J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
1Ae31.
Soft S-Paracompact Space Vishwanath Kumasi1 .
Baiju Thankachan2O Department of Mathematics.
Manipal Institute of Technology.
Manipal Academy of Higher Education.
India vishwanathkumasi@gmail.
com, 2 baiju.
t@manipal.
Abstract.
This paper introduces a new topological class called soft S-paracompact These spaces generalize the concept of soft paracompact spaces.
A soft topological space is considered soft S-paracompact if every soft open cover has a locally finite soft semi-open refinement.
We explore the key properties of soft Sparacompact spaces and investigate their relationships with other well-established soft topological spaces.
We depict an application of soft S-paracompactness in the decision-making problem.
Key words and phrases: soft set, soft semi-open set, soft semi-closed set, soft topology, soft paracompactness.
INTRODUCTION
In 1965.
Zadeh .
presented the idea of fuzzy set theory, which has since grown to be a useful mathematical tool for expressing uncertainty and has been essential in decision-making-related problem-solving.
Molodtsov.
introduced the novel concept of soft set theory in 1999.
Soft set theory extends fuzzy set theory designed to address uncertainty issues parametrically.
The Aysoft setAy refers to a parameterized family of sets where the parameter defines the soft setAos boundary.
Despite the limitations of existing theories such as probability theory, rough sets, vague sets, and fuzzy sets.
Molodtsov pointed out that these can be effectively utilized to manage uncertainty problems.
Furthermore.
Molodtsov presented the initial findings of soft set theory, which is not restricted by the limitations above and can handle uncertainty-related problems.
Maji.
and colleagues also applied soft set theory using rough techniques in decision-making scenarios.
Later, in 2011.
Shabir and Naz .
defined soft topology and introduced a set of parameters on the initial universe.
The concepts of soft open sets, soft closed sets, soft closure, soft interior points, soft neighborhoods of a point, and soft separation axioms were O Corresponding author 2020 Mathematics Subject Classification:54A40, 03E72.
Received: 06-12-2024, accepted: 19-04-2025.
established, and their fundamental characteristics were examined.
Subsequently, in the same year.
Camgma and colleagues .
, .
defined soft topology within the context of a soft set and explored its associated properties, thereby laying the groundwork for the theory of soft topological spaces.
Levin .
introduced the concept of the semi-open set in topological space.
Al-Zoubi .
delved into the features of the S-paracompact space in the topological space, characterized the S-paracompact spaces with extremely disconnected .
and S-closed space.
, .
, and also studied basic properties of S-paracompact space.
Ergun .
, introduced nearly paracompactness in topological space.
In 2010.
Baiju and Sunil .
studied the behavior of various types of noncompact covering properties such as paracompactness, metacompactness, subparacompactness, submetacompactness under various types of fuzzy mappings such as open maps, closed maps, and perfect maps in L-topological spaces.
In 2012.
Chen .
defined soft semi-open sets, soft semi-closed sets, soft semi-interior, and soft semi-closure in soft topological space and examined the characterizations of soft semi-open sets, soft semi-closed sets, soft semi-interior, and soft semi-closure in soft topological space.
In 2013.
Lin .
defined soft paracompact and explored its characteristics in soft topological space.
In the present paper, we studied soft S-locally finite, soft S-refinement, soft extremely disconnected space, soft S-expandable space, and soft S-closed spaces in soft topological space.
In Section 3, we introduce and study the concept of soft S-paracompact spaces within soft topological spaces.
Soft S-paracompact defines every soft open cover of a soft topological space as having a locally finite semi-open We prove some characterizations of soft S-paracompact space and investigate the relation between soft S-paracompact space and some well-known spaces.
In Section 4, we define soft S-paracompact space, soft sg-closed space, soft -open set, soft -closed set, soft s -open set, and soft s -closed set in soft topological Section 5 studies some basic properties of soft S-paracompact space, i.
subspace, sum, and product.
Section 6 discusses the benefits and drawbacks of soft S-paracompactness in relation to different soft topological spaces.
In the last section, we estimate the application of soft S-paracompactness in decision-making PRELIMINARY This section provides a foundation for our work by reviewing relevant existing We discuss key concepts and findings related to a fixed set of parameters, which will serve as a basis for exploring our contributions.
Let UE represent the universal set and E denote the collection of parameters concerning UE , where parameters represent the properties or characteristics of objectives UE .
Consider AE OI E and P(UE ) to be the power set of UE .
(F.
E) denotes the soft set.
Let (UE .
EE .
E) be the soft topological space, where EE be the collection of soft sets.
The soft closure, soft interior, and relative soft topological space are denoted by Cl(F.
AE).
Int(F.
AE) and EEAE , respectively.
The soft semi-closure and soft semi-interior are denoted by Cls (F.
AE) and Ints (F.
AE) respectively.
Definition 2.
A soft set over UE is a pair (F.
E), where F is a function defined as F : E Ie P(UE ).
In other words, a soft set over UE can also be described as a parameterized family of subsets of the universe UE .
For any e OO E.
F .
represents the set of e-approximate elements of the soft set (F.
E).
Example 2.
The collection UE = .
, b, c, d, .
represents a group of five motorcycles being analyzed.
In contrast, the set E = .
1 , e2 , e3 } encompasses the parameters of interest, which are top speed .
1 ), fuel efficiency .
2 ), and weight .
3 ) of each To characterize these motorcycles, a soft set (F.
E) is established.
This soft set (F.
AE), where AE = E, is defined by F .
1 ) = .
, d, .
F .
2 ) = .
, c, .
, and F .
3 ) = .
, c, .
This formulation provides a representation of the motorcycleAos The parameterized family {F .
i ) : i = 1, 2, .
associated with the motorcycles in the set UE is identified as the soft set (F.
AE), which offers various approximations for describing the objects.
Definition 2.
Let (F.
AE) and (G.
BE) represent two soft sets defined over a common universal set UE , we say that (F.
AE) is a subset of (G.
BE).
If AE OI BE and for every element e OO AE, the approximations provided by F .
and G.
are the same.
Definition 2.
Let (F.
E) represent a soft set defined over UE .
The collection of all subsets of the soft set (F.
E) is known as the soft power set of (F.
E).
|P(F.
E)| = 2
eOOE |F .
| where |F .
| is a cardinal of F .
Example 2.
Let us consider the UE = .
1 , a2 , a3 , a4 } universal set, and E = .
1 , e2 } is the collection of parameters and AE = E.
Then the soft set (F.
AE) is defined as, (F.
AE) = {.
1 , .
1 }), .
2 , .
1 , a2 })} and the power set of a soft set (F.
AE) are given as follows:
(F.
AE1 ) = {.
1 , .
1 })} (F.
AE2 ) = {.
1 , .
1 }), .
2 , .
1 })} (F.
AE3 ) = {.
1 , .
1 }), .
2 , .
2 })} (F.
AE4 ) = {.
2 , .
1 })} (F.
AE5 ) = {.
2 , .
2 })} (F.
AE6 ) = {.
2 , .
1 , a2 })} (F.
AE7 ) = {.
1 , .
1 }), .
2 , .
1 , a2 })} = (F.
AE) (F.
AE8 ) = i = (F.
AEi ) Definition 2.
Let us consider the soft set (F.
AE) defined over the universe UE .
The complement of this soft set is denoted as (F.
AE)c , which is expressed as (F.
AE)c = (F c .
AAE).
Here F c : AAE OeIe P(UE) represents a mapping defined by F c (A.
= UE Oe F .
for every Ae OO AAE.
Definition 2.
Let us consider two soft sets (F.
AE) and (G.
BE) over the universe UE .
The union of these two soft sets over the common universe UE is a soft set (H.
CE), where CE = AE O BE and every element e OO CE.
if e OO A Oe B F .
= G.
, if e OO B Oe A F .
O G.
, if e OO A O B then (F.
AE) O (G.
BE) = (H.
CE).
Definition 2.
Let us consider two soft sets (F.
AE) and (G.
BE) over the universe UE .
The intersection of these two soft sets over common universe UE is a soft set (H.
CE), then CE = AE O BE and F .
O G.
= H.
for every element e OO CE, we have (F.
AE) O (G.
BE) = (H.
CE).
Definition 2.
Let (F.
AE) represent a soft set defined over the universe UE .
Then (F.
AE) is said to be an absolute soft set if ei OO AE and F .
i ) = UE .
The notation used for absolute soft set is UEAE .
Note 2.
To facilitate our discussion, let us consider a soft set (F.
E) defined on a universe UE .
The family of soft set defined over UE is denoted by SS(UE )E .
Definition 2.
Let (F.
AE) be the soft sets over UE .
A soft topology on (F.
AE) is represented by EE , which consists collection of subsets of (F.
AE) satisfying the following conditions.
(F.
AEi ), (F.
AE) OO EE .
The arbitrary union of elements from EE belongs to EE .
The intersection of a finite number of soft sets belonging to EE is also contained within EE .
Then (UE .
EE .
E) is called a soft topological space.
Example 2.
Let us consider UE = .
, b, c, .
is the universal set and E = .
1 , e2 } is the collection of parameters and AE = E.
Then the soft set (F.
AE) is defined as, (F.
AE) = {.
1 , .
, .
), .
2 , .
, .
)}.
Then the power set of a soft set (F.
AE) is given as, (F.
AE1 ) = {.
1 , .
)}, (F.
AE2 ) = {.
1 , .
)}, (F.
AE3 ) = {.
1 , .
, .
)}, (F.
AE4 ) = {.
2 , .
)}, (F.
AE5 ) = {.
2 , .
)}, (F.
AE6 ) = {.
2 , .
, .
)}, (F.
AE7 ) = {.
1 , .
, .
), .
2 , .
)}, (F.
AE8 ) = {.
1 , .
), .
2 , .
, .
)}, (F.
AE9 ) = {.
1 , .
), .
2 , .
)}, (F.
AE10 ) = {.
1 , .
), .
2 , .
, .
)}, (F.
AE11 ) = {.
1 , .
), .
2 , .
)}, (F.
AE12 ) = {.
1 , .
), .
2 , .
)}, (F.
AE13 ) = {.
1 , .
, .
), .
2 , .
)}, (F.
AE14 ) = {.
1 , .
), .
2 , .
)}, (F.
AE15 ) = {.
1 , .
, .
), .
2 , .
, .
)} = (F.
AE), (F.
AE16 ) = .
= (F.
AEi ) .
hich is called null soft se.
Let us consider the collection of soft subsets of (F.
AE) is denoted by EE and it is defined as EE = {(F.
AEi ), (F.
AE), (F.
AE2 ), (F.
AE11 ), (F.
AE13 )}.
Consequently, the structure (UE .
EE .
E) is identified as a soft topological space.
Definition 2.
Let (UE .
EE .
E) is a soft topological space.
The elements of EE are called soft open sets.
If the complement of a soft set is a soft open set, it is called a soft closed set.
Definition 2.
The soft set (F.
E) OO SS(UE )E is called a soft point in UEE if there exist elements x OO UE and e OO E that satisfy the following conditions:
F .
= .
, and for every eA = E Oe .
F .
A ) = i.
The notation ex is used to represent the soft point (F.
E).
Definition 2.
Let (UE .
EE .
E) is the soft topological space and (F.
AE) is the soft set in (UE .
EE .
E).
If (F.
AE) is said to be soft semi-open set if and only if there exists a soft open set (F.
OE) , such that (F.
OE) OC (F.
AE) OC Cl(F.
OE).
Definition 2.
Let (UE .
EE .
E) is the soft topological space and (F.
AE) is the soft set in (UE .
EE .
EE).
If (F.
AE) is said to be soft semi-closed set if and only if there exists a soft closed set (F.
DE), such that Int(F.
DE) OC (F.
AE) OC (F.
DE).
The relative complement of soft semi-open set is called as soft semi-closed Definition 2.
Consider the soft topological space (UE .
EE .
E).
Let (F.
AE) is a soft set over UE .
Then the soft semi-interior of a soft set (F.
AE) on UEE is defined as the union of all soft semi-open sets contained within (F.
AE).
Definition 2.
Consider the soft topological space (UE .
EE .
E).
Let (F.
AE) is a soft set over UE .
Then the soft semi-closure of (F.
AE) over UEE is defined as the intersection of all soft semi-closed set containing (F.
AE).
Definition 2.
If the pair (F.
AE) is classified as soft regular open and soft regular closed, it is defined by the conditions (F.
AE) = Int(Cl(F.
AE)) for soft regular open and (F.
AE) = Cl(Int((F.
AE)) for soft regular closed.
Definition 2.
A soft topological space (UE .
EE .
E) is considered soft almost regular if, for any soft regular closed set (F.
AE) and any soft point ex not contained in (F.
AE), there exist separate soft open sets that contain (F.
AE) and ex respectively.
In a soft topological space (UE .
EE .
E), the collections of all subsets that are soft semi-open, soft regular open, soft semi-closed, and soft regular closed are represented by SSO(UE .
EE .
E).
SRO(UE .
EE .
E).
SSC(UE .
EE .
E), and SRC(UE .
EE .
E), respectively.
It is established that for such a space, the set SRO(UE .
EE .
E) forms a base for a soft topology EES on UE , where EES OI EE .
The resulting soft topological space (UE .
EuS .
E) is referred to as the soft semi-regularization of (UE .
EE .
E).
Definition 2.
Consider the soft topological space (UE .
EE .
E) and let (F.
AE) is said to be the soft neighborhood of the soft set (F.
BE) if there exists a soft open set (F.
OE) such that (F.
BE) OI(F.
OE) OI (F.
AE).
Definition 2.
Let (UE .
EE .
E) is the soft topological space.
(F.
AE) is the soft set over UE and ex OO UEE .
Consequently, (F.
AE) is termed as a soft semi neighborhood of ex if there exist soft semi-open set (F.
OE) such that ex OO (F.
OE) OI (F.
AE).
Definition 2.
Let (UE .
EE .
E) is the soft topological space.
If it is called soft expandable if for each soft locally finite collection (F.
CE) = {C .
OO I} of subset of UEE , there is a soft locally finite collection (F.
KE) = {K .
OO I} of soft open subsets of UEE satisfying for each OO I that K OI C .
Definition 2.
Let (UE .
EE .
E) is the soft topological space and (F.
AE) OI UuE .
Then (F.
AE) is said to be the soft preopen set if (F.
AE) OC Int(Cl(F.
AE)).
Soft preopen set is denoted by SP O(UE .
EE .
E) and if (F.
AE) is said to be soft semi preopen set if (F.
AE) OC Cl(Int(F.
AE)), is denoted as SP Os (UE .
EE .
E).
Definition 2.
Let (UE .
EE .
E) is the soft topological space and (F.
AE) be the soft set is called soft -open set if (F.
AE) OC Int(Cl(Int(F.
AE))).
The complement of soft -open set is called the soft -closed set.
Definition 2.
, .
Consider a soft topological space (UE .
EE .
E).
A collection (F.
CE) = {C .
OO I} of subset of (UE .
EE .
E) is said to be soft locally finite .
r soft S-locally finit.
if for every ex OO UEE , there exists (F.
OE) OO EE .
r (F.
OE) OO SSO(UE .
EE .
E)) that contains ex and intersects with only a finite number of elements from (F.
CE).
Lemma 2.
Let (UE .
EE .
E) is the soft topological space.
Consider the collection (F.
CE) = {C .
OO I} of soft subsets of the soft topological space (UE .
EE .
E).
Then (F.
CE) is soft S-locally finite if and only if {Cls (C ).
OO I} is soft S-locally Lemma 2.
Let (UE .
EE .
E) is the soft topological space.
Consider the collection (F.
CE) = {C .
OO I} of soft subsets of the soft topological space (UE .
EE .
E).
(F.
CE) is soft locally finite, then OOI Cls (C ) = Cls ( OOI C ).
It is clear that a family (F.
CE) = {C .
OO I} of subsets within a soft topological space (UE .
EE .
E) is classified as soft locally finite if and only if the family {Cls (C ).
OO I} is also soft locally finite.
Definition 2.
Let (UE .
EE .
E) is the soft topological space is said to be soft S-expandable if, for every soft S-locally finite collection (F.
CE) = {C .
OO I} of subsets of (UE .
EE .
E), there is a soft S-locally finite collection (F.
KE) = {K .
OO I}
of soft open subsets of (UE .
EE .
E) such that for each OO I.
K OI C .
Definition 2.
Let (UE .
EE .
E) is the soft topological space.
It is called soft extremely disconnected if the soft closure of each soft open set in soft topological space (UE .
EE .
E) is soft open or if the soft interior of each soft closed set in soft topological space (UE .
EE .
E) is soft closed.
Lemma 2.
Let (UE .
EE .
E) is the soft extremely disconnected space, then Cls (M ) = Cl(M ) for every M OO SSO(UE .
EE .
E).
Proposition 2.
A soft topological space (UE .
EE .
E) is soft extremely disconnected if and only if SSO(UE .
EE .
E) OC SP O(UE .
EE .
E) Definition 2.
Let (UE .
EE .
E) is the soft topological space.
Soft paracompact is called if every soft open cover has a locally finite soft open refinement.
Definition 2.
In a soft topological space (UE .
EE .
E), the characteristic of being soft nearly paracompact is established when every soft regularly open cover has a locally finite soft open refinement.
A subset (F.
AE) is deemed soft near paracompact if the topology it inherits demonstrates this same property of soft near
SOFT S-PARACOMPACT SPACE
In this section, we investigate some characteristics of soft semi-open refinement, soft S-locally finite, and soft S-expandable in soft topological space, and we present the notion of a soft S-paracompact space within the context of soft topological spaces.
Furthermore, we explored the relation between soft S-paracompact spaces and various soft separation axioms, soft S-closed spaces, soft -open sets, and soft extremely disconnected spaces.
Definition 3.
Consider a soft topological space (UE .
EE .
E), where (F.
AE) and (F.
BE) represent collections of soft sets within UEE .
We define (F.
BE) as a soft refinement .
r soft S-refinemen.
of (F.
AE) if for each element B in (F.
BE) .
r SSO(F.
BE) respectivel.
, there exists an element A in (F.
AE) .
r SSO(F.
AE) respectivel.
such that A encompasses B.
In cases where the elements of (F.
BE) are soft open sets .
r soft semi-open set.
, we may refer to (F.
BE) as a soft open refinement .
r soft semi-open refinemen.
of (F.
AE).
When the elements of (F.
BE) are soft closed sets .
r soft semi-closed set.
, we may refer to (F.
BE) as a soft closed refinement .
r soft semi-closed refinemen.
of (F.
AE).
Definition 3.
Let (UE .
EE .
E) denote the soft topological space and collection (F.
CE) = {C .
OO I} of subsets of (UE .
EE .
E) is said to be:
Soft discrete if every soft point ex OO UEE has a soft neighborhood that intersects at most one of the soft sets in (F.
CE).
Soft point finite if every soft point ex OO UEE is contained in at most a finite number of the soft sets in (F.
CE).
Lemma 3.
Let (UE .
EE .
E) is the soft topological space and (F.
CE) be the soft open covering of UEE , .
If (F.
CE) has a soft point finite soft semi-open refinement, then it also has a precise soft point finite soft semi-open refinement.
If (F.
CE) possesses a soft s-locally finite soft semi-open .
r close.
refinement, it necessarily follows that it also has a precise soft s-locally finite soft semi-open .
r close.
Proposition 3.
Let (UE .
EE .
E) is soft extremely disconnected semi-regular space.
Then (UE .
EE .
E) is soft expandable if and only if it is soft S-expandable.
Proof.
The proof follows from theorem 4.
4 of .
n Proposition 3.
Let (UE .
EE .
E) is the soft extremely disconnected semi-regular Then .
SSO(UE .
EE .
E) = EE .
(UE .
EE .
E) is soft regular.
Lemma 3.
Let (UE .
EE .
E) be the soft topological space.
If (F.
AE) is the soft open subset in a soft topological space (UE .
EE .
E) and (F.
VE ) OO SSO(UE .
EE .
E) then (F.
AE) (F.
VE ) OO SSO(UE .
EE .
E).
(UuS .
EEy .
E) represents the subspace of the soft topological space (UE .
EE .
E).
Consider the subset (F.
BE) OI UuS .
If it holds that (F.
BE) OO SSO(UE .
EE .
E), then it follows that (F.
BE) OO SSO(UuS .
EEy .
E).
(UuS .
EEy .
E) represents the subspace of a soft topological space (UE .
EE .
E).
Furthermore, if we consider a subset (F.
BE) OI UuS , and it holds that UuS OO EE , along with the condition that (F.
BE) OO SSO(UuS .
EEy .
E), then it follows that (F.
BE) OO SSO(UE .
EE .
E).
Lemma 3.
Let (UE .
EE .
E) is the soft topological space and (F.
AE) be the subset in a soft topological space (UE .
EE .
E).
Then (F.
AE) OO SP O(UE .
EE .
E) if and only if Cls (F.
AE) = Int(Cl(F.
AE)).
Definition 3.
Let (UE .
EE .
E) be the soft topological space.
It is said to be a soft S-paracompact space if every soft open cover has a locally finite soft semi-open Example 3.
Consider UE = .
, b, .
is the universal set and E = .
1 , e2 } is the collection of parameters and AE = E.
Then the soft set (F.
AE) is defined as (F.
AE) = {.
1 , .
, b, .
), .
2 , .
, b, .
)}, and the soft subset of (F.
AE) are:
(F.
AE1 ) = {.
1 , .
, .
), .
2 , .
, .
)}, (F.
AE2 ) = {.
1 , .
), .
2 , .
, .
)}, (F.
AE3 ) = {.
1 , .
, .
), .
2 , .
)}, (F.
AE4 ) = {.
1 , .
), .
2 , .
)}, (F.
AE5 ) = {.
1 , .
, .
), .
2 , .
, b, .
)}, (F.
AE6 ) = {.
1 , .
, b, .
), .
2 , .
, .
)}, (F.
AE7 ) = {.
1 , .
, .
), .
2 , .
, .
)}.
Consider a soft topology EE is defeined as EE = {(F.
AEi ), (F.
AE), (F.
AE1 ), (F.
AE2 ), (F.
AE3 ), (F.
AE4 ), (F.
AE5 ), (F.
AE6 ), (F.
AE7 )}.
Suppose (F.
CE) = {(F.
AE5 ), (F.
AE6 ), (F.
AE7 )} is a soft open cover of (F.
AE) and (F.
DE) = {(F.
AE1 ), (F.
AE2 ), (F.
AE3 ), (F.
AE4 ), (F.
AE5 ), (F.
AE6 ), (F.
AE7 )} is a soft semi-open refinement set of (F.
CE), as each element in (F.
DE) is a subset of a soft open cover (F.
CE).
(F.
DE) includes soft semi-open sets, such as (F.
AE7 ), which satisfies (F.
AE3 ) OI (F.
AE7 ) OI Cl(F.
AE3 ).
For a soft point .
) OO (F.
AE), there exists a soft open set (F.
AE3 ) containing .
), and (F.
AE7 ) serves as a soft neighborhood of .
) since .
) OO (F.
AE3 ) OI (F.
AE7 ) and (F.
AE7 ) O (F.
AE6 ) = (F.
AE3 ) OO (F.
DE), hence (F.
DE) encompasses locally finite soft semi-open sets.
Theorem 3.
Every soft S-paracompact T2 space is soft regular.
Proof.
Consider the soft S-paracompact T2 space (UE .
EE .
E).
Let ex OO UEE and let (F.
DE) is the soft closed set of UEE , which is disjoint from ex .
By the soft T2 condition, for every soft point ey of (F.
DE), there exists a soft open set (F.
Buey ) about ey , such that its closure is disjoint from ex .
Let A = {(F.
Buey ).
ey OO (F.
DE)} (F.
DE)c .
A be the soft open covering of UEE .
Since (UE .
EE .
E) is soft S-paracompact, there is a locally finite soft semi-open refinement C that covers UEE .
Let H be a subcollection of C consisting of all elements of C that intersect (F.
DE).
Let V be defined as union of H i.
V = {H OO C.
H O(F.
DE) = .
Then V is a soft semi-open set containing (F.
DE) and Cl(V) = {Cl(H).
H OO C and H O (F.
DE) = .
, where the soft closure of V is disjoint from ex .
Consequently, (F.
PE ) = UEE Oe Cl(V) is a soft open set containing ex , such that (F.
PE ) O (F.
Buey ) = i.
n Theorem 3.
Every soft S-paracompact soft regular space is soft normal.
Proof.
Consider the soft S-paracompact soft regular space (UE .
EE .
E).
Let ex , ey OO UEE and let (F.
CE) and (F.
DE) be the soft closed set of UEE , which are disjoint from each other, where ex OO (F.
CE) and ey OO (F.
DE).
By the soft regular condition, for every soft point ey of (F.
DE), there exists a soft open set (F.
Buey ) about ey and disjoint Soft open set (F.
Puex ), such that ey OO (F.
DE) OI (F.
Buey ) and ex OO (F.
CE) OI (F.
Puex ) which is implies that (F.
CE) O (F.
Buey ) = i as every point (F.
CE) has a neighborhood (F.
Puex ) that is disjoint from the (F.
Buey ).
Since (UE .
EE .
E) is soft S-paracompact, there is a locally finite soft semi-open refinement C that covers UEE .
Let H be a subcollection of C consisting of all elements of C that intersect (F.
DE).
Let V be defined as union of H i.
V = {H OO C.
H O (F.
DE) = .
Then a soft open neighborhood of (F.
DE), whose soft closure is disjoint from (F.
CE).
Then Cl(V) = {Cl(H).
H OO C and H O (F.
DE) = .
, where the soft closure of V is disjoint from ex .
We completed this proof as mentioned in Theorem3.
10, in the same way to get normality.
Instead of ex OO / Cl(F.
Buey ), we used (F.
CE) O Cl(F.
Buey ) = i, whenever relevant.
n Theorem 3.
Each soft S-paracompact T2 space is soft semiregular i.
EE = EuS .
Proof.
Consider the soft S-paracompact T2 space (UE .
EE .
E).
Let a soft point ex OO UEE and (F.
DE) be the soft closed set of UEE , which is disjoint from ex .
For every soft point ey of (F.
DE), there exists a soft semi-open set (F.
Vuey )s for which ey OO (F.
Vuey )s and its closure is disjoint from ex i.
, ex OO / Cl(F.
Vuey )s .
Let A = {(F.
Vuey )s .
ey OO (F.
DE)} (F.
DE) .
A be the soft open covering of UEE .
Since (UE .
EE .
E) is soft Sparacompact, there is a locally finite soft semi-open refinement C that covers UEE .
Let H be a subcollection of C consisting of all elements of C that intersect (F.
DE), .
C = {H OO C.
H O (F.
DE) = .
Then C is a soft semi-open set containing (F.
DE).
From Lemma 3.
7 Cls (C) = Int(Cl(C)).
Then Cls (C) = {Cls (H).
H OO C and H O (F.
DE) = .
, where the soft semi-closure of C is disjoint from ex .
Consequently, (F.
PE ) = UEE Oe Cls (C) is a soft semi-open set containing ex and Int(Cl((F.
PE ))) is soft regular open for every soft open subset (F.
PE ) of UuE , such that (F.
PE ) O (F.
Vuey )s = i.
n Theorem 3.
Each soft extremely disconnected soft S-paracompact T2 space is soft regular.
Proof.
Consider the soft S-paracompact T2 space (UE .
EE .
E).
Let ex OO UEE and (F.
DE) be the soft closed set of UEE , which is disjoint from ex .
By the soft T2 condition, for every soft point ey of (F.
DE), there exists a soft open set (F.
Buey ) about ey , such that its closure is disjoint from ex .
Let A = {(F.
Buey ).
ey OO (F.
DE)} (F.
DE)c .
A be the soft open covering of UEE .
Since (UE .
EE .
E) is soft extremely disconnected soft Sparacompact space, there is a locally finite soft semi-open refinement C that covers UEE .
Let H be a subcollection of C consisting ofSall elements of C that intersect (F.
DE).
Let V be defined as union of H i.
V = {H OO C.
H O (F.
DE) = .
Then V is a soft semi-open set containing (F.
DE) and Cls (V) = Cl(V) .
y Lemma 2.
Cls (V) = {Cls (H).
H OO C and H O (F.
DE) = .
, where the soft semi-closure of V is soft semi-open and disjoint from ex .
Consequently, (F.
PEex ) = UEE Oe Cls (V) is a soft semi-open set containing ex , such that (F.
PEex ) O (F.
Buey ) = i.
n Theorem 3.
Every soft closed subspace of a soft S-paracompact space is a soft S-paracompact.
Proof.
Consider a soft S-paracompact space (UE .
EE .
E) and any soft closed set (F.
DE) within UEE .
A soft open covering (F.
CE) of (F.
DE) exists, where (F.
CE) = i=1 (F.
Ci ) for each (F.
Ci ) OO (F.
CE), encompassing (F.
DE).
For every (F.
Ci ) OO (F.
CE), there is a corresponding (F.
Ci )C such that (F.
Ci ) = (F.
Ci )C (F.
DE).
The soft semiopen set (F.
Ci )C , in conjunction with the soft semi-open set (F.
DE)C , forms a cover for UEE .
A soft locally finite soft semi-open T refinement B of this soft covering exists, which covers UEE .
The set G = {(F.
BE) (F.
DE).
(F.
BE) OO B} constitutes the required locally finite soft semi-open refinement of (F.
CE).
n Theorem 3.
Let (UE .
EE .
E) is the soft extremely disconnected soft regular space.
If each soft open cover of UEE has a soft S-locally finite soft semi-open refinement.
Every soft open cover of UEE has a soft locally finite soft open refinement.
Proof.
Let (F.
PE ) denote the soft open cover of UEE , where each ex OO UEE corresponds to a selected member Pex OO (F.
PE ) and by soft regularity and the existence of a soft open subset Vex OO EE such that ex OO Vex OI Cl(Vex ) OI Pex .
Consequently, the collection V = {Vex .
ex OO UEE } forms a soft open cover of UEE and, as assumed, possesses a soft S-locally finite soft semi-open refinement denoted by W = {Wey .
ey OO (F.
BE)}.
For each ey OO (F.
BE), select a soft open set Hey such that Hey OI Wey OI Cl(Hey ).
However, if ey OO (F.
BE), then Cl(Hey ) = Cl(Wey ) OI Cl(Vex ) for some Vex OO V, implying that Cl(Hey ) OI P for some P OO (F.
PE ).
Conversely, as (UE .
EE .
E) is soft extremely disconnected.
Cl(Hey ) OO EE for every ey OO (F.
BE).
We now demonstrate that the set H = Cl(Hey ).
ey OO (F.
BE) is soft locally finite.
Given ex OO UEE , according to Lemma 2.
26, the set {Cls (Wey ).
ey OO (F.
BE)} is locally finite.
Therefore, the set H is soft S-locally finite (Cl(Hey ) = Cl(Wey ) = Cls (Wey ) for all ey OO (F.
BE) according to Lemma 2.
Choose Oex OO SSO(UE .
EE .
E) such that ex OO Oex and Oex intersects at most finitely many members of H.
Select Aex OO EE such that Aex OI Oex OI Cl(Aex ).
Given that (UE .
EE .
E) is soft extremely disconnected, it follows that Cl(Aex ) is a soft open set that includes ex and satisfies Cl(Aex ) O Cl(Hex ) = i if and only if Oex O Hex = i.
Consequently.
H serves as a locally finite soft open refinement of (F.
PE ).
n Corollary 3.
Each soft, extremely disconnected soft S-paracompact T2 space is a soft paracompact.
Definition 3.
Let (UE .
EE .
E) is said to be the soft S-compact space if every soft cover of UEE by semi-open sets has a finite subcover.
Definition 3.
Let (UE .
EE .
E) is said to be the soft S-Lindelof space if every cover of UEE by soft semi-open subsets of UEE has a countable subcover.
Proposition 3.
Every soft S-compact space is a soft S-Lindelof space, and every soft S-Lindelof space is a soft S-paracompact space.
Proposition 3.
Let (UE .
EE .
E) be the soft S-paracompact space and if AE = .
, then (UE .
EE .
E) is soft S-paracompact if and only if the collection (F.
CE) = {(F.
AEi ).
i OO I.
(F.
AEi ) OO EE } is a soft paracompact on UEE .
Based on Proposition 3.
20, it can be inferred that not every soft S-compact space is necessarily a soft S-paracompact space.
For instance, a universal set UE , a soft topological space (UE .
EEdis .
E) can be a soft S-paracompact space without being a soft S-compact space.
Definition 3.
A soft topological space (UE .
EE .
E) is defined as soft S-closed if every soft semi-open cover of UEE has finite subcover whose soft closures covers UEE .
Theorem 3.
In countably soft S-closed spaces, every soft S-locally finite collection of soft semi-open sets is finite.
Proof.
Let (UE .
EE .
E) be the countably soft S-closed space.
Assume A = {Ai .
i OO I} is an infinite soft s-locally finite collection of soft semi-open in space (UE .
EE .
E).
For each n OO I, we define Fn = j=n Cls (Aj ) = Cls j=n Aj .
rom Lemma 2.
Since j=n Aj is an element of SSO(UE .
EE .
E) .
s noted in Theorem 3.
), we can conclude that Cl(Fn ) belongs to SRC(UE .
EE .
E).
Thus, we have the following of inclusions: Cl(F1 ) ON Cl(F2 ) ON Cl(F3 ) ON .
Our objective is to prove that {Int(Cl(Fn )).
n OO I} = OI.
T will prove this by contradiction.
Suppose there is a point t in the intersection {Int(Cl(Fn )).
n OO I}.
For each natural number n, we can find Vn .
in EE such that t is in Vn .
and Vn .
is contained in the closure of Fn .
Because A is soft s-locally finite, there exists Ot OO (UE .
EE .
E) containing t that intersects only finitely many members of A.
We can choose Ut OO EE satisfying Ut OI Ot OI Cl(Ut ).
Then, for every nTOO I.
Vn .
O Ut is non-empty, allowing us to select Kn OO Vn .
O Ut OI Cl(Fn ) Ut .
As a result, (Fn ) O Ut = i, which T that Ot intersects numerous elements of A.
Therefore, we can conclude that {Int(Cl(Fn )).
n OO I} = i.
This contradicts the assertion that for a space (UE .
EE .
E) to be countably soft S-closed, any countable cover consisting of soft semiopen sets must have a finite subset whose members closures collectively cover UuE .
n Corollary 3.
Each soft S-paracompact countably soft S-closed space is soft Theorem 3.
Every soft, extremely disconnected, soft compact space is soft Sclosed.
Proof.
Let us consider (UE .
EE .
E) is the soft extremely disconnected space, and we know that by the definition of soft extremely disconnected space that is soft closure of any soft open set is soft open.
Then, the interior of the soft semi-open set is dense in it.
Then we will assume {Int(Cl(Vt )).
t OO EE } instead of given soft semi-open set.
In other words, we consider soft pre-open sets rather than soft semi-open sets.
n Corollary 3.
Let (UE .
EE .
E) be the soft extremely disconnected space the following are equivalent:
(UE .
EE .
E) is soft S-paracompact and soft S-closed.
(UE .
EE .
E) is soft compact.
Note 3.
From the Definition 2.
The family of all soft -open set of a space (UE .
EE .
E) is denoted as EE , forms a topology on UuE and it is finer than the EE such that EE OI EE OI SSO(UE .
EE .
E) and SSO(UE .
EE .
E) = SSO(UE .
EE .
E).
Theorem 3.
If the soft -open space (UE .
EE .
E) is soft S-paracompact space then (UE .
EE .
E) is soft S-paracompact space.
Proof.
Consider a soft S-paracompact space (UE .
EE .
E) and a soft open cover (F.
CE) of the soft S-paracompact space (UE .
EE .
E).
As noted in Note 3.
EE OI EE , so (F.
CE) is also a soft open cover of (UE .
EE .
E).
Consequently, (F.
CE) possesses a locally finite soft semi-open refinement (F.
DE) in (UE .
EE .
E).
Note 3.
1 also states that SSO(UE .
EE .
E) = SSO(UE .
EE .
E).
Our objective is to demonstrate that (F.
DE) is locally finite in (UE .
EE .
E).
For any ex OO UuE , there exists a soft open subset (F.
ME ) OO EE that intersects only a finite number of members of (F.
DE), denoted as {D1 .
D2 .
D3 .
Dn }.
For each D OO (F.
CE), we can find WD OO EE such that WD OI D OI Cl(WD ).
We claim that Int(Cl.
nt(F.
GE))) is a soft open set in (UE .
EE .
E) containing ex and satisfying Int(Cl.
nt(F.
GE))) O D = i for all D OO (F.
DE) Oe {D1 .
D2 .
D3 .
Dn }.
If we assume Int(Cl.
nt(F.
GE))) O D = i, then Int(Cl.
nt(F.
GE))) O Cl(WD ) = i and (F.
GE) O WD = i, implying (F.
GE) O D = i, and thus D OO {D1 .
D2 .
D3 .
Dn }.
Therefore, we can conclude that (F.
DE) is a locally finite soft semi-open refinement of (F.
CE) in (UE .
EE .
E), establishing that (UE .
EE .
E) is indeed a soft S-paracompact space.
n The converse of Theorem 3.
26 is given in the following example.
Example 3.
Consider UE = .
, b, c, .
is the universal set and E = .
1 , e2 , e3 } is the collection of parameters and AE = E.
Then the soft set (F.
AE) is defined as (F.
AE) = {.
1 , .
, b, c, .
), .
2 , .
, b, c, .
), .
3 , .
, b, c, .
)} and subsets of (F.
AE) (F.
AE1 ) = {.
1 , .
), .
2 , .
, .
), .
3 , .
, .
)}, (F.
AE2 ) = {.
1 , .
, .
), .
2 , .
, c, .
), .
3 , .
, b, .
)}, (F.
AE3 ) = {.
2 , .
), .
3 , .
)}, (F.
AE4 ) = {.
1 , .
, b, .
), .
2 , .
, b, c, .
), .
3 , .
, b, c, .
)}, (F.
AE5 ) = {.
1 , .
, .
), .
2 , .
, .
), .
3 , .
)}, (F.
AE6 ) = {.
1 , .
), .
2 , .
)}, (F.
AE7 ) = {.
1 , .
, .
), .
2 , .
, c, .
), .
3 , .
, b, .
)}, (F.
AE8 ) = {.
2 , .
), .
3 , .
)}, (F.
AE9 ) = {.
1 , .
, b, c, .
), .
2 , .
, b, c, .
), .
3 , .
, b, .
)}, (F.
AE10 ) = {.
1 , .
, .
), .
2 , .
, c, .
), .
3 , .
, .
)}, (F.
AE11 ) = {.
1 , .
, c, .
), .
2 , .
, b, c, .
), .
3 , .
, b, .
)}, (F.
AE12 ) = {.
1 , .
), .
2 , .
, c, .
), .
3 , .
, b, .
)}, (F.
AE13 ) = {.
1 , .
), .
2 , .
, .
), .
3 , .
)}, (F.
AE14 ) = {.
1 , .
, .
), .
2 , .
, .
)}, (F.
AE15 ) = {.
1 , .
), .
2 , .
, .
), .
3 , .
)}, (F.
AE16 ) = {.
1 , .
), .
2 , .
), .
3 , .
, .
)}.
Let us consider a soft topology EE = {(F.
AEi ), (F.
AE), (F.
AE1 ), (F.
AE2 ), (F.
AE3 ), (F.
AE4 ), (F.
AE5 ), (F.
AE6 ), (F.
AE7 ), (F.
AE8 ), (F.
AE9 ), (F.
AE10 ), (F.
AE11 ), (F.
AE12 ), (F.
AE13 ), (F.
AE14 ), (F.
AE15 )}.
Let (F.
CE) = {(F.
AE4 ), (F.
AE9 )} represent the soft open cover of (F.
AE).
Suppose (F.
DE) = {(F.
AE1 ), (F.
AE2 ), (F.
AE10 ), {(F.
AE11 )} denotes the soft semi-open refinement set of (F.
CE).
This set fulfills the S-paracompactness condition.
Thus (UE .
EE .
E) is classified as a soft S-paracompact space.
On the other hand, if EE denotes the soft topology on (F.
AE), then (UE .
EE .
E) does not qualify as a soft S-paracompact space.
Assume (F.
ME ) = {(F.
AE4 ), (F.
AE9 )} represents the soft open cover of (F.
AE).
Let (F.
NE ) = {(F.
AE1 ), (F.
AE2 ), (F.
AE3 ), (F.
AE16 )} be the soft semi-open refinement of (F.
ME ).
For (F.
AE16 ), we have Int(F.
AE16 ) = (F.
AE3 ) and Cl(Int(F.
AE16 )) = (F.
AE5 )c .
However, (F.
AE16 ) OO Int(Cl(Int(F.
AE16 ))), indicating that it is not a soft -open set.
Given that EE OI EE and (UE .
EE .
E) is soft S-paracompact, (UE .
EE .
E) fails to be a soft S-paracompact space.
This is because the collection of soft open covers of (UE .
EE .
E) does not allow for locally finite semi-open refinements in (UE .
EE .
E), as (F.
AE16 ) belongs to EE but is not a soft -open set.
Let us consider (UE .
EE .
E) is the soft topological space and EEsso be the soft topology on UEE .
And it has soft subbase SSO(UE .
EE .
E).
The collection SSO(UE .
EE .
E) is soft topology on UEE if and only if SSO(UE .
EE .
E) is soft extremely disconnected.
In such cases EEsso = SSO(UE .
EE .
E).
Corollary 3.
Let (UE .
EE .
E) be a soft extremely disconnected space.
Then (UE .
EE .
E) is soft S-paracompact if (UE .
EEsso .
E) is soft S-paracompact.
Proof.
Let (UE .
EE .
E) be the soft extremely disconnected space.
Then SSO(UE .
EE .
E) OI SP O(UE .
EE .
E).
ased on Proposition 2.
, hence EE = SP O(UE .
EE .
E)OSSO(UE .
EE .
E) .
)= SSO(UE .
EE .
E) = EEsso .
Then from the Theorem 3.
26 we conclude that (UE .
EE .
E) is soft S-paracompact if (UE .
EEsso .
E) is soft S-paracompact.
n The converse of Corollary 3.
28 can be demonstrated using Example 3.
1 from .
It is established that every soft open set in soft topological space qualifies as a soft semi-open set.
however, not every soft semi-open set is classified as a soft open set in soft topological space.
Theorem 3.
If (UE .
EE .
E) is the soft T2 space, then (UE .
EE .
E) is soft S-paracompact space if and only if each soft open cover (F.
BE) of UEE has a locally finite soft semiclosed refinement (F.
CE) .
hat is C OO SSC(UE .
EE .
E) for every C OO (F.
CE)).
Proof.
To establish the necessary condition, consider that (F.
BE) constitutes a soft open cover of UEE , and for each ex OO UEE , we choose a member Bex OO (F.
BE).
According to Theorem 3.
10, there exists a soft open subset Cex OO EE such that ex OO Cex OI Cls (Cex ) OI Bex .
Thus, (F.
CE) = {Cex .
ex OO UEE } functions as a soft open cover of UEE and, as assumed, it has a locally finite soft semi-open refinement, labeled D = {Dey .
ey OO I}.
Examine the set Cls (D) = {Cls (Dey ).
ey OO I} (Lemma Clearly.
Cls (D) creates a locally finite group of soft semi-closed subsets of (UE .
EE .
E).
For each ey OO I.
Cls (Dey ) OI Cls (Cex ) OI Bex ) for a particular Bex OO (F.
BE), showing that Cls (D) acts as a soft refinement of (F.
BE).
To demonstrate the sufficient condition, let (F.
BE) form a soft open cover of UEE and (F.
CE) be the locally finite soft semi-closed refinement of (F.
BE).
For each ex OO UEE , let Dex be the soft open set containing ex that intersects with at most a finite number of elements from (F.
CE).
Let (F.
GE) be the soft semiclosed locally refinement of D = {Dex .
ex OO UEE }.
For every C OO (F.
CE), define C A = UEE Oe {G OO (F.
GE).
G O C = .
The set {C A : C OO (F.
CE)} then forms a soft semi-open cover of UEE .
Next, for each C OO (F.
CE), pick BC OO (F.
BE) such that C OI BC .
Consequently, the collection {BC O C A : V OO (F.
CE)} serves as a locally finite soft semi-open (Lemma 3.
refinement of F.
BE), thereby proving that (UE .
EE .
E) is a soft S-paracompact space.
n Theorem 3.
If (UE .
EE .
E) be the soft regular space, then (UE .
EE .
E) is soft Sparacompact space if and only if each soft open cover (F.
BE) of UEE has a locally finite soft semi-closed refinement (F.
CE) .
hat is C OO SRC(UE .
EE .
E) for every C OO (F.
CE)).
Proof.
Let us establish the necessary conditions.
Suppose (F.
BE) forms a soft open cover of UEE , and for each ex OO UEE , we select a member Bex OO (F.
BE).
According to Theorem 3.
10, there exists a soft open subset Cex OO EE such that ex OO Cex OI Cls (Cex ) OI Bex .
Consequently, (F.
CE) = {Cex .
ex OO UEE } serves as a soft open cover of UEE and, as assumed, it possesses a locally finite soft semi-open refinement, denoted as D = {Dey .
ey OO I}.
Consider the set Cl(D) = {Cl(Dey ).
ey OO I} (Lemma 2.
It is evident that Cl(D) forms a locally finite group of soft semiclosed subsets of (UE .
EE .
E).
For each ey OO I.
Cl(Dey ) OI Cls (Cex ) OI Bex ) for a certain Bex OO (F.
BE).
Cl(D) OO SRC(UE .
EE .
E) for each D OO SSO(UE .
EE .
E), indicating that Cl(D) serves as a soft refinement of (F.
BE).
Now we will establish the sufficient condition.
(F.
BE) forms a soft open cover of UEE and (F.
CE) be the a locally finite soft semi-closed refinement of (F.
BE) or each ex OO UEE .
Let Dex be the soft open set containing ex intersecting at most finitely many element of (F.
CE).
Let (F.
GE) be the soft semi-closed locally refinement of D = {Dex .
ex OO UEE }.
For every C OO (F.
CE), define C A = UEE Oe {G OO (F.
GE).
G O
C = .
Then the set {C A : C OO (F.
CE)} forms a soft semi-open cover of UEE .
Subsequently, for each C OO (F.
CE), select BC OO (F.
BE) such that C OI BC .
Hence, the collection {BC OC A : V OO (F.
CE)} serves as a locally finite soft semi-open(Lemma .
refinement of F.
BE) and therefore (UE .
EE .
E) is a soft S-paracompact space.
n Theorem 3.
Let (UE .
EE .
E) be a soft, extremely disconnected, and semi-regular Then the following are equivalent:
(UE .
EE .
E) is soft nearly paracompact .
(UE .
EE .
E) is soft paracompact .
(UE .
EE .
E) is soft S-paracompact Proof.
OeIe .
: Let (UE .
EE .
E) represent a soft nearly paracompact space, and let (F.
CE) = {C .
OO I} denote a soft open cover for its soft semi-regularization space (UE .
EES .
E).
For each set C , there exists an index set J such that C = OOJ W , where W is soft regularly open in UuE for every OO J .
Let U = {Ur .
r OO I O } be a soft regularly open and soft locally finite refinement of the collection {W .
S .
OO I}.
For any r OO J , there exists an r OO I such that Ur OI W OI OOJ W OI C .
Since the soft regularly open envelope of any soft neighborhood disjoint with almost all Ur also has this property, the family U can be accepted as the soft locally finite refinement of (F.
CE) is the space(UE .
EES .
E) that is (UE .
EES .
E) is soft paracompact.
Conversely (UE .
EES .
E) be the soft paracompact and consequently a soft nearly paracompact and let (F.
CE)O be a soft regular open cover of UuE .
Since (F.
CE)O is also a soft regular open cover of (UE .
EES .
E) is being soft regular open in UuE .
It has a soft, regularly open, soft, locally finite refinement U O by Theorem 1 of .
Consequently.
U O is soft regularly open, soft locally finite refinement of (F.
CE)O in UuE .
So (UE .
EE .
E) verify the necessary and sufficient conditions for being soft nearly paracompact.
OeIe .
: The proof is obvious as every soft open refinement is a soft semi-open n SOFT S-PARACOMPACT SPACE This section introduces the notion of a soft S-paracompact space and examines the fundamental characteristics of soft S-paracompact spaces.
We define soft semi-generalized closed set.
g-close.
, soft -open set, soft -closed set, soft s -open set,soft s -closed set in soft topological space.
Definition 4.
Consider a soft topological space denoted by (UE .
EE .
E).
Within this space, let (F.
AE) represent a soft subset.
This subset (F.
AE) is defined as a soft S-paracompact space within (UE .
EE .
E) if any cover of (F.
AE) composed of soft open subsets of (UE .
EE .
E) possesses a locally finite soft semi-open refinement in (UE .
EE .
E).
From .
, we recall soft g-closed set.
Definition 4.
A soft set (F.
AE) is said to be soft generalized closed set .
oft g-close.
in a soft topological space (UE .
EE .
E) if Cl(F.
AE) OC (F.
BE), whenever (F.
AE) OC (F.
BE) and (F.
BE) OC (UE .
EE .
E).
This allows us to define the following terms.
Definition 4.
A soft set (F.
AE) is said to be soft semi-generalized closed set .
oft sg-close.
in a soft topological space (UE .
EE .
E) if Cls (F.
AE) OC (F.
BE), whenever (F.
AE) OC (F.
BE) and (F.
BE) OC SSO(UE .
EE .
E).
Theorem 4.
Every soft g-closed subset of a soft S-paracompact space is soft S-paracompact space.
Proof.
Consider a soft S-paracompact space (UE .
EE .
E) and an element ex OO UuE .
Suppose (F.
AE) is a soft g-closed subset of this space.
Consider a collection of soft open subsets (F.
CE) = {C .
OO I} in UuE that covers (F.
AE), such that (F.
AE) OI {C .
OO I}.
As (F.
AE) is soft g-closed, we have Cl(F.
AE) OI {C .
OO I}.
For any ex in UuE not in Cl(F.
AE), there exists a soft open set (F.
WEex ) in UuE where (F.
AE) O (F.
WEex ) = i.
We can now define (F.
OE) = {C .
OO I} {WEex .
ex OO Cl(F.
AE)}, which forms a soft open cover of (UE .
EE .
E).
Let (F.
HE) = {H .
(F.
BE)} be a soft locally finite soft semi-open refinement of (F.
OE).
For each OO (F.
BE), either H OI C() for some () OO I or H OI Wex () for some ex () OO I.
We can then define (F.
BE)A = { OO (F.
BE).
H OI C() }.
Thus, (F.
HE)A = {H .
(F.
BE)A } is a soft locally finite soft semi-open refinement of (F.
CE) and (F.
AE) OI {H .
OO (F.
BE)}.
As a result, (F.
AE) is classified as a soft S-paracompact n Theorem 4.
Every soft open subset of a soft S-paracompact space of (UE .
EE .
E) is soft S-paracompact.
Proof.
Consider a soft S-paracompact space (UE .
EE .
E) and a soft open subset (F.
AE) within it.
Suppose (F.
CE) = {C .
OO I} is a cover of (F.
AE) consisting of soft open subsets of the subspace (Uus .
EEy .
AE).
Given that (F.
AE) is soft open and (F.
CE) covers it with soft open subsets of (UE .
EE .
E), there exists a soft locally finite soft semi-open refinement (F.
WE ) in (UE .
EE .
E).
Consequently, (F.
WEAE ) = {(F.
W ) O (F.
AE).
W OO (F.
WE )} forms a soft locally finite soft semi-open refinement of (F.
CE) within (Uus .
EEy .
AE).
n Theorem 4.
Let (UE .
EE .
E) be the soft topological space and (F.
AE) is the soft clopen subspace of a soft space (UE .
EE .
E).
Then (F.
AE) is soft S-paracompact space if and only if it is soft S-paracompact.
Proof.
To establish the necessary condition, consider a soft S-paracompact space (UE .
EE .
E) and a soft open subset (F.
AE) within it.
Let (F.
CE) = {C .
OO I} be a cover of (F.
AE) consisting of soft open subsets of the subspace (Uus .
EEy .
AE).
Given that (F.
AE) is soft open and (F.
CE) covers it with soft open subsets of (UE .
EE .
E), there exists a soft locally finite soft semi-open refinement (F.
WE ) in (UE .
EE .
E).
Consequently, (F.
WEAE ) = {(F.
W ) O (F.
AE).
W OO (F.
WE )} forms a soft locally finite soft semi-open refinement of (F.
CE) in (Uus .
EEy .
AE).
This demonstrates that (F.
AE) is soft S-paracompact.
We will now establish a sufficient condition.
Consider (F.
CE) = {C .
OO I} as the cover of (F.
AE) by the soft open subset of (UE .
EE .
E).
Consequently, (F.
CE)A = {(F.
AE) O C .
OO I} forms a soft open cover of the soft S-paracompact subspace (Uus .
EEy .
AE), which possesses a soft locally finite soft semi-open refinement (F.
WE ) in (Uus .
EEy .
AE).
According to Lemma 3.
, for every W OO (F.
WE ).
W OO SSO(UE .
EE .
E).
Our objective is to demonstrate that (F.
WE ) is soft locally finite in (UE .
EE .
E).
Let ex OO UuE .
If ex OO (F.
AE), there exists (F.
OEex ) OO EEA OI EE containing ex such that (F.
OEex ) intersects with at most a finite number of members of (F.
WE ).
Alternatively, (F.
AE)c is a soft open set in (UE .
EE .
E) containing ex that does not intersect with any member of (F.
WE ).
Thus, (F.
WE ) is locally finite in (UE .
EE .
E) such that (F.
AE) OC {W .
W OO (F.
WE )}.
Therefore (F.
AE) is soft S-paracompact.
n Corollary 4.
Each soft clopen subspace of soft S-paracompact space is soft Sparacompact.
Definition 4.
Let (F.
AE) be the soft subset of of a soft space (UE .
EE .
E) is called soft -open if for each ex OO (F.
AE), there exist on soft open subset (F.
OE) of (UE .
EE .
E) such that ex OO (F.
OE) OI Cl(F.
OE) OI (F.
AE).
The complement of soft -open set is called soft -closed set.
Definition 4.
Let (F.
AE) be the soft subset of of a soft space (UE .
EE .
E) is called soft s -open if for each ex OO (F.
AE), there exist on soft open subset (F.
OE) of (UE .
EE .
E) such that ex OO (F.
OE) OI Cls (F.
OE) OI (F.
AE).
The complement of soft s -open set is called soft s -closed set.
Note 4.
Soft s -closed Ne soft closed Ne soft g-closed.
Theorem 4.
Consider (UE .
EE .
E) is the soft T2 space and (F.
AE) is the soft Sparacompact space, then (F.
AE) is soft s - closed.
Proof.
Consider (UE .
EE .
E) as a soft T2 space and (F.
AE) as a soft S-paracompact Assume ex OO / (F.
AE) and ey OO (F.
AE).
There exists a soft open set (F.
Bey ) where ey OO (F.
Bey ) and ex OO / (F.
AE).
The collection of such open sets (F.
BE) = {(F.
Bey ).
ey OO (F.
AE)} forms a soft open cover of the soft S-paracompact subset (F.
AE) of UuE .
Let (F.
CE) be the soft locally finite soft semi open refinement in (F.
BE) of (UE .
EE .
E).
Define (F.
D) = O{C.
C OO (F.
CE)} and (F.
DE)c = UuE Oe Cl(FE .
DE).
Then (F.
DE) OO SSO(UE .
EE .
E) and (F.
DE)c OO EE .
Additionally, ex OO (F.
DE)c OI Cls (F.
DE)c OI UuE Oe (F.
AE), demonstrating that UuE Oe (F.
AE) is soft s -open.
Hence (F.
AE) is soft s -open.
n Corollary 4.
Let (UE .
EE .
E) be the soft S-paracompact T2 space and (F.
AE) be the soft subset of UuE .
Then, the following are equivalent:
(F.
AE) is soft S-paracompact space, .
(F.
AE) is the soft s -closed space, .
(F.
AE) is soft closed, .
(F.
AE) is soft g-closed.
Proposition 4.
Consider (F.
AE) as any soft sg-closed subspace within (UE .
EE .
E), and (F.
BE) as any soft subset of UuE .
Suppose (F.
AE) is a soft S-paracompact space and (F.
AE) OI (F.
BE) OI Cls (F.
AE).
Under these conditions, (F.
BE) will also be a soft S-paracompact space in (UE .
EE .
E).
Proposition 4.
Consider two soft subsets (F.
AE) and (F.
BE) of (UE .
EE .
E), where (F.
AE) is contained within (F.
BE), and (F.
BE) is a soft open set.
Under these conditions, (F.
AE) is a soft S-paracompact space in (UES .
EEB .
BE) if and only if it is also a soft S-paracompact space in (UE .
EE .
E).
SUM AND PRODUCT OF SOFT S-PARACOMPACT SPACE
From .
, 27, .
, we recall the definition of sum, product, and mappings of soft topological space.
Definition 5.
Let { (Uu .
EE .
E) : OO I } be a family of pairwise disjoint soft topological space and UE = OnOOI Uu .
Then the collection EE = {(F.
E)over OnOOI Uu :
Uu O (F.
E) = .
F .
) O Uu : e OO E} is a soft open set in (Uu .
EE .
E) for every OO I} defines a soft topology on UE with a fixed set of parameters E.
Then the soft topological space (UE .
EE .
E)is said to be sum of soft topological space and it is denoted by (OiOOI Uu .
EE .
E).
Theorem 5.
Let (UE .
EE .
E) be the soft topological space.
Then sum of soft topological space (OiOOI Uu .
EE .
E) is soft S-paracompact space if and only if (Uu .
EE .
E) is soft S-paracompact for each OO I.
Proof.
To prove necessary condition, let (OiOOI Uu .
EE .
E) is soft S-paracompact From .
, we have, all soft sets UuE are soft clopen in (OiOOI Uu .
EE .
E).
Since (Uu .
EE .
E) is a soft clopen subspace of (OiI Uu .
EE .
E), it follows from Corollary 4.
that (Uu .
EE .
E) is also a soft S-paracompact space.
To prove sufficient condition, let (F.
CE) = {C .
OO I} be a soft open cover of OiOOI Uu .
For every OO I, the collection C = {V O Uu : V OO (F.
CE)} forms a soft open cover of the soft S-paracompact space (Uu .
EE .
E).
Consequently.
C possesses a locally finite soft semi-open refinement W within (Uu .
EE .
E).
Define WE = oI W .
Evidently.
WE constitutes a locally finite soft semi-open refinement of (F.
CE) such that W OO SSO(UE .
EE .
E) for every W OO WE .
Hence.
OiOOI Uu exhibits soft S-paracompactness.
n We recall from .
, the function f : (UE .
EE .
E) OeIe (VE .
EE.
E) is said to be soft irresoluteness if inverse image of every soft semi-open set is soft semi-open.
Every soft continuous open surjective function is soft irresolute.
Theorem 5.
Let (UE .
EE .
E) be compact and (VE .
EE .
E) be the soft S-paracompact space then product of (UE .
EE .
E) y (VE .
EE.
E) is soft S-paracompact space.
Proof.
Consider (F.
CE) = {C : OO I} as a soft open cover for the product soft topological space (UE .
EE .
E) y (VE .
EE.
E).
This implies that (F.
CE) also serves as a soft open cover for the soft compact subspace AVEOe1 .
= UE y .
for each v OO VE , where AVE denotes the natural projection from (UE .
EE .
E) y (VE .
EE.
E) onto (VE .
EE.
E).
There exists a finite subset I.
of I such that AVEOe1 .
OI OOI.
C = Cv , and Cv is soft open.
Given that AVE is a soft closed function, for each v OO VE , we can identify a soft open subset Dv of VE such that v is an element of Dv and AVEOe1 (Dv ) OI Cv .
Consequently, the collection DE = {Dv .
v OO VE } forms a soft open cover of the soft S-paracompact space (VE .
EE.
E), which therefore possesses a locally finite soft semiopen refinement, denoted as WE = {W .
OO I}.
Since AVE is both soft continuous and soft irresolute, the family AVEOe1 (WE ) = {AVEOe1 (W ).
OO I} constitutes a soft semiopen locally finite cover of (UE .
EE .
E)y(VE .
EE.
E), such that for each OO I.
AVEOe1 (W ) = Cv for some v OO VE .
Ultimately, the collection {AVEOe1 (W ) O Cv .
v OO VE .
OO I} provides a locally finite soft semi-open refinement of (F.
CE), where AVEOe1 (W ) O Cv = {AVEOe1 (W ) O C.
: OO I, .
OO I.
Thus, (UE .
EE .
E) y (VE .
EE.
E) is confirmed to be a soft S-paracompact space.
n ADVANTAGE & LIMITATIONS OF SOFT S-PARACOMPACTNESS In this section, we examine the characteristics of soft S-paracompactness.
In soft topology, soft paracompactness is essential for broadening the applicability of compact-like attributes to a wider range of soft topological spaces.
Soft Sparacompactness, which pertains to soft open covers with locally finite semi-open refinements rather than entirely open ones, presents a more flexible alternative to soft paracompactness.
This comparative analysis will highlight the connections between S-paracompactness and recognized soft topological properties and their advantages and limitations.
Advantage of soft S-paracompactness A A condition less strict than soft paracompactness: Soft S- paracompactness represents a less stringent criterion than full soft paracompactness, allowing for its application to a broader spectrum of soft topological spaces.
A Useful in generalizing theorems: The enhanced adaptability of soft semi-open sets over soft open sets enables the extension of results applicable to soft paracompact spaces to a wider variety of soft topological spaces under the framework of soft S-paracompactness.
A Applications in the field of analysis and functional spaces:
The notion of soft semi-open covers proves beneficial in examining functional spaces, particularly in scenarios where full soft paracompactness may be excessively stringent.
A Bridges the gap between soft compactness and soft paracompactness: It offers a middle ground for spaces that do not fully meet the criteria for soft paracompactness.
Limitations of soft S-paracompactness The main drawback of soft S-paracompactness is that it does not inherently ensure normality, unlike paracompactness, which offers a stronger separation property.
Moreover, since its refinements rely on soft semi-open sets rather than soft open sets, fundamental results like partitions of unity may not always apply.
This weaker nature also reduces its effectiveness in establishing deeper soft topological results.
SOFT S-PARACOMPACT SPACE IN DECISION MAKING
This section presents an application of soft S-paracompact space in decisionmaking.
Molodtsov .
presented some applications of soft set theory in several directions: studies of smoothness function, game theory, operation research.
Riemann integration.
Perron integration, probability, theory of measurement, and other wellknown theories.
Maji .
applied the soft set theory to solve a decision-making problem using rough mathematics.
Mareay .
gives a decision-making application of the theory of soft set.
Atef .
investigated covering soft and rough sets and their topological properties with application, and Sanjitha and Baiju .
proposed ordered weighted aggregation operators on multiple sets with their application on decision-making problems.
Choosing from the available options can become tedious in many real-world The selection process is complicated because it must consider all characteristics and factors.
This situation often arises during the selection of cricket Soft set theory appears to be an essential tool for developing a framework that accommodates the vague evaluations involved in the selection process.
Here, we apply soft S-paracompactness in a decision-making problem using a rough approach.
Algorithm.
Step 1: Define a soft set denoted as (F.
E) within the universe UE .
The input parameters AE and BE serve as choice parameters for the selectors XE and YE , respectively, both of which are subsets of E.
Step 2: The evaluation of the playerAos skills differs based on the selection of parameters chosen by each selector represented as (F.
AE), (F.
BE) OI (F.
E).
Step 3: Consider alternative selection of various type of selectors represented as (F.
Ei ) OI (F.
E).
Step 4: Compute the soft topological space EE = {(F.
Ei ) : (F.
Ei ) OI (F.
E).
OAi = 1, 2, 3, .
Step 5: Create a soft open cover that has all the playerAos skills and integrates the selectorAos choices.
Step 6: The choice value of an objective hi OO UE is si , given by where hi represents players and hij are the entries in the table of the players with skills.
Step 7: Construct weighted-table .
i ) from the weightage of the attributes .
i ) given by the selectors in such a way that where tij = wi y hij .
Step 8: Select the players with the maximum choice value from a pool of aggregate We will get sk = max.
i ), then hk Aos are the optimal choices of the respected selectors.
Step 9: Determine the result based on the decisions made by both selectors XE and YE .
A Illustration:
Let UE = .
, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, .
, .
represent 20 cricket players who are waiting for their selection in a cricket team, and let E={ Better batting average.
Good strike rate.
All-rounder.
Best fielder.
Good bowling econom.
denotes the set of parameters.
Consider the soft set (F.
E) that outlines the AuskillsAy of these cricket players, defined as (F.
E) = { Better batting average.
1 ) = .
, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, .
Good strike rate.
2 )=.
, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, .
All rounder.
3 )=.
, 5, 7, 8, 10, 12, 13, 14, 15, 16, .
Best fielder.
4 )=.
, 4, 6, 8, 10, 12, .
Good bowling economy .
5 )=.
, 8, 12, 14, 16, 17, 18, 19, .
Suppose selector XE wants to choose players based on his preferred parameters, which include Aobetter batting averageAo.
Aogood strike rateAo.
Aoall rounderAo, and Aogood bowling economyAo, forming the subset AE = {Better batting average.
Good strike rate.
All rounder.
Good bowling econom.
of the set E.
This meant that from the available players in UE , the selector selected the best ones who meet all the parameters in set AE.
Consider another selector.
YE , aims to pick players based on his set of parameters, which includes Aobetter batting averageAo.
Aobest fieldersAo.
Aoall rounderAo, and Aogood bowling economyAo.
These parameters form the subset BE = {Better batting average.
Best fielders.
All rounder.
Good bowling econom.
of the set E.
The challenge is identifying the most appropriate players using XE and YE selection parameters.
Players deemed the best by XE may not necessarily be the top choice for YE , as each selectorAos decision is influenced by their specific set of parameters.
A Consider the reduct of a soft set (F.
E), which represents alternative selections of various selectors.
Those are as follows:
(F.
E1 ) = { .
1 ,.
, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, .
)}.
(F.
E2 ) = { .
2 ,.
, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, .
)}.
(F.
E3 ) = { .
3 ,.
, 5, 7, 8, 10, 12, 13, 14, 15, 16, .
)}.
(F.
E4 ) = { .
, 4, 6, 8, 10, 12, .
)}.
(F.
E5 ) = { .
5 ,.
, 8, 12, 14, 16, 17, 18, 19, .
)}.
(F.
E6 ) = { .
1 ,.
, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, .
), .
2 ,.
, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, .
)}.
(F.
E7 ) = { .
1 ,.
, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, .
),.
3 ,.
, 5, 7, 8, 10, 12, 13, 14, 15, 16, .
)}.
(F.
E8 ) = {.
2 ,.
, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, .
), .
3 ,.
, 5, 7, 8, 10, 12, 13, 14, 15, 16, .
)}.
(F.
E9 ) = { .
1 ,.
, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, .
), .
2 ,.
, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, .
), .
3 ,.
, 5, 7, 8, 10, 12, 13, 14, 15, 16, .
)}.
(F.
E10 ) = { .
1 ,.
, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, .
), .
4 ,.
, 4, 6, 8, 10, 12, .
)}.
(F.
E11 ) = {.
2 ,.
, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, .
), .
4 ,.
, 4, 6, 8, 10, 12, .
)}.
(F.
E12 ) = { .
1 ,.
, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, .
), .
2 ,.
, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, .
), .
4 ,.
, 4, 6, 8, 10, 12, .
)}.
(F.
E13 ) = { .
3 ,.
, 5, 7, 8, 10, 12, 13, 14, 15, 16, .
), .
, 4, 6, 8, 10, 12, .
)}.
(F.
E14 ) = { .
1 ,.
, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, .
), .
3 ,.
, 5, 7, 8, 10, 12, 13, 14, 15, 16, .
), .
4 ,.
, 4, 6, 8, 10, 12, .
)}.
(F.
E15 ) = {.
2 ,.
, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, .
), .
3 ,.
, 5, 7, 8, 10, 12, 13, 14, 15, 16, .
), .
4 ,.
, 4, 6, 8, 10, 12, .
)}.
(F.
E16 ) = { .
1 ,.
, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, .
), .
2 ,.
, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, .
), .
3 ,.
, 5, 7, 8, 10, 12, 13, 14, 15, 16, .
), .
4 ,.
, 4, 6, 8, 10, 12, .
)}.
(F.
E17 ) = { .
1 ,.
, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, .
), .
5 ,.
, 8, 12, 14, 16, 17, 18, 19, .
)}.
(F.
E18 ) = {.
2 ,.
, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, .
), .
5 ,.
, 8, 12, 14, 16, 17, 18, 19, .
)}.
(F.
E19 ) = { .
1 ,.
, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, .
), .
2 ,.
, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, .
), .
5 ,.
, 8, 12, 14, 16, 17, 18, 19, .
)}.
(F.
E20 ) = { .
3 ,.
, 5, 7, 8, 10, 12, 13, 14, 15, 16, .
), .
5 ,.
, 8, 12, 14, 16, 17, 18, 19, .
)}.
(F.
E21 ) = { .
1 ,.
, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, .
),.
3 ,.
, 5, 7, 8, 10, 12, 13, 14, 15, 16, .
), .
5 ,.
, 8, 12, 14, 16, 17, 18, 19, .
)}.
(F.
E22 ) = {.
2 ,.
, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, .
), .
3 ,.
, 5, 7, 8, 10, 12, 13, 14, 15, 16, .
), .
5 ,.
, 8, 12, 14, 16, 17, 18, 19, .
)}.
(F.
E23 ) = { .
1 ,.
, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, .
), .
2 ,.
, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, .
), .
3 ,.
, 5, 7, 8, 10, 12, 13, 14, 15, 16, .
), .
5 ,.
, 8, 12, 14, 16, 17, 18, 19, .
)}.
(F.
E24 ) = { .
4 ,.
, 4, 6, 8, 10, 12, .
), .
5 ,.
, 8, 12, 14, 16, 17, 18, 19, .
)}.
(F.
E24 ) = { .
1 ,.
, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, .
), .
4 ,.
, 4, 6, 8, 10, 12, .
), .
5 ,.
, 8, 12, 14, 16, 17, 18, 19, .
)}.
(F.
E25 ) = { .
2 ,.
, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, .
), .
4 ,.
, 4, 6, 8, 10, 12, .
), .
5 ,.
, 8, 12, 14, 16, 17, 18, 19, .
)}.
(F.
E26 ) = { .
1 ,.
, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, .
), .
2 ,.
, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, .
), .
4 ,.
, 4, 6, 8, 10, 12, .
), .
5 ,.
, 8, 12, 14, 16, 17, 18, 19, .
)}.
(F.
E27 ) = { .
3 ,.
, 5, 7, 8, 10, 12, 13, 14, 15, 16, .
), .
4 ,.
, 4, 6, 8, 10, 12, .
), .
5 ,.
, 8, 12, 14, 16, 17, 18, 19, .
)}.
(F.
E28 ) = { .
1 ,.
, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, .
),.
3 ,.
, 5, 7, 8, 10, 12, 13, 14, 15, 16, .
), .
4 ,.
, 4, 6, 8, 10, 12, .
), .
5 ,.
, 8, 12, 14, 16, 17, 18, 19, .
)}.
(F.
E29 ) = { .
2 ,.
, 2, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, .
),.
3 ,.
, 5, 7, 8, 10, 12, 13, 14, 15, 16, .
), .
4 ,.
, 4, 6, 8, 10, 12, .
), .
5 ,.
, 8, 12, 14, 16, 17, 18, 19, .
)}.
A Here, we define soft topology EE .
EE = {(F, .
, (F.
E), (F.
AE), (F.
BE), (F.
Eu1 ), (F.
Eu2 ), (F.
Eu3 ), .
, (F.
Eu29 )} where, (F.
AE) and (F.
BE) are the selected parameters of the selectors XE, and YE .
A Let us consider a soft open cover (F.
P ) defined as (F.
P ) = {(F.
AE), (F.
BE)}.
A Now, we must find out the locally finite soft semi-open refinement of (F.
P ).
We obtain the outstanding 11 players from the XE selection and YE .
A Tabular representation of soft set Lin.
and Yao .
previously introduced a tabular format for presenting soft sets.
We offer a similar representation using a binary table.
To do this, consider the soft set (F.
E) based on the parameters E.
This soft set can be depicted below.
Such a representation is advantageous for storing a soft set in computer memory.
If hi OO F .
, where hi denotes the players numbers .
= 1, 2, .
, .
, then the playerAos skill is indicated by hij = 1.
otherwise, it is hij = 0.
Table 1: Professional Skill Evaluation of Players UE A Choice value of an objective hi The choice P5 value of an objective hi OO UE is si OAi = 1, 2, 3, .
, 20, given by si = j=1 hij where hij are the entries in the table of the players with From Table 1.
Table 2: Choice Value of Players UE j=1 hij A Weighted-table choice of an objective The weightage choice value of an object hi OO UE is si OAi = 1, 2, 3, .
, 20,
si = j=1 tij where tij = wi y hij .
A Imposing weights on the choice of selector XE.
Suppose selector XE sets the following weights for the parameters AE: for the parameter Aobetter batting averageAo w1 = 0.
Aogood strike rateAo w2 = 0.
Aoall rounderAo w3 = 0.
6, and Aogood bowling economyAo w5 = 0.
Table 3: Choice Value si and Weight-Table .
ij ) for The Selector UE e1 e2 e3 e5 si = j=1 hij si = j=1 tij h1 1 1 0 0 h2 1 1 1 0 h3 1 0 0 0 h4 1 1 0 0 h5 1 1 1 0 h6 1 1 0 1 h7 0 1 1 0 h8 1 1 1 1 h9 0 1 0 0 h10 1 1 1 0 h11 1 0 0 0 h12 1 1 1 1 h13 1 1 1 0 h14 0 1 1 1 h15 1 1 1 0 h16 0 1 1 1 h17 0 0 0 1 h18 0 0 0 1 h19 0 0 0 1 h20 0 1 1 1 From the Table 3, max.
i ) = s2 , s5 , s6 , s8 , s10 , s12 , s13 , s14 , s15 , s16 , and s20 .
Decision: The selector XE can choose 11 players, specifically h2 , h5 , h6 , h8 , h10 , h12 , h13 , h14 , h15 , h16 , and h20 .
A Now, imposing weights on the choice of selector YE .
Suppose selector YE sets the following weights for the parameters BE: for the parameter Aobetter batting averageAo w1 = 0.
Aoall rounderAo w3 = 0.
Aobest fieldersAo w4 = 0.
5 and Aogood bowling economyAo w5 = 0.
Table 4: Choice Value .
i ) and Weight-Table.
ij ) for Selector YE j=1 hij j=1 tij From the Table 4, max.
i ) = s2 , s5 , s6 , s8 , s10 , s12 , s13 , s14 , s15 , s16 and s20 .
Decision: The selector YE can choose 11 players, specifically h2 , h5 , h6 , h8 , h10 , h12 , h13 , h14 , h15 , h16 and h20 .
A The selectors XE and YE chose 11 players according to the criteria AE and BE.
The players finally chosen are: h2 , h5 , h6 , h8 , h10 , h12 , h13 , h14 , h15 , h16 , and h20 .
CONCLUSION
This paper investigates some characteristics of soft semi-locally finite sets, soft semi-refinement sets, soft s-expandable spaces, and soft extremely disconnected spaces in soft topological spaces.
We extend and generalize existing topological notions using the parameterized family of topological spaces induced by the soft Additionally, we introduce soft S-paracompact spaces, a generalization of soft paracompact spaces, and explore their characteristics.
Furthermore, we investigate concepts such as soft sg-closed sets, soft -open sets, soft -closed sets, and soft s -open sets, and soft s -closed sets in the context of soft topological These contributions expand the theoretical foundations of soft topology and provide a solid framework for future research.
We investigated a comparative analysis of soft S-paracompactness with concepts like soft compact and soft paracompactness in soft topological spaces.
apply soft S-paracompactness in decision-making, establishing a robust foundation for future investigations in soft topological spaces.
REFERENCES