About Neutrosophic Countably Compactness

2020·0.29 MB·English

by  Murad Arar

Neutrosophic Sets and Systems, Vol. 36,2020
University of New Mexico
About Neutrosophic Countably Comapctness
Murad Arar ∗
Prince Sattam Bin Abdulaziz University ; [email protected]
∗Correspondence: [email protected]
Abstract. We answer the following question: Are neutrosophic µ-compactness and neutrosophic µ-countably
compactness equivalent? which posted in [10]. Since every neutrosophic topology is neutrosophic µ-topology,
weanswerthequestionforneutrosophictopologicalspaces,moreprecisely,wegiveanexampleofneutrosophic
topology which is neutrosophic countably comapact but not neutrosophic compact
Keywords: Neutrosophic topological spaces; Neutrosophic compact; Neutrosophic Lindelo¨f; Neutrosophic
countably compact space
—————————————————————————————————————————-
1. Introduction
Neutrosophic sets first introduced in [25,27] as a generalization of intuitionistic fuzzy
sets [14], where each element x ∈ X has a degree of indeterminacy with the degree of mem-
bership and the degree of non-membership . Operations on neutrosophic sets are investigated
after that. Neutrosophic topological spaces are studied by Smarandache [27], Lupianez [19,20]
and Salama [23]. The interior , closure, exterior and boundary of neutrosophic sets can be
found in [26]. Neutrosophic sets applied to generalize many notaions about soft topology and
applications [18], [22], [15], generalized open and closed sets [28] , fixed point theorems [18] ,
graphtheory[17]androughtopologyandapplications[21]. Neutrosophyhasmanyapplications
specially in decision making, for more details about new trends of neutrosophic applications
one can consult [1]- [7].
Generalized topology and continuity introduced in 2002 in [13], where many generalized open
sets in general topology become examples in generalized topological spaces, and it become one
of the most important generalization in topology which has different properties than general
topology, see for example [9], [11] and [12]. There are a lot of studies about neutrosophic
topological spaces that shows the importance of studying neutrosophic topology where it has
Murad Arar, About Neutrosophic Countably Comapctness
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possible applications, see for example [24], Neutrosophic µ-topological spaces first introduced
in [10], and since Neutrosophic µ-topological space is a generalization of neutrosophic topo-
logical space it guarantees generalized results that are still true for neutrosophic topological
spaces,seeforexampleTheorem2.30in[10]whichshowsthatneutrosophicµ-compactnessand
neutrosophic µ-countably compactness are equivalent, and this is not true in crisp topology,
but it becomes true for neutrosophic topological spaces since every neutrosophic µ-topological
space is neutrosophic topological space, another thing about the importance of neutrosophic
µ-topological space is that some existing notations about neutrosophic topology can be con-
sidered as examples of neurosophic µ-topological spaces, see for example Theorem 2.9 in [10]
which shows the relationship between µ-topological space and previous studies where we can
consider all neutrosophic α-open sets over (X; τ ) and all neutrosophic pre-closed sets in (X;
τ ) (introduced in [8]) as examples of strong neutrosophic µ-topology over X. The following
question appeared in [10].
Definition 1.1. [25]: A set A is said neutrosophic on X if A = {(cid:104)x,µ (x),σ (x),ν (x)(cid:105);x ∈
A A A
X}; µ,σ,ν : X →]−0,1+[ and −0 ≤ µ(x)+σ(x)+ν(x) ≤ 3+.
The class of all neutrosophic set on the universe X is by N(X). We will exhibit the
basic neutrosophic operations definitions (union, intersection and complement. Since there
are different definitions of neutrosophic operations, we will organize the existing definitions
into two types, in each type these operation will be consistent and functional.
Definition 1.2. [24][Neutrosophic sets operations] Let A,A ,B ∈ N(X) such that α ∈ ∆.
α
Then we define the neutrsophic:
(1) (Inclusion): A (cid:118) B If µ (x) ≤ µ (x), σ (x) ≥ σ (x) and ν (x) ≥ ν (x).
A B A B A B
(2) (Equality): A = B ⇔ A (cid:118) B∧B (cid:118) A.
(3) (Intersection) (cid:117) A (x) = {(cid:104)x, ∧ µ (x), ∨ σ (x), ∨ ν (x)(cid:105);x ∈ X}.
α∈∆ α α∈∆ Aα α∈∆ A α∈∆ A
(4) (Union) (cid:116) A (x) = {(cid:104)x, ∨ µ (x), ∧ σ (x), ∧ ν (x)(cid:105);x ∈ X}.
α∈∆ α α∈∆ Aα α∈∆ A α∈∆ A
(5) (Complement) Ac = {(cid:104)x,ν (x),1−σ (x),µ (x)(cid:105);x ∈ X}
A A A
(6) (Universal set) 1 = {(cid:104)x,1,0,0(cid:105);x ∈ X}; will be called the neutrosophic universal set.
X
(7) (Empty set) 0 = {(cid:104)x,0,1,1(cid:105);x ∈ X}; will be called the neutrosophic empty set.
X
Proposition 1.3. [24] For A,A ∈ N(X) for every α ∈ ∆ we have:
α
(1) A(cid:117)( (cid:116) A ) = (cid:116) (A(cid:117)A ).
α α
α∈∆ α∈∆
(2) A(cid:116)( (cid:117) A ) = (cid:117) (A(cid:116)A ).
α α
α∈∆ α∈∆
Definition 1.4. [24] [Neutrosophic Topology] τ ⊂ N(X) is called a neutrosophic topology for
X if
Murad Arar, About Neutrosophic Countably Comapctness
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(1) 0 ,1 ∈ τ.
X X
(2) If A ∈ τ for every α ∈ ∆, then (cid:116) A ∈ τ,
α α
α∈∆
(3) For every A,B ∈ τ, we have A,B ∈ τ.
The ordered pair (X,τ) will be said a neutrosophic space over X. The elements of τ will be
called neutrosophic open sets. For any A ∈ N(X), If Ac ∈ τ, then we say A is neutrosophic
closed.
2. Neutrosophic Countably Compact Spaces
Definition 2.1. [10] Let X be nonempty, 0 < α,β,γ < 1. Then A ∈ N(X) is said a
neutrosophic point iff there exists x ∈ X such that A = {(cid:104)x,α,β,γ(cid:105)}∪{(cid:104)x´,0,1,1(cid:105);x´ (cid:54)= x}.
Neutrosophic points will be denoted by x .
α,β,γ
Definition 2.2. [10] We say x in the neutrosophic set A -in symbols x ∈ A- iff
α,β,γ α,β,γ
α < µ (x),β > σ (x) and γ > ν (x).
A A A
Lemma 2.3. [10] Let A ∈ N(X) and suppose that for every x ∈ A there exists
α,β,γ
B(x ) ∈ N(X) such that x ∈ B(x ) (cid:118) A. Then A = (cid:116){B(x );x ∈ A}.
α,β,γ α,β,γ α,β,γ α,β,γ α,β,γ
Corollary 2.4. [10] A ∈ N(X) is neutrosophic open in (X,τ) iff for every x ∈ A there
α,β,γ
exists a neutrosophic set B(x ) ∈ τ; x ∈ B(x ) (cid:118) A.
α,β,γ α,β,γ α,β,γ
Definition 2.5. [10] Let (X,τ) be a neutrosophic topology on X. A sub-collection B ⊆ τ is
calledaneutrosophicbaseforτ ifforanyU ∈ τ thereexistsB´⊆ BsuchthatU = (cid:116){B;B ∈ B´}.
Definition 2.6. [10] Consider the neutrosophic space (X,τ) . We say the collection U from
τ is a neutrosophic open cover of X, if 1 = (cid:116){U;U ∈ U}.
X
Definition 2.7. [10] Consider the space (X,τ) and the neutrosophic open cover U of X.
Then we say the sub-collection U´ ⊆ N(X) is a neutrosophic subcover of X from U, if U´ is
neutrosophic covers X and U´ ⊆ U.
The following is an immediate result of Corollary2.4.
Corollary 2.8. [10] A sub-collection U from the neutrosophic space (X,τ) is an open cover
of X iff for every x in X there exists U ∈ U such that x ∈ U.
α,β,γ α,β,γ
Theorem 2.9. Consider the collection B of neutrosophic sets on the universe X. Then B is
a neutrosophic base for some neutrosophic topology on X iff
(1) For every U ∈ τ and every x ∈ U there exists B ∈ B such that x ∈ B (cid:118) U.
α,β,γ α,β,γ
(2) For every A,B ∈ B we have A(cid:117)B is a union of elements from B.
Murad Arar, About Neutrosophic Countably Comapctness
Neutrosophic Sets and Systems, Vol. 36, 2020 249
Proof. →) Obvious!
←)SupposeBsatisfiesthetwoconditionsinthetheorem. Letτ(B)beallpossibleneutrosophic
unions of elements from B with 0 . It suffices to show that τ(B) is a neutrosophic topology
X
on X. From the first condition and the construction of τ(B) we have 0 ,1 ∈ τ(B). Now let
X X
H,K ∈ τ(B). Then H = (cid:116)H and K = (cid:116)K where H ,K ∈ B for every i,j. So we have (by
i j i j
i j
parts (3) and (4) of Proposition1.3)
H (cid:117)K = ((cid:116)H )(cid:117)((cid:116)K ) = (cid:116)(((cid:116)H )(cid:117)K ) = (cid:116)(cid:116)(H (cid:117)K )
i j i j i j
i j j i j i
Since H ,K ∈ B for every i,j, we have H (cid:117)K ∈ τ(B). The proof that the union of elements
i j
from τ(B) is an element from τ(B) is easy! And we done.
τ(B) will be called the neutrosophic topology generated by the neutrosophic base B on X .
Definition 2.10. [10] (X,τ) is said to be neutrosophic compact if each neutrosophic open
(in τ) cover of X has a finite neutrosophic subcover.
Theorem 2.11. [10] Consider the space (X,τ), and let B be a neutrosophic base for τ. Then
(X,τ) is a neutrosophic compact space iff every neutrosophic open cover of X from B has a
finite neutrosophic subcover.
Definition 2.12. [10] A neutrosophic space (X,τ) is said:
(1) A neutrosophic Lindelo¨f space if each neutrosophic open cover of X from τ has a
countable neutrosophic subcover of X.
(2) A neutrosophic countably compact space if each neutrosophic open countable cover of
X from τ has a finite neutrosophic subcover of X.
The following thee results have proofs similar to their correspondings about neutrosophic
µ-topological spaces in [10].
Theorem 2.13. Every neutrosophic space with a countable neutrosophic base is neutrosophic
Lindelo¨f.
Theorem 2.14. Every neutrosophic Lindelo¨f and countably compact space is compact.
Corollary 2.15. Every neutrosophic countably compact space with a neutrosophic countable
base is neutrosophic compact.
The following example show that neutrosophic Lindelo¨f spaces are not neutrosophic count-
ably compact.
Example 2.16. Let Y = {a,b} and let B = {A ;n = 1,2,3,...} where A = {(cid:104)y,1 −
n n
1 , 1 , 1 (cid:105);y ∈ X}. We will show that B is a base for some neutrosophic topology on Y ;
2n 2n 2n
Murad Arar, About Neutrosophic Countably Comapctness
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i.e. we want to show B satisfies (1) and (2) in Theorem2.9.
First condtion: B neutrosophic covers Y, actually:
(cid:116)B = (cid:116){A ;n = 1,2,3,...} = {(cid:104)y,∨∞1− 1 ,∧∞ 1 ,∧∞ 1 (cid:105);y ∈ Y} = {(cid:104)y,1,0,0(cid:105);y ∈ Y} =
n 1 2n 1 2n 1 2n
1 .
Y
Second condition: The neutrosophic intersection of two elements from B is the neutrosophic
union of elements from B, but is clear that for any A and A in B we have A (cid:117)A = A
n m n m t
where t = max{n,m} which an element of B, so that B is a neutrosophic base form some
neutrosophic topology τ(B) on Y. Since τ(B) has a countable base, τ(B) is neutrosophic
Lindelo¨f. Now, we will show that τ(B) is not neutrosophic countably paracompact (which im-
plies it is not neutrosophic compact). By contrapositive, suppose Y is neutrosophic countably
paracompact. Then U = B is a countable neutrosophic open cover of Y. But Y is a neutro-
sophic countably paracompact space, so that we have U has a neutrosophic finite subcover,
say U∗ = {A ,A ,...,A }. But A (cid:116)A (cid:116)...(cid:116)A = A where t = max{n ,n ,...,n },
n1 n2 nk n1 n2 nk t 1 2 k
and A = {(cid:104)y,1− 1, 1, 1(cid:105);y ∈ Y} =(cid:54) 1 , a contradiction. So Y is not neutrosophic countably
t 2t 2t 2t Y
paracompact and hence it is not neutrosophic compact.
Thefollowingtheoremshowsthatneutrosophiccompactspacesandneutrusophiccountably
compact spaces are equivalent if the universe of discourse is countable, which is not true for
topological spaces.
Theorem 2.17. For every countable neutrosophic topological space Y, the following two state-
ments are equivalent :
(1) Y is neutrosophic compact.
(2) Y is neutrosophic countably compact.
Proof. ⇒) Obvious!
⇐) Suppose that Y is a countable neutrosophic countably compact space, and let U be a
neutrosophic open cover of Y. For every y ∈ Y we define the following three subsets of [0,1].
(1) Dy = {µ (y);A ∈ U}.
µ A
(2) Dy = {σ (y);A ∈ U}.
σ A
(3) Dy = {ν (y);A ∈ U}.
ν A
Let Dy, Dy and Dy be three countable dense subsets of Dy, Dy and Dy respectively in the
1 2 3 µ σ ν
usual sense (the usual topology on the unit interval). Since U is a neutrosophic µ-open cover
of Y, we have supDy = supDy = 1 ,infDy = infDy = 0 and infDy = infDy = 0. Let
1 µ 2 σ 3 ν
U(y) = {A ∈ U;µ (y) ∈ Dy,σ (y) ∈ Dy or ν (y) ∈ Dy}. It is clear that U(y) is countable.
A 1 A 2 A 3
Let U∗ = ∪{U(y);y ∈ Y}. Since Y is countable, U∗ is a countable sub-collection from U. We
will show that U∗ is a neutrosophic cover of Y. Set B = (cid:116)U∗. For every y ∈ Y we have:
(1) µ (y) = ∨{µ (y);A ∈ B} ≥ ∨{µ (y);A ∈ Dy} = supDy = 1.
B A A 1 1
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(2) σ (y) = ∧{σ (y);A ∈ B} ≥ ∧{σ (y);A ∈ Dy} = infDy = 0.
B A A 1 2
(3) ν (y) = ∧{ν (y);A ∈ B} ≥ ∧{∨ (y);A ∈ Dy} = infDy = 0.
B A A 1 3
Which implies that B = 1 and U∗ is a neutrosophic countable open cover. Since Y is a
Y
neutrosophic µ-countably compact space, U∗ has a finite subcover , that is Y is compact.
The following example shows that neutrosophic compactness and neutrosophic countably
compactness are not equivalent.
Example 2.18. Consider the set of all countable ordinals W with the usual ordering. Let
0
β = {[s,t),s,t < ω (the first uncountable ordinal)}. We know that β is a base for some topol-
1
ogy τ on Y = W . For every [s,t) ∈ β define the neutrosophic set
0

(y,1,0,0) if y ∈ [s,t)
A =
[s,t)
(y,0,1,1) if y ∈/ [s,t)
Set β´= {A ;[s,t) ∈ β}. We will show that β´is a base for some neutrosophic topology on Y.
[s,t)
First we show it is a neutrosophic cover for Y. Let A = (cid:116)β ; it suffices to show that A = 1 .
Y
But for every y ∈ Y, we have y ∈ [s,y) for some s < y, so that µ (y) = ∨{µ (y);C ∈ β´} ≥
A C
µ = 1, σ (y) = ∧{σ (y);C ∈ β´} ≤ σ = 0, and ν (y) = ∧{ν (y);C ∈ β´} ≤ ν = 0,
[s,y) A C [s,y) A C [s,y)
that means A = 1 and β covers Y. Now, we will show that the intersection of any two
Y
elements from β is empty or an element of β. Let A and A be two neutrosophic
[s1,t1) [s2,t2)
sets in β and set C = A (cid:117)A , if [s ,t )∩[s ,t ) = ∅, then for every y ∈ Y we have
[s1,t1) [s2,t2) 1 1 2 2
y ∈/ [s ,t ) or y ∈/ [s ,t ), which implies µ = µ ∧µ = 0, σ = σ ∨σ = 1
1 1 2 2 C [s1,t1) [s2,t2) C [s1,t1) [s2,t2)
and ν = ν ∨ ν = 1 and that means A (cid:117) A = 0 . Now, suppose that
C [s1,t1) [s2,t2) [s1,t1) [s2,t2) Y
[s ,t )∩[s ,t ) (cid:54)= ∅. Then for every y < max{s ,s } or y ≥ min{t ,t } we have y ∈/ [s ,t ) or
1 1 2 2 1 2 1 2 1 1
y ∈/ [s ,t ), which means µ = 0, σ = 1 and ν = 1, and if max{s ,s } ≤ y < min{t ,t },
2 2 C C C 1 2 1 2
then y ∈ [s ,t ) and y ∈ [s ,t ), that is µ = 1, σ = 0 and ν = 0 , so that we have
1 1 2 2 C C C

(y,1,0,0) if y ∈ [s,t)
A (cid:117)A = A = ∈ β
[s1,t1) [s2,t2) [s,t)
(y,0,1,1) if y ∈/ [s,t)
where s = max{s ,s } and t = max{t ,t }. Let τ(β) be the neutrosophic topology gen-
1 2 1 2
erated on Y by β. Then τ(β) is a neutrosophic countably compact space: We will prove
this by showing τ(β) has no countable cover form β. Let C = {A = [s ,t );n = 1,2,3,...}
n n n
be any countable subset from β, it suffices to show that C does not cover Y; by contaposi-
∞
tive, suppose C covers Y, then D = (cid:116)C = (cid:116) A = 1 . So that for every y ∈ Y we have
n Y
i=1
Murad Arar, About Neutrosophic Countably Comapctness
Neutrosophic Sets and Systems, Vol. 36, 2020 252
∞
µ = ∨ µ = 1; since µ = 1 or 0 for every n = 1,2,3,..., there exist i such that µ = 1,
C i=1 An An Ai
∞
that is y ∈ A = [s ,t ), which implies Y = ∪ [s ,t ), a contradiction, since Y is uncountable
i i i n n
i=1
∞
and ∪ [s ,t ) is countable, so β has no countable cover for Y, and so Y is neutrosophic count-
n n
i=1
ably compact. Now, to show that Y is not neutrosophic compact. But β is a neutrosophic
open cover of Y and has no countable, and hence no finite, subcover, that means Y is not
neutrosophic compact.
Corollary 2.19. There is a neutrosophic µ−topological spaces which is neutrosophic countably
compact but not neutrosophic compact.
Proof. Since every neutrosophic space is µ−topological space, we have Example2.18 is an
example of a neutrosophic µ−topological spaces which is neutrosophic countably compact but
not neutrosophic compact.
The approach we used in Example2.18 can be generalized to get more counterexample for
neutrosophic topological spaces as follows.
Theorem 2.20. Let (X,τ) be a topological space and for every U ∈ τ set

(x,1,0,0) if x ∈ U
A =
U
(x,0,1,1) if x ∈/ U
and let Neut(τ) = {A ;U ∈ τ}. Then (X,Neut(τ)) is a neutrosophic topological space.
U
Proof. Since ∅,X ∈ τ, we have A ,A ∈ Neut(τ), but
∅ X
 
(x,1,0,0) if x ∈ ∅ (x,1,0,0) if x ∈ ∅
A = = = 0
∅ X
(x,0,1,1) if x ∈/ ∅ (x,0,1,1) if x ∈ X
 
(x,1,0,0) if x ∈ X (x,1,0,0) if x ∈ X
A = = = 1
X X
(x,0,1,1) if x ∈/ X (x,0,1,1) if x ∈ ∅
So we have 0 ,1 ∈ Neut(τ). Now, let H = A (cid:117)A where A ,A ∈ Neut(τ). Then
X X U V U V
  
1 if x ∈ U 1 if x ∈ V 1 if x ∈ U ∩V
µ (x) = ∧ = = µ
H A
(U∩V)(x)
0 if x ∈/ U 0 if x ∈/ V 0 if x ∈/ U ∩V
Murad Arar, About Neutrosophic Countably Comapctness
Neutrosophic Sets and Systems, Vol. 36, 2020 253
  
0 if x ∈ U 0 if x ∈ V 0 ifx ∈ U ∩V
σ (x) = ∨ = = σ (x)
H A
(U∩V)
1 if x ∈/ U 1 if x ∈/ V 1 ifx ∈/ U ∩V
  
0 if x ∈ U 0 if x ∈ V 0 ifx ∈ U ∩V
ν (x) = ∨ = = ν (x)
H A
(U∩V)
1 if x ∈/ U 1 if x ∈/ V 1 ifx ∈/ U ∩V
SowehaveA (cid:117)A = A ∈ Neut(τ). Similarlyweshowthat (cid:116) A ∈ Neut(τ)whenever
U V (U∩H) α
α∈∆
A ∈ Neut(τ) for every α ∈ ∆.
α
3. Applications and further studies
This paper is a completion part of [10] and gives an answer for the following question: Are
neutrosophic µ-compactness and neutrosophic µ-countably compactness equivalent? which
posted in [10]. We give an example to show that the answer is no! the approach is used to
give such example can be generalized to give many counter examples in neutrosophic topology
using those existing in general topology. This paper, also, studied more advanced notations
about neutrosophic topology such as neutrosophic comapactness and neutrosophic Lindelo¨f,
which opens doors for more studies about neutrosophic topology, such as neutrosophic para-
compactness, and other covering properties
Funding: This Project was supported by the Deanship of Scientific research at Prince
Sattam Bin Abdulaziz University under the research project (cid:93) 2019/01/9633.
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Received: June 9, 2020 / Accepted: Aug 16, 2020
Murad Arar, About Neutrosophic Countably Comapctness
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