International Journal of Engineering Trends and Technology (IJETT) – Volume 12 Number 7 - Jun 2014
Soft Pre Generalized - Closed Sets in a Soft
Topological Space
J.Subhashinin1, Dr.C.Sekar 2
1
2
Department of Mathematics, VV College of Engineering, Tisayanvilai- INDIA.
Department of Mathematics, Aditanar College of Arts and Science, Tirchendur- INDIA.
Abstract
This paper introduces soft pre-generalized closed set in soft topological spaces. The notations of soft pre interior and soft pre
closure are generalized using these sets. In a soft topological space, a soft set F
p(F ) ⊆ F whenever F ⊆ F
is said to soft pre generalised –closed if
where F is a soft P-open set . A detail study is carried out on properties of soft Pre generalized
closed sets and soft PT1/2 Space.
AMS subject classification (2010): 54A10, 54A05 and 06D72.
Keywords: Soft pre generalized-closed set, Soft pre generalized –open sets, Soft pre generalized interior, soft pre-
generalized closure, soft PT1/2-Space .
Introduction. Soft set theory was first introduced by Molodtsov [5] in 1999 as a general mathematical tool for
dealing uncertain fuzzy, not clearly defined objects. He has shown several applications of this theory in solving many
practical problems in economics, engineering, social science, medical science, and so on. Modern topology depends
strongly on the ideas of set theory. In 2010 Muhammad shabir, Munazza Naz [6] used soft sets to define a topology
namely Soft topology. J.Subhashini and C.Sekar defined soft pre-open sets [11] in a soft topological space. In
General topology the concept of generalized closed set was introduced by Levine [3] plays a significant. This
notation has been studied extensively in recent years by many topologies. The investigation of generalized closed sets
has led to several new and interesting concepts, e.g. new covering properties and new separation axioms. Some of
these separation axioms have been found to be useful in computer science and digital topology. Soft generalised
closed set was introduced by K.Kannan[2] in 2012. In this paper we introduce soft pre-generalized closed set and soft
PT1/2 – Space in a soft topological space and a detailed study of some of its properties.
2. Preliminaries
For basic notations and definitions not given here, the reader can refer [1-12].
2.1. Definition. [11]
,
( ) ∈ ( ) , where E is a set of parameters, A ⊆ E , P (U) is the power set of U, and
( ) = ∅ if
=
A soft set FA on the universe U is defined by the set of ordered pairs
∉ . Here,
:
,
( ) : ∈
→ ( ) such that
is called an approximate function of the soft set FA. The value of
( ) may be
arbitrary, some of them may be empty, some may have non-empty intersection. Note that the set of all soft set over U
is denoted by S (U).
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International Journal of Engineering Trends and Technology (IJETT) – Volume 12 Number 7 - Jun 2014
2.2. Example. Suppose that there are five cars in the universe. Let U={c1, c2, c3, c4, c5} under consideration,and that
=
,
,
,
,
,
,
,
is a set of decision parameters. The
( = 1,2,3,4,5,6,7,8) stand for the parameters
“expensive”, “beautiful”, “manual gear”, “cheap”, “automatic gear”, “in good repair”, “in bad repair” and "costly”
respectively. In this case, to define a soft set means to point out expensive cars, beautiful cars and so on. It means
that, Consider the mapping
given by “cars (.)”, where (.) is to be filled in by one of the parameters
∈ . For
( ) means “car (expensive)”, and its functional value is the set {c∈ : c is an expensive car} and so,
instance,
⊆ , the soft set FA that describes the “attractiveness of the cars” in the opinion of a buyer say Ram,
Let
may be defined like
A={ ,
,
,
,
},
( )= {c2, c3, c5},
( )= {c2, c4 },
( )= {c1},
( )= {U} and
( )= {c3,c5}. We can view this soft set FA as consisting of the following collection of approximations:
=
,
,
,( ,{ ,
,,
}), ( , { }), ( , { }), ( , { ,
∈ ( ) is called a soft point in FE, denoted by (
2.3. Definition. [11] The soft set
( ) ≠ ∅ and
and
(
) ∈
( ) = ∅ for all
∈
if for the element
2.4.Definition. [10] Let ∈
− { }.The soft point (
( ) ⊆
and
∈ |
( )|
=
= FA,
={(e1,{,h 1})},
={(e1,{h1}),(e2,{h1,h2})},
={(e1,{h2}),(e2,{h2})},
⊆
A={e1,e2} and
:
⊆
,
⊆
} and its
( ).
= {( , {ℎ , ℎ }), ( , {ℎ , ℎ })}. Then
={(e1,{h2})},
={(e1,{h2}),(e2,{h1,h2}},
={(e1,{h1}),(e2,{h1})},
={(e2,{h2})},
={(e1,{h1,h2}),(e2,{h2})},
={(e2,{h1,h 2})},
={(e1,{h2,h3}),(e2,{h1})},
={(e1,{h2}),(e2,{h1})},
= {(e1,{h2}), (e2,{h2})} are all soft subset of FA. So
2.6. Definition. [11] Let FE ∈ ( ). If
set) denoted by
, denoted by
∅,
={(e1,{h1,h2})},
={(e2,{h1})},
∈
( ).
, where | ( )| is the cardinality of
2.5. Example. [10] Let U={u1,u2}, E={e1,e2,e3},
), if for the element
) is said to be in the soft set
∈ (U). The soft power set of FA is defined by ( ) = {
( ) = 2∑
cardinality is defined by
}) .
( )=
for all
( ) = 24 = 16.
∈ , then FE is called a soft universal set (soft absolute
or .
2.7. Definition. [6] Let
∈ ( ). A soft topology on
denoted by ̃ is a collection of soft subsets of
having
the following properties:
(i).
∅ ,
∈ ̃
(ii).
⊆
: ∈ ⊆
(iii).
⊆
:1 ≤ ≤ ,
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⊆ ⟹⋃
∈
⊆ ⟹⋂
̃
̃ .
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International Journal of Engineering Trends and Technology (IJETT) – Volume 12 Number 7 - Jun 2014
The pair (
, ̃)is called a soft topological space.
2.8. Example. Let us consider the soft subsets of FA that are given in Example 2.5. Then
̃ =
∅,
,
,
= { ( )}
2.9. Definition. [5] Let (
Clearly
∅
and ̃
be a soft set over U and
( ) for all ∈ . Not that for any
2.12. Definition. [1] The soft set
) . Let
⊆
,
,
. Then FC
is soft open in FE. The collection of all soft closed set is denoted by
2.10. Definition. A soft topology whose soft open sets are all soft subsets of
by
,
̃ is called a soft open set.
(
are soft open sets. The collection of all soft open set is denoted by
is said to be soft closed if the soft set
whenever ∈
∅,
are soft topologies on FA.
, ̃ ) be a soft topological space .Then, every element of
2.11. Definition. [1] Let
̃ =
∈ , we say that
∈ ,
over U such that
∉
if
∉
( ) = { } for all
(
) .
is called soft discrete topology.
∈
read as belongs to the soft set
( ) for some
∈
∈ .
is called soft singleton and is denoted
.
2.13. Definition. [5] Let ( , ̃ ) be a soft topological space over U,
Then the soft subset of
=
words
∩
over V denoted by
∈ ( ) and Let V be a non null subset of U.
, is defined as follows:
( )=
⊆
( ) , for all ∈ . In other
.
2.14. Definition. [9] Let ( , ̃ ) be a soft topological space over U. A soft set
U if
∩
whenever
⊆
and
is called a soft generalized closed in
is soft open in U.
2.15. Definition. [12] Let ( , ̃ ) be a soft topological space, a soft set FA is said to be soft pre-open set (soft P-open)
if there exists a soft open set
(
or
(
) . Then
̃
(
. The set of all soft P-open set of FE is denoted by
is said to be soft pre-closed. The set of all soft P-closed set of FE is denoted by
(
, ̃)
, ̃ ) or
).
2.16. Remark. [11] A soft set
∅
such that FA ⊆ FO ⊆
and
which is both soft P-open and soft P-closed is known as soft P-clopen set. Clearly
are soft P-clopen sets.
2.17. Proposition. [11]
(i) Every soft open set is a soft pre-open set.
(ii) Every soft closed set is a soft pre-closed set.
2.18. Theorem. [10]
(i) Arbitrary soft union of soft P-open sets is a soft P-open set.
(ii) The soft intersection of any two soft P-open set need not be a soft P-open set.
(iii) Arbitrary soft intersection of soft P-closed sets is soft P-closed set.
(iv) The soft union of any two soft P-closed set need not be a soft P-closed set.
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2.19. Definition. [11] Let ( , ̃ ) be a soft topological space and
⊆
. Then the soft pre- interior (soft P-interior)
( ) is defined as the soft union of all soft P-open subsets of FA. Note that ( ) is the biggest
of FA denoted by
soft P- open set that contained in FA.
2.20. Definition. [11 ] Let (
, ̃ ) be a soft topological space and
⊆
.Then the soft pre closure (soft P-closure)
( ) is defined as the soft intersection of all soft P-closed supersets of FA . Note that,
of FA denoted by
( )
is the smallest soft P- closed set that containing FA.
( ) =
2.21. Proposition. [11] A soft set is soft P-open iff
( )=
A soft set is soft P-closed iff
.
.
2.22. Definition. [3] A subset B of a topological space X is said to be:
⊆
(i) a generalized closed set if
⊆
whenever
(ii) a pre generalized closed set if ( ) ⊆
and U is open in X.
whenever
⊆
and U is pre-open in X.
3. Soft P-generalized closed set
⊆
whenever
( )⊆
∈ ( ) is said to be soft pre generalized closed set (soft Pg-closed set) if
3.1 Definition. Let
( ). The collection of all soft Pg- closed sets is denoted by
∈
and
( ).
3.2. Example. Let as consider the soft subsets in Example 2.5. Let ( , ̃ ) be a soft topological space where
{ℎ , ℎ },
( )=
( )=
={ ,
},
∅,
,
,
,
,
,
∅,
,
⊆
whenever
= {( , {ℎ , ℎ }), ( , {ℎ , ℎ })}
,
,
,
,
,
,
( ). Therefore the soft subsets
∈
,
,
,
. ̃ =
∅,
, . Clearly for the soft sub sets
and
contained in any soft P-open set. Therefore
∅,
,
,
=
,
.
( ) and for the soft subset
∈
and
,
̃=
and
̃
=
,
⊆
=
⊆
⊆
whenever
are soft Pg-closed sets but the soft set
=
,
and
is not
is not a soft Pg- closed set.
3.3. Proposition. Every soft P-closed set is a soft Pg-closed set.
Proof: Let
be a soft P-closed set, then for every soft P-closed set
= (
,
), we have every soft p closed set
is soft Pg-closed set. But the converse is not true in general.
3.4. Remark. A soft Pg-closed set need not be a soft P-closed set. The following example supports our claim.
3.5. Example. Let as consider the soft subsets in Example 2.5. Let ( , ̃ ) be a soft topological space where
{ℎ , ℎ },
∅,
={ ,
,
,
,
},
= {( , {ℎ , ℎ }), ( , {ℎ , ℎ })} and ̃ =
,
,
Example 3.2 soft subset
,
,
.Then
( )=
∅,
∅,
,
,
,
̃
,
,
. ̃ =
,
,
∅,
,
,
,
,
=
( )=
,
,
. Then by
is a soft P-closed set. Then
∩
is a soft pre
is a soft Pg-closed set but it is not a soft P- closed set.
3.6. Proposition.
(i)If
(ii)Let
and
are soft pre generalized closed sets then so is
∩
be a soft pre generalized closed set and suppose that
.
generalized closed set.
3.7. Definition. Let (
, ̃ ) be a soft topological space and
closure) of FA denoted by
⊆
.Then the soft pre generalized closure (soft Pg-
( ) is defined as the soft intersection of all soft pre generalized closed supersets of
FA .
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International Journal of Engineering Trends and Technology (IJETT) – Volume 12 Number 7 - Jun 2014
3.8. Remark. Since the arbitrary soft intersection of soft pre generalized closed sets is a soft pre generalized closed
( ) is soft pre generalized closed set. Note that,
set,
( ) is the smallest soft pre generalized closed set that
containing FA.
3.9. Theorem. Let ( , ̃ ) be a soft topological space and let FA and FB be a soft sets over U. Then
⊆ (a)
(b)
( )
=
FA is soft pre generalized closed iff
(c)
⊆
(d)
( ∅) = (e) (f)
( )⊆
, then
∅
and
∩
⊆ ∪
=
)
( ) =
.
( )∩ ( ) ∪ ( ) =
(g) (
(
(
( )
)
)
( )
Proof: Refer theorem 4.16 [11]
of a soft topological space ( , ̃ ) is soft Pg-closed set iff ( )\
3.10. Theorem. A soft subset
any non-empty soft pre-closed set. (That is ( )\
Proof: Necessity, Suppose that
P-closed, we have its relative complement
⊆
̃
=
∅.
Hence ( )\
Sufficiency, Let
)≠
̃
⊆
. Hence
∅.
⊆
contains only null soft pre-closed set.)
is soft Pg-closed set. Let
̃
. Consequently ( ) ⊆
does not contain
be a soft P-closed and
⊆ ( )\
is soft P-open. Since
̃
. [Since
⊆ ( )\ . Since
⊆ ( ) and
, we have
⊆
is soft Pg-closed set]. Therefore,
is soft
̃
. Hence
contains only null soft P-closed set.
, where
is soft pre open in ( , ̃ ). If ( ) is not contained in
, then ( ) ∩ (
\
Now, since
( ) ∩(
\
and therefore
) ⊆ ( )\
and ( ) ∩ (
\
) is a non-empty soft pre closed set, then we obtain a contradiction
is soft Pg-closed set.
3.11. Theorem (Sufficient condition for soft Pg-closed set is soft P-closed set.)
If
is a soft Pg-closed set of a soft topological space (
(i)
, ̃ ), then the following are equivalent
is soft P-closed set.
(ii)
( )\
is soft P-closed set.
Proof:
Let (
, ̃ ) be a soft topological space. Let
be soft Pg-closed set.
(i)→(ii)
Let
be soft P-closed set also it is a soft Pg-closed set, and then by Theorem 3.10, ( )\
=
∅
which is soft P-
closed set.
(ii)→(i)
Let ( )\
be a soft P-closed set and
soft P-closed subset. Since, ( )\
be soft Pg-closed. Then by Theorem 3.10, ( )\
is soft P-closed and ( )\
3.12. Theorem. For a soft topological space (
(i)Every soft subset of
(ii)
( )=
=
∅.
This shows that
contains only null
is soft P-closed set.
, ̃ ), the following are equivalent
is soft Pg-closed set.
( ).
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International Journal of Engineering Trends and Technology (IJETT) – Volume 12 Number 7 - Jun 2014
Proof: (i)→(ii)
∈
(
Therefore
∈
Let
). Then by hypothesis,
(
\
and then
). Also let
(
∈
(
∈
), thus
(
∈
(
is soft Pg-closed which implies that
). Then
\
)⊂
( ), hence by hypothesis
∈
) according above we have
(
)=
(
. So
\
(
)=
,
is soft Pg-closed
).
(ii)→(i)
If
, ̃) such that
is a soft subset of a soft topological space (
therefore ( ) ⊆
which shows that
3.13. Theorem . If
is soft Pg-closed in (
, ̃ ) and
⊆ ( ), then
⊆
is soft Pg- closed set in ( , ̃ ) and
(
and
⊆
⊆ ( ), we have (
⊆
(
∈
where
),then
( ) and
∈
is soft Pg-closed set.
Proof: Suppose that
, ̃). Since
⊆
)⊆ ( )⊆
⊆ ( ), Let
⊆
. Hence ( ) ⊆
⊆
, we have
is soft Pg- closed set.
. Therefore,
⊆
(since
and
is soft P-open in
is soft Pg-closed set). Since
is soft Pg- closed set.
4. Soft pre generalized open set
4.1. Definition. Let ( , ̃ ) be a soft topological space. A soft set
̃
open set) if the complement
(
is soft Pg-closed set in ( , ̃ ). The collection of all soft Pg-open sets is denoted by
). Equivalently, a soft set
⊆ ( ) whenever
⊆
is called a soft pre generalized open set (soft Pg-
is called a soft Pg-open in a soft topological space (
is soft P-closed set in (
and
, ̃ ) if and only if
, ̃ ).
4.2. Example. Let as consider the soft subsets in Example 2.5. Let ( , ̃ ) be a soft topological space where
{ℎ , ℎ },
̃=
∅,
,
subset
,
. ̃ ̃ =
,
∅
⊆
},
∅,
,
,
whenever
⊆
whenever
={ ,
= {( , {ℎ , ℎ }), ( , {ℎ , ℎ })}
(
,
⊆
)=
and
∅
( ). Therefore
∈
and
∅
=
∅,
,
,
,
,
,
,
,
( ) and for the soft subset
∈
, and
and
. Clearly for the soft
,
⊆
=
are soft Pg-open sets but the soft subset
is not
a soft Pg- open set.
4.3. Proposition. Every soft P-open set is a soft Pg-open set
Proof: Let
soft P-open set, then for every soft P-open set
,
= (
) , we have every soft P-open set is soft
Pg- open set. But the converse is not true in general.
4.4. Remark. A soft Pg-open set need not be soft P-open set. The following example supports our claim.
4.5. Example. Let as consider the soft subsets in Example 2.5. Let ( , ̃ ) be a soft topological space in Example 4.2
= {ℎ , ℎ },
where
. ̃ ̃ =
∅,
,
,
( )=
,
={ ,
( )=
,
∅,
,
,
,
},
∅,
,
,
,
= {( , {ℎ , ℎ }), ( , {ℎ , ℎ })}
,
,
,
,
,
,
,
,
and
̃=
∅,
.
. The soft set
,
,
Then
is soft Pg-open set but it is not a soft P-
open set.
4.6. Proposition
(i)If
and
∪
are soft Pg-open sets then so is
.
(ii)The soft union of two soft Pg-open set is generally not a soft Pg- open set.
4.7. Proposition
( , ̃) ⊆
Proof: Let
( , ̃ ).
be any soft P-open set. Then,
Pg-open. Hence
(
, ̃) ⊆
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(
̃
is soft P-closed and hence soft Pg-closed. This implies that
is soft
, ̃ ).
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4.8. Definition. Let (
, ̃ ) be a soft topological space and
⊆
. Then the soft pre generalized interior (soft Pg-
( ) is defined as the soft union of all soft pre generalized open subsets of FA .
interior) of FA denoted by
( ) is
4.9. Remark. Since arbitrary soft union of soft pre generalized-open sets is soft pre generalized open set,
( ) is the biggest soft pre generalized open set that contained in FA.
a soft pre generalized open set. Note that
4.10. Theorem. Let ( , ̃ ) be a soft topological space and FA, FB be a soft sets over U. Then
(a)
( ) ⊆
.
=
(b) FA is soft pre generalized open iff
(c)
⊆
( ) ⊆
(d)
( ∅ ) =
(e)
(
∩
) =
( ) ∩ (f) (
∪
) ⊇ ( ) ∪ , then
(g) (
(h)
(i) (
( )
∅
=
.
(
) .
(
) .
( ) .
( )) ̃ =
(ii) (
) .
( ) =
and
)
(
( ) .
̃
(
( ) ) ̃ =
)
̃
(
)
Proof: Refer Theorem 3.23 [11].
4.11. Theorem .A soft set
Proof: Suppose that
is soft pre generalized open in (
⊆
closed set. Since
, ̃ ) and ( ) ⊆
is soft Pg-open in (
⊆
and
open). Since ( ) ⊆
⊆
, we have
⊆ ( ) ⊆ (
, we have
⊆
, then
, ̃) and ( ) ⊆
is soft Pg- open set.
⊆
. Let
⊆ ( ) . (Since
. Hence
) . Therefore,
⊆
and
is soft p-
is soft pre generalized
is soft pre generalized open.
4.12. Note. Soft Pg-open set implies soft P-open set if it is a soft PT1\2 space.
5. Soft PT1/2 – Spaces
In this section, we introduce the new soft lower separation axioms, namely soft pre T1/2-space in soft topological
space with the help of soft Pg- generalized closed sets.
, ̃) is said to be a soft pre T1/2 Space (PT1/2 –space) if every soft Pg-
5.1. Definition. A soft topological space (
closed set is soft P-closed set. The spaces where the class of soft P-closed sets and the soft Pg-closed sets coincide.
5.2. Example. Any soft discrete topological space( , ̃ ) is soft PT1/2 -space. Since every soft subset of
closed sets in a soft discrete topology. Hence every soft subset of
is a soft P-
is a soft Pg-closed set in a soft discrete
topological space.
= {ℎ , ℎ },
5.3. Example. Consider a soft topological space ( , ̃ ) in Example 2.5. Where
{( , {ℎ , ℎ }), ( , {ℎ , ℎ })} and ̃ =
{( , { })}.
∅,
,
,
.
=
,
∅,
,
,
,
,
,
∅,
,
,
,
,
,
,
,
,
={ ,
,
,
,
,
,
,
,
,
,
,
,
,
=
= {( , { })},
. The soft sets are defined as follows
,
},
,
.
=
=
,
Therefore ( , ̃ ) is a soft PT1/2 -space over U.
5.4. Remark
Every soft PT1/2 -space is a soft topological space.
5.5. Remark.
A soft topological space need not be a soft PT1/2 -space.
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5.6. Example
= {ℎ , ℎ },
Let as consider the soft subsets in Example 2.5. Let ( , ̃ ) be a soft topological space where
{ ,
},
= {( , {ℎ , ℎ }), ( , {ℎ , ℎ })} and
̃=
∅,
,
∅,
,
,
(
. ̃ ̃ =
,
,
)=
,
,
∅,
,
∅,
,
,
,
,
,
,
,
because the soft Pg- closed
(
,
)
. Then
,
,
,
∅,
,
,
( )=
∅,
,
,
,
,
,
,
,
,
,
,
(
.
,
,
,
=
)=
,
.
. The soft topological space ( , ̃) is not a soft PT1/2 space,
is not a soft P-closed set.
5.7. Theorem. A soft topological space (
, ̃ ) is a soft PT1/2-space if and only if
(
)=
( ) holds.
Proof: Necessity, Since the soft P-closed sets and the soft Pg-closed sets coincide by the assumption, ( ) =
(
) holds for every soft subsets
( )=
Sufficiency, we have that
̃
hence
( ). Thus
∈
of (
(
) . Let
(
)=
( ).
(
=
be a soft Pg closed set. Then, we have
is soft P-closed set. Therefore (
5.8. Theorem. A soft topological space (
singleton
, ̃). Therefore, we have that
) and
, ̃ ) is soft PT1/2 –space.
, ̃ ) over U is a soft PT1/2 -space if and only if, for each
∈ , the soft
is either soft P-open or soft P-closed.
∈ ,
Poof: Necessity, Suppose that for some
containing
̃
̃
, the soft set
is not soft P-closed. Since
is the only soft P-open set
is soft Pg-closed and so it is soft P-closed in the soft PT1/2 –space (
, ̃ ). Therefore
is soft P-open set.
( )⊆
Sufficiency, Since
( ). Suppose that
∈
̃
∈
̃
⊃
and
∉
̃
(
(
∉
̃
=
) holds, by Theorem 5.7, it is enough to prove that
̃
). Then,
. Since
̃
∉
. By the hypothesis, the soft singleton
=
̃
̃
and
̃
≠
( )⊆
(
hold. There exists
there exist a soft Pg-closed set
such that
). Let
∈
such that
∉
and
⊂
̃
is soft P-open or soft P-closed set.
Case (i)
The soft singleton
̃
∉
̃
is soft P-closed set: Since
̃
. This contradicts the fact that ∈
̃
is a soft P-closed set with
. Therefore
∈
(
⊂
̃
,
̃
we have
. That is
).
Case (ii)
If the soft singleton
have
̃
∈
5.9. Theorem. If (
(i)(
̃
⊃ ( )⊃
cases, we have
̃
is soft P-closed: Since
(
. Therefore ∉
), that is
̃
( )⊆
is a soft P-open set containing the soft Pg-closed set
. This is a contradiction. Therefore
(
∈
(
⊃
̃
, we
) . Hence in both
).
, ̃ )is a soft topological space, then the following statements are equivalent
, ̃ ) is a soft PT1/2 –space.
(ii)Every soft subset of S (U) is the soft intersection of soft P- open and soft P-closed set containing it.
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Page 363
International Journal of Engineering Trends and Technology (IJETT) – Volume 12 Number 7 - Jun 2014
Proof: (i)→(ii)
If
is a soft PT1/2 -space with
⊂
, then
=∩ { \
:
∉
} is the intersection of soft P-open and soft P-
closed set containing it.
(ii)→(i)
For each
∈ ,
\
is the soft intersection of all soft P-open and soft P-closed sets containing it. Hence
either soft P-open or soft P-closed. Therefore by Theorem 5.8, (
\
is
, ̃) is a soft PT1/2 -space.
5.10. Proposition. The soft subspace of a soft PT1/2 –space is soft PT1/2-space.
Proof: Let V is a soft subspace of a soft pre-T1/2 space over U and
closed in(
, ̃ ). (by Theorem 5.8) Therefore
∈
⊆ . Then
is soft p-open or soft pre
is either soft P-open or soft P-closed set in V. Hence V is a soft PT1/2
-space.
Conclusion: The initiation of notation of soft topological space was introduced by D.Molodtsov [5] in 1999. Many
Mathematicians turned their attention to the various concepts of soft topological space. By this way, [6] Muhammad
Shabir and Munazza Naz introduced the concept soft topological spaces. In 2013 J.Subhashini and C.Sekar [11]
introduced the concept of soft pre-open sets. In this paper, we continue this work and introduce soft pre generalised closed sets taking help of the soft pre open sets. Also we introduce soft PT1/2 -Space. We discuss the relation between
soft pre generalised closed sets and soft PT1/2 spaces. Finally we prove every soft topological space need not be a soft
PT1/2 – Spaces.
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