z, Τ1)
– R0 space is a W-dΨR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a W- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of mi-gcl{x} is empty set
xϵZ by fact, every mi-gcl(Z, Τ1)is a micl(Z, T1) and then
every micl(Z, T1) is
Ψcl set Then βcl{x} contained in mi-gcl{x} x∈Z.
That implies the intersection of Ψcl{x} contained in migcl{x}. But the intersection of mi-gcl{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 W-d- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of d- mi-gcl{x} is empty
xϵZ by fact, every d- mi-gcl(Z, Τ1)is a d-micl(Z, T1) and
then every d-micl(Z, T1) is
dΨcl set Then dΨcl{x} contained in d-mi-gcl{x} x∈Z.
That implies the intersection of dΨcl{x} contained in d- migcl{x}. But the intersection of d-mi-gcl{x} empty set x∈Z.
we get the intersection of dΨcl{x} is empty set x∈Z
Hence (Z, Τ) is W-dΨR0 space.
EXAMPLE 3.56: Let Z={ ζ₁, δ₂, Ω₃ } and T= { Ф, Z, {δ₂ },
{Ω₃], {δ₂, Ω₃} }
mi-gcl(Z, Τ1) are Ф, Z
Ψcl sets are Ф, Z, {ζ₁}, {δ₂}, {Ω₃}, {ζ₁, δ₂} {ζ₁, Ω₃}
mi-gcl{ζ₁} is Z
mi-gcl{δ₂} is Z
mi-gcl{Ω₃} is Z
The intersection of mi-gcl{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} }.
mi-gcl(Z, Τ1) are Ф, Z
d-mi-gcl(Z, Τ1) are Ф, Z
Ψcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}.
dΨcl sets are Ф, Z, {δ₂}.
d-mi-gcl{ζ₁} is Z
d-mi-gcl{δ₂} is Z
d-mi-gcl{Ω₃} is Z
The intersection of d-mi-gcl{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 W-b- mi-gcl(Z, Τ1)
– R0 space is a W-bΨR0 space but not converse.
THEOREM 3.65: In a TOS (Z, Ƭ, ≤), every W-i- mi-gcl(Z, Τ1)
– R0 space is a W-dΨR0 space but not converse.
Proof. Suppose (Z, Ƭ ) be a W-i- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of i- mi-gcl{x} is empty
xϵZ by fact, every i- mi-gcl(Z, Τ1)is a d-micl(Z, T1) and
then every d-micl(Z, T1) is
dΨcl set Then dΨcl{x} contained in i-mi-gcl{x} x∈Z.
That implies the intersection of dΨcl{x} contained in i- migcl{x}. But the intersection of i-mi-gcl{x} empty set x∈Z.
we get the intersection of dΨcl{x} is empty set x∈Z
Hence (Z, Τ) is W-dΨR0 space.
Proof: Suppose (Z, Ƭ ) be a W-b- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of b- mi-gcl{x} is empty
xϵZ by fact, every b- mi-gcl(Z, Τ1)is a b-micl(Z, T1) and
then every b-micl(Z, T1) is
bΨcl set Then bΨcl{x} contained in b-mi-gcl{x} x∈Z.
That implies the intersection of bΨcl{x} contained in b- migcl{x}. But the intersection of b-mi-gcl{x} empty set x∈Z.
we get the intersection of bΨcl{x} is empty set x∈Z
Hence (Z, Τ) is W-bΨR0 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) } }.
mi-gcl(Z, Τ1) are Ф, Z
b-mi-gcl(Z, Τ1) are Ф, Z
Ψcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}
bΨcl sets are Ф, Z, {δ₂}.
b-mi-gcl{ζ₁} is Z
b-mi-gcl{δ₂} is Z
b-mi-gcl{Ω₃} is Z
The intersection of b-mi-gcl{x} is Z x∈Z.
bΨcl{ζ₁} is Z
bΨcl{δ₂} is {δ₂}
bΨcl{Ω₃} is Z
The intersection of b-Ψcl{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) }
mi-gcl(Z, Τ1) are Ф, Z
i-mi-gcl(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}
i-mi-gcl{ζ₁} is Z
i-mi-gcl{δ₂} is Z
i-mi-gcl{Ω₃} is Z
The intersection of i-mi-gcl{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 W-i- mi-gcl(Z, Τ1)
– R0 space is a W-bΨR0 space but not converse.
THEOREM 3.67: In a TOS (Z, Ƭ, ≤), every W-b- mi-gcl(Z, Τ1)
– R0 space is a W-iΨR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a W-i- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of i- mi-gcl{x} is empty
xϵZ by fact, every i- mi-gcl(Z, Τ1)is a b-micl(Z, T1) and
then every b-micl(Z, T1) is
bΨcl set Then bΨcl{x} contained in i-mi-gcl{x} x∈Z.
That implies the intersection of bΨcl{x} contained in i- migcl{x}. But the intersection of i-mi-gcl{x} empty set x∈Z.
we get the intersection of bΨcl{x} is empty set x∈Z
Hence (Z, Τ) is W-bΨR0 space.
Proof: Suppose (Z, Ƭ ) be a W-b- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of b- mi-gcl{x} is empty
xϵZ by fact, every b- mi-gcl(Z, Τ1)is a i-micl(Z, T1) and
then every i-micl(Z, T1) is
iΨcl set Then iΨcl{x} contained in b-mi-gcl{x} x∈Z.
That implies the intersection of iΨcl{x} contained in b- migcl{x}. But the intersection of b-mi-gcl{x}
empty set x∈Z.
we get the intersection of iΨcl{x} is empty set x∈Z Hence
(Z, Τ) is W-iΨR0 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) }.
mi-gcl(Z, Τ1) are Ф, Z, {δ₂], {Ω₃], {ζ₁, Ω₃}
i-mi-gcl(Z, Τ1) are Ф, Z, {Ω₃}
Ψcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃}
bΨcl sets are Ф, Z
i-mi-gcl{ζ₁} is Z
i-mi-gcl{δ₂} is Z
i-mi-gcl{Ω₃} is {Ω₃}
The intersection of i-mi-gcl{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) }.
mi-gcl(Z, Τ1) are Ф, Z, {ζ₁}, {δ₂, Ω₃}
b-mi-gcl(Z, Τ1) are Ф, Z
Ψcl sets are Ф, Z, {ζ₁}, {δ₂, Ω₃}
iΨcl sets are Ф, Z, {δ₂, Ω₃}
b-mig-gcl{ζ₁} is Z
b-mi-gcl{δ₂} is Z
b-mi-gcl{Ω₃} is Z
The intersection of b-mi-gcl{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 W-b- mi-gcl(Z, Τ1)
– R0 space is a W-dΨR0 space but not converse.
THEOREM 3.73: In a TOS (Z, Ƭ, ≤), every W-d- mi-gcl(Z, Τ1)
– R0 space is a W-bβR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a W-b- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of b- mi-gcl{x} is empty
xϵZ by fact, every b- mi-gcl(Z, Τ1)is a d-micl(Z, T1) and
then every d-micl(Z, T1) is
dΨcl set Then dΨcl{x} contained in b-mi-gcl{x} x∈Z.
That implies the intersection of dΨcl{x} contained in b- migcl{x}. But the intersection of b-mi-gcl{x}
empty set x∈Z.
we get the intersection of dΨcl{x} is empty set x∈Z
Hence (Z, Τ) is W-dΨR0 space.
Proof: Suppose (Z, Ƭ ) be a W-d- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of d- mi-gcl{x} is empty
xϵZ by fact, every d- mi-gcl(Z, Τ1)is a b-micl(Z, T1) and
then every b-micl(Z, T1) is
bΨcl set Then bΨcl{x} contained in d-mi-gcl{x} x∈Z.
That implies the intersection of bΨcl{x} contained in d- migcl{x}. But the intersection of d-mi-gcl{x}
empty set x∈Z.
we get the intersection of bΨcl{x} is empty set x∈Z
Hence (Z, Τ) is W-bΨR0 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) }
mi-gcl(Z, Τ1) are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃}
b-mi-gcl(Z, Τ1) are Ф,Z
Ψcl sets are Ф, Z, {ζ₁}, {Ω₃}, {δ₂, Ω₃}, {ζ₁, Ω₃}
dΨcl sets are Ф, Z, {ζ₁}
b-mi-gcl{ζ₁} is Z
b-mi-gcl{δ₂} is Z
b-mi-gcl{Ω₃} is Z
The intersection of b-mi-gcl{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) }.
mi-gcl(Z, Τ1) are Ф, Z, {ζ₁}, {δ₂, Ω₃}
d-mi-gcl(Z, Τ1) are Ф,Z, {δ₂, Ω₃]
Ψcl sets are Ф, Z, {δ₂}, {Ω₃}, {δ₂, Ω₃}
bΨcl sets are Ф, Z, {Ω₃}
d-mi-gcl{ζ₁} is Z
bΨcl{ζ₁} is Z
cl
d-mi-g {δ₂} is {δ₂, Ω₃}
bΨcl{δ₂} is Z
cl
d-mi-g {Ω₃} is {δ₂, Ω₃}
bΨcl{Ω₃} is
{Ω₃}
The intersection of d-mi-gcl{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 W-d- mi-gcl(Z, Τ1)
– R0 space is a W-iΨR0 space but not converse.
Proof: Suppose (Z, Ƭ ) be a W-d- mi-gcl(Z, Τ1) – R0 space.
Then the intersection of d- mi-gcl{x} is empty
xϵZ by fact, every d- mi-gcl(Z, Τ1)is a i-micl(Z, T1) and
then every i-micl(Z, T1) is
iΨcl set Then iΨcl{x} contained in d-mi-gcl{x} x∈Z.
That implies the intersection of iΨcl{x} contained in d- migcl{x}. But the intersection of d-mi-gcl{x}
empty set x∈Z.
we get the intersection of iΨcl{x} is empty set x∈Z Hence
(Z, Τ) is W-iΨR0 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) }.
mi-gcl(Z, Τ1) are Ф, Z
d-mi-gcl(Z, Τ1) are Ф, Z
Ψcl are Ф, Z, {ζ₁}, {δ₂}, {Ω₃}, {ζ₁, δ₂}, {ζ₁, Ω₃}
iΨcl sets are Ф, Z, {δ₂}, {Ω₃}
d-mi-gcl{ζ₁} is Z
d-mi-gcl{δ₂} is Z
d-mi-gcl{Ω₃} is Z
The intersections of d-mi-gcl{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), 961-970.
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