A degree approach to relationship among fuzzy convex structures, fuzzy closure systems and fuzzy Alexandrov topologies

Definition 2.1

([25]). An M-fuzzifying convex structure on X is a mapping 𝒞 : 2^X ⟶ M which satisfies:

1. 𝒞(∅) = 𝒞(X) = ⊤;

2. 𝒞(⋂_i∈I A_i) ≥ ⋀_i∈Ω 𝒞(A_i);

3. 𝒞( ⋃i∈Id A_i) ≥ ⋀_i∈Ω 𝒞(A_i),

   where ⋃i∈Id A_i means that {A_i : i ∈ I} ⊆d 2^X, i.e., {A_i : i ∈ I} is an up-directed subfamily of 2^X, and ⋃i∈Id A_i = ⋃_i∈I A_i. For an M-fuzzifying convex structure 𝒞 on X, the pair (X, 𝒞) is called an M-fuzzifying convex space.

Definition 2.2

([35]). An M-fuzzifying closure system on X is a mapping 𝓢 : 2^X ⟶ M which satisfies:

1. 𝓢(∅) = 𝓢(X) = ⊤;

2. 𝓢(⋂_i∈I A_i) ⩾ ⋀_i∈I 𝓢(A_i).

For an M-fuzzifying closure system 𝓢 on X, the pair (X, 𝓢) is called an M-fuzzifying closure space.

Definition 2.3

([5, 36]). An M-fuzzifying Alexandrov topology on X is a mapping τ : 2^X ⟶ M which satisfies:

1. τ(∅) = τ(X) = ⊤;

2. τ(⋂_i∈I A_i) ≥ ⋀_i∈I τ(A_i);

3. τ(⋃_i∈I A_i) ≥ ⋀_i∈I τ(A_i).

   For an M-fuzzifying Alexandrov topology τ on X, the pair (X, τ) is called an M-fuzzifying Alexandrov topological space.

Definition 3.1

For each 𝒞 ∈ P_M(2^X), define D^con(X, 𝒞) as follows:

Then D^con(X, 𝒞) is called the approximate degree to which 𝒞 is an M-fuzzifying convex structure on X.

Proposition 3.2

Let {𝒞_t}_t∈T be a family of mappings from 2^X to M and define ⋀_t∈T 𝒞_t : 2^X ⟶ M by

Then D^con(X, ⋀_t∈T 𝒞_t) ≥ ⋀_t∈T D^con(X, 𝒞_t).

Proof

We first verify the following three inequalities.

1. D_⊤(X, ⋀_t∈T 𝒞_t) = ⋀_t∈T D_⊤(X, 𝒞_t). This can be shown by

   D⊤(X,⋀t∈TCt)=(⋀t∈TCt)(X)∧(⋀t∈TCt)(∅)=⋀t∈T(Ct(X)∧Ct(∅))=⋀t∈TD⊤(X,Ct).

2. D_⋂(X, ⋀_t∈T 𝒞_t) ≥ ⋀_t∈T D_⋂(X, 𝒞_t). This can be shown by

   D⋂(X,⋀t∈TCt)=⋀{Ak:k∈K}⊆2X(⋀k∈K(⋀t∈TCt)(Ak)→(⋀t∈TCt)(⋂k∈KAk))=⋀{Ak:k∈K}⊆2X(⋀t∈T⋀k∈KCt(Ak)→⋀t∈TCt(⋂k∈KAk))≥⋀{Ak:k∈K}⊆2X⋀t∈T(⋀k∈KCt(Ak)→Ct(⋂k∈KAk))=⋀t∈T⋀{Ak:k∈K}⊆2X(⋀k∈KCt(Ak)→Ct(⋂k∈KAk))=⋀t∈TD⋂(X,Ct).

3. The proof of D_⋃^d(X, ⋀_t∈T 𝒞_t) ≥ ⋀_t∈T D_⋃^d(X, 𝒞_t) is similar to (2).

   Therefore,

   Dcon(X,⋀t∈TCt)=D⊤(X,⋀t∈TCt)∧D⋂(X,⋀t∈TCt)∧D⋃d(X,⋀t∈TCt)  ≥⋀t∈TD⊤(X,Ct)∧⋀t∈TD⋂(X,Ct)∧⋀t∈TD⋃d(X,Ct)  =⋀t∈T(D⊤(X,Ct)∧D⋂(X,Ct)∧D⋃d(X,Ct))  =⋀t∈TDcon(X,Ct).

   □

Proposition 3.3

Let (Y, 𝒟) be an M-fuzzifying convex space and f : X ⟶ Y be a surjective mapping. Define a mapping f^–1(𝒟) : 2^X ⟶ M by

Then (X, f^–1(𝒟)) is an M-fuzzifying convex space.

Proposition 3.4

Let f : X ⟶ Y be a surjective mapping and 𝒟 ∈ P_M(2^Y). Define f^–1(𝒟) : 2^X ⟶ M by

Then D^con(X, f^–1(𝒟)) ≥ D^con(Y, 𝒟).

Proof

We first verify the following three inequalities.

1. D_⊤(X, f^–1(𝒟)) ≥ D_⊤(Y, 𝒟). This can be shown by

   D⊤(X,f−1(D))=f−1(D)(X)∧f−1(D)(∅)=⋁f−1(B)=XD(B)∧⋁f−1(B)=∅D(B)≥D(Y)∧D(∅)=D⊤(Y,D).

2. D_⋂(X, f^–1(𝒟)) ≥ D_⋂(Y, 𝒟). That is,

   ⋀{Ak:k∈K}⊆2X(⋀k∈Kf−1(D)(Ak)→f−1(D)(⋂k∈KAk))≥⋀{Bk:k∈K}⊆2Y(⋀k∈KD(Bk)→D(⋂k∈KBk)).

   Take any α ∈ M such that

   α≤⋀{Bk:k∈K}⊆2Y(⋀k∈KD(Bk)→D(⋂k∈KBk)).

   Then for each {B_k : k ∈ K} ⊆ 2^Y, it follows that α ∧ ⋀_k∈K 𝒟(B_k) ≤ 𝒟(⋂_k∈K B_k). In order to show

   α≤⋀{Ak:k∈K}⊆2X(⋀k∈Kf−1(D)(Ak)→f−1(D)(⋂k∈KAk)),

   we need only show that for each {A_k : k ∈ K} ⊆ 2^X,

   α∧⋀k∈Kf−1(D)(Ak)≤f−1(D)(⋂k∈KAk),

   i.e.,

   α∧⋀k∈K⋁f−1(B)=AkD(B)≤⋁f−1(B)=⋂k∈KAkD(B).

   Take each β ∈ M such that β ≺ α ∧ ⋀_k∈K ⋁_f^–1(B)=Ak 𝒟(B). Then β ≤ α and for each k ∈ K, there exists B_k ∈ 2^Y such that f^–1(B_k) = A_k and 𝒟(B_k) ≥ β. This implies ⋀_k∈K𝒟(B_k) ≥ β. Put C = ⋂_k∈K B_k. Then

   f−1(C)=f−1(⋂k∈KBk)=⋂k∈Kf−1(Bk)=⋂k∈KAk.

   Further, it follows that

   ⋁f−1(B)=⋂k∈KAkD(B)≥D(C)=D(⋂k∈KBk)≥α∧⋀k∈KD(Bk)≥β.

   By the arbitrariness of β, we obtain that for each {A_k : k ∈ K} ⊆ 2^X,

   α∧⋀k∈Kf−1(D)(Ak)≤f−1(D)(⋂k∈KAk).

   This means that

   α≤⋀{Ak:k∈K}⊆2X(⋀k∈Kf−1(D)(Ak)→f−1(D)(⋂k∈KAk)).

   By the arbitrariness of α, we obtain

   ⋀{Ak:k∈K}⊆2X(⋀k∈Kf−1(D)(Ak)→f−1(D)(⋂k∈KAk))≥⋀{Bk:k∈K}⊆2Y(⋀k∈KD(Bk)→D(⋂k∈KBk)),

   as desired.

3. The proof of D_⋃^d(X, f^–1(𝒟)) ≥ D_⋃^d(Y, 𝒟) is similar to (2).

   As a result, we get

   Dcon(X,f−1(D))=D⊤(X,f−1(D))∧D⋂(X,f−1(D))∧D⋃d(X,f−1(D)) ≥D⊤(Y,D)∧D⋂(Y,D)∧D⋃d(Y,D) =Dcon(Y,D).

   □

Definition 3.5

([25]). Let (X, 𝒞) be an M-fuzzifying convex space, ∅ ≠ Y ⊆ X. Then (Y, 𝒞|_Y) is an M-fuzzifying convex space on Y, where

We call (Y, 𝒞|_Y) a subspace of (X, 𝒞).

Definition 3.6

([25]). Let (X, 𝒞_X) be an M-fuzzifying convex space and let f : X ⟶ Y be a surjective mapping. Define 𝒞_Y : 2^Y ⟶ M by

Then 𝒞_Y is an M-fuzzifying convex structure on Y. The pair (Y, 𝒞_Y) is called the quotient space of (X, 𝒞_X) with respect to f.

Definition 3.7

([28]). Let {(X_i, 𝒞_i)}_i∈I be a family of M-fuzzifying convex spaces, where I = {1, 2, …n}, and let X be the product of {X_i}_i∈I, that is, X = ∏_i∈I X_i. Define ∏i=1n 𝒞_i : 2^X ⟶ M by

Then ∏i=1n 𝒞_i is an M-fuzzifying convex structure on X. The pair (X, ∏i=1n 𝒞_i) is called the product of {(X_i, 𝒞_i)}_i∈I.

Proposition 3.8

Let 𝒞 ∈ P_M(2^X) and ∅ ≠ Y ⊆ X. Define 𝒞|_Y by

Then D^con(Y, 𝒞|_Y) ≥ D^con(X, 𝒞).

Proof

It is enough to prove the following three inequalities.

1. D_⊤(Y, 𝒞|_Y) ≥ D_⊤(X, 𝒞). By the definition of 𝒞|_Y, it follows that

   D⊤(Y,C|Y)=C|Y(Y)∧C|Y(∅) =⋁B∈2X, B∩Y=YC(B)∧⋁B∈2X, B∩Y=∅C(B) ≥C(X)∧C(∅) =D⊤(X,C).

2. D_⋂(Y, 𝒞|_Y) ≥ D_⋂(X, 𝒞). That is,

   ⋀{Ak:k∈K}⊆2Y(⋀k∈KC|Y(Ak)→C|Y(⋂k∈KAk))≥⋀{Bk:k∈K}⊆2X(⋀k∈KC(Bk)→C(⋂k∈KBk)).

   Take any α ∈ M such that

   α≤⋀{Bk:k∈K}⊆2X(⋀k∈KC(Bk)→C(⋂k∈KBk)).

   Then for each {B_k : k ∈ K} ⊆ 2^X, it follows that α ∧ ⋀_k∈K 𝒞(B_k) ≤ 𝒞(⋂_k∈K B_k). In order to show

   α≤⋀{Ak:k∈K}⊆2Y(⋀k∈KC|Y(Ak)→C|Y(⋂k∈KAk)),

   we need only show that for each {A_k : k ∈ K} ⊆ 2^Y,

   α∧⋀k∈KC|Y(Ak)≤C|Y(⋂k∈KAk),

   i.e.,

   α∧⋀k∈K⋁B∈2X,B∩Y=AkC(B)≤⋁B∈2X,B∩Y=⋂k∈KAkC(B).

   Take each β ∈ M such that β ≺ α ∧ ⋀_k∈K ⋁_{B∈2^X, B∩Y=Ak} 𝒞(B). Then β ≤ α and for each k ∈ K, there exists B_k ∈ 2^X such that B_k ∩ Y = A_k and 𝒞(B_k) ≥ β. This implies ⋀_k∈K 𝒞(B_k) ≥ β. Put C = ⋂_k∈K B_k. Then

   C∩Y=(⋂k∈KBk)∩Y=⋂k∈K(Bk∩Y)=⋂k∈KAk.

   Furthermore, it follows that

   ⋁B∈2X,B∩Y=⋂k∈KAkC(B)≥C(C)=C(⋂k∈KBk)≥α∧⋀k∈KC(Bk)≥β.

   By the arbitrariness of β, we obtain that for each {A_k : k ∈ K} ⊆ 2^X, it follows that α ∧ ⋀_k∈K 𝒞|_Y(A_k) ≤ 𝒞|_Y(⋂_k∈K A_k). This means that

   α≤⋀{Ak:k∈K}⊆2X(⋀k∈KC|Y(Ak)→C|Y(⋂k∈KAk)).

   By the arbitrariness of α, we get

   ⋀{Ak:k∈K}⊆2Y(⋀k∈KC|Y(Ak)→C|Y(⋂k∈KAk))≥⋀{Bk:k∈K}⊆2X(⋀k∈KC(Bk)→C(⋂k∈KBk)),

   as desired.

3. The proof of D_⋃^d(Y, 𝒞|_Y) ≥ D_⋃^d(X, 𝒞) is similar to (2).

   As a result, we get

   Dcon(Y,C|Y)=D⊤(Y,C|Y)∧D⋂(Y,C|Y)∧D⋃d(Y,C|Y)≥D⊤(X,C)∧D⋂(X,C)∧D⋃d(X,C)=Dcon(X,C).

   □

Proposition 3.9

Let 𝒞_X ∈ P_M(2^X) and f : X ⟶ Y be a surjective mapping. Define 𝒞_Y : 2^Y ⟶ M by

Then D^con(Y, 𝒞_Y) ≥ D^con(X, 𝒞_X).

Proof

It is enough to prove the following three inequalities.

1. D_⊤(Y, 𝒞_Y) ≥ D_⊤(X, 𝒞_X). By the definition of 𝒞_Y, it follows that

   D⊤(Y,CY)=CY(Y)∧CY(∅)=CX(f−1(Y))∧CX(f−1(∅))≥CX(X)∧CX(∅)=D⊤(X,CX).

2. D_⋂(Y, 𝒞_Y) ≥ D_⋂(X, 𝒞_X). It can be checked as follows:

   D⋂(Y,CY)=⋀{Bk:k∈K}⊆2Y(⋀k∈KCY(Bk)→CY(⋂k∈KBk))=⋀{Bk:k∈K}⊆2Y(⋀k∈KCX(f−1(Bk))→CX(f−1(⋂k∈KBk))=⋀{Bk:k∈K}⊆2Y(⋀k∈KCX(f−1(Bk))→CX(⋂k∈Kf−1(Bk))≥⋀{Ak:k∈K}⊆2X(⋀k∈KCX(Ak)→CX(⋂k∈KAk))=D⋂(X,CX).

3. The proof of D_⋃^d(Y, 𝒞_Y) ≥ D_⋃^d(X, 𝒞_X) is similar to (2).

   As a result, we get

   Dcon(Y,CY)=D⊤(Y,CY)∧D⋂(Y,CY)∧D⋃d(Y,CY)≥D⊤(X,CX)∧D⋂(X,CX)∧D⋃d(X,CX)=Dcon(X,CX).

   □

Proposition 3.10

Let {X_i}_i∈I be a family of nonempty sets, where I = {1, 2, –n}, let 𝒞_i ∈ P_M(2^X_i) for each i ∈ I and let X be the product of {X_i}_i∈I, that is, X = ∏_i∈I X_i. Define 𝒞 : 2^X ⟶ M by

Then D^con(X, 𝒞) ≥ ⋀_i∈I D^con(X_i, 𝒞_i).

Proof

Suppose that {p_i : X ⟶ X_i}_i∈I is the family of projection mappings. Next we verify the following three inequalities.

1. D_⊤(X, 𝒞) ≥ ⋀_i∈I D_⊤(X_i, 𝒞_i). By the definition of 𝒞, it follows that

   D⊤(X,C)=C(X)∧C(∅) =⋁∏i∈IAi=X⋀i∈ICi(Ai)∧⋁∏i∈IAi=∅⋀i∈ICi(Ai) ≥⋀i∈ICi(Xi)∧⋀i∈ICi(∅) =⋀i∈I(Ci(Xi)∧Ci(∅)) ≥⋀i∈ID⊤(Xi,Ci).

2. D_⋂(X, 𝒞) ≥ ⋀_i∈I D_⋂(X_i, 𝒞_i). It suffices to show the following inequality:

   ⋀{Ak:k∈K}⊆2X(⋀k∈KC(Ak)→C(⋂k∈KAk))≥⋀i∈I⋀{Bk,i:k∈Ki}⊆2Xi(⋀k∈KiCi(Bk,i)→Ci(⋂k∈KiBk,i)).

   Take any α ∈ M such that

   α≤⋀i∈I⋀{Bk,i:k∈Ki}⊆2Xi(⋀k∈KiCi(Bk,i)→Ci(⋂k∈KiBk,i)).

   Then for each i ∈ I and {B_k,i : k ∈ K_i} ⊆ 2^X_i, it follows that

   α∧⋀k∈KiCi(Bk,i)≤Ci(⋂k∈KiBk,i).

   In order to show

   α≤⋀{Ak:k∈K}⊆2X(⋀k∈KC(Ak)→C(⋂k∈KAk)),

   we need only show that for each {A_k : k ∈ K} ⊆ 2^X,

   α∧⋀k∈KC(Ak)≤C(⋂k∈KAk),

   i.e.,

   α∧⋀k∈K⋁∏i∈IkBk,i=Ak⋀i∈ICi(Bk,i)≤⋁∏i∈IAi=⋂k∈KAk⋀i∈ICi(Ai).

   Take each β ∈ M such that

   β≺α∧⋀k∈K⋁∏i∈IBk,i=Ak⋀i∈ICi(Bk,i).

   Then β ≤ α and for each k ∈ K, there exists {B_k,i : i ∈ I} such that ∏_i∈I B_k,i = A_k and ⋀_i∈I 𝒞_i(B_k,i) ≥ β. This implies

   ⋀i∈I⋀k∈KCi(Bk,i)=⋀k∈K⋀i∈ICi(Bk,i)≥β.

   Then it follows that

   ⋂k∈KAk=⋂k∈K∏i∈IkBk,i=⋂k∈K⋂i∈Ikpi←(Bk,i)=⋂i∈I⋂k∈Kpi←(Bk,i)=⋂i∈Ipi←(⋂k∈KBk,i)=∏i∈I⋂k∈KBk,i.

   This implies that

   ⋁∏i∈IAi=⋂k∈KAk⋀i∈ICi(Ai)≥⋀i∈ICi(⋂k∈KBk,i)≥α∧⋀i∈I⋀k∈KCi(Bk,i)≥β.

   By the arbitrariness of β, we obtain that for each {A_k : k ∈ K} ⊆ 2^X,

   α∧⋀k∈K⋁∏i∈IkBk,i=Ak⋀i∈ICi(Bk,i)≤⋁∏i∈IAi=⋂k∈KAk⋀i∈ICi(Ai).

   This means that

   α≤⋀{Ak:k∈K}⊆2X(⋀k∈KC(Ak)→C(⋂k∈KAk)).

   By the arbitrariness of α, we obtain

   ⋀{Ak:k∈K}⊆2X(⋀k∈KC(Ak)→C(⋂k∈KAk))≥⋀i∈I⋀{Bk,i:k∈Ki}⊆2Xi(⋀k∈KiCi(Bk,i)→Ci(⋂k∈KiBk,i)),

   as desired.

3. The proof of D_⋃^d(X, 𝒞) ≥ ⋀_i∈I D_⋃^d(X_i, 𝒞_i) is similar to (2).

   Therefore, we get

   Dcon(X,C)=D⊤(X,C)∧D⋂(X,C)∧D⋃d(X,C)≥⋀i∈ID⊤(Xi,Ci)∧⋀i∈ID⋂(Xi,Ci)∧⋀i∈ID⋃d(Xi,Ci)=⋀i∈I(D⊤(Xi,Ci)∧D⋂(Xi,Ci)∧D⋃d(Xi,Ci)=⋀i∈IDcon(Xi,Ci).

   □

Definition 4.1

For each 𝒞 ∈ P_M(2^X), define D^clo(X, 𝒞) as follows:

Then D^clo(X, 𝒞) is called the approximate degree to which 𝒞 is an M-fuzzifying closure system on X. Obviously, 𝒞 is an M-fuzzifying closure system on X if and only if D^clo(X, 𝒞) = ⊤.

If D^clo(X, 𝒞) = ⊤, then D_⊤(X, 𝒞) = ⊤ and D_⋂(X, 𝒞) = ⊤. This implies (MFYS1) and (MFYS2) hold. Conversely, For each 𝒞 ∈ P_M(2^X), if it satisfies (LFYC1) and (LFYC2), then D^clo(X, 𝒞) = ⊤. Hence we obain

Proposition 4.2

For each 𝒞 ∈ P_M(2^X), 𝒞 is an M-fuzzifying closure system on X if and only if D^clo(X, 𝒞) = ⊤.

Proposition 4.3

For each 𝒞 ∈ P_M(2^X), D^con(X, 𝒞) ≤ D^clo(X, 𝒞).

Actually, there are close relations between M-fuzzifying closure systems and M-fuzzifying convex structures. In [29], Pang and Xiu provided a transforming method from M-fuzzifying closure systems to M-fuzzifying convex structures in the following way.

Proposition 4.4

([29]). Let (X, 𝒞) be an M-fuzzifying closure space. Define a mapping 𝒞^* : 2^X ⟶ M by

Then 𝒞^* is an M-fuzzifying convex structure on X.

Now let us give an approximate degree description for the above proposition.

Proposition 4.5

Let 𝒞 ∈ P_M(2^X) and define 𝒞^* : 2^X ⟶ M by

Then D^con(X, 𝒞^*) ≥ D^clo(X, 𝒞).

Proof

We first verify three inequalities in the following.

1. D_⊤(X, 𝒞^*) ≥ D_⊤(X, 𝒞). By the definition of 𝒞^*, we have

   D⊤(X,C∗)=C∗(X)∧C∗(∅)=⋁⋃λ∈ΛdBλ=X⋀λ∈ΛC(Bλ)∧⋁⋃λ∈ΛdBλ=∅⋀λ∈ΛC(Bλ)≥C(X)∧C(∅)=D⊤(X,C).

2. D_⋂(X, 𝒞^*) ≥ D_⋂(X, 𝒞). It suffices to show the following inequality.

   ⋀{Ak:k∈K}⊆2X(⋀k∈K⋁⋃j∈JkdBk,j=Ak⋀j∈JkC(Bk,j)→⋁⋃t∈TdBt=⋂k∈KAk⋀t∈TC(Bt))≥⋀{Bk:k∈K}⊆2X(⋀k∈KC(Bk)→C(⋂k∈KBk)).

   Take any α ∈ M such that

   α≤⋀{Bk:k∈K}⊆2X(⋀k∈KC(Bk)→C(⋂k∈KBk)).

   Then for each {B_k : k ∈ K} ⊆ 2^X, it follows that α ∧ ⋀_k∈K𝒞(B_k) ≤ 𝒞(⋂_k∈KB_k). Now we need only show that for each {A_k : k ∈ K} ⊆ 2^X,

   α∧⋀k∈K⋁⋃j∈JkdBk,j=Ak⋀j∈JkC(Bk,j)≤⋁⋃t∈TdBt=⋂k∈KAk⋀t∈TC(Bt).

   Take each β ∈ M such that

   β≺α∧⋀k∈K⋁⋃j∈JkdBk,j=Ak⋀j∈JkC(Bk,j).

   Then β ≤ α and for each k ∈ K, there exists an up-directed set {B_k,j : j ∈ J_k} such that ⋃j∈Jkd B_k,j = A_k and for each j ∈ J_k, 𝒞 (B_k,j) ≥ β. By the completely distributive law, it follows that

   ⋂k∈KAk=⋂k∈K⋃j∈JkdBk,j=⋃f∈∏k∈KJk⋂k∈KBk,f(k).

   Put C_f = ⋂_k∈K B_k,f(k) for each f ∈ ∏_k∈K J_k. Since {B_k,j : j ∈ J_k} is up-directed, it is trivial to verify that {C_f : f ∈ ∏_k∈K J_k} is up-directed. Then for each f ∈ ∏_k∈K J_k, it follows that

   C(Cf)=C(⋂k∈KBk,f(k))≥α∧⋀k∈KC(Bk,f(k))≥β.

   This implies ⋀_{f∈∏k∈KJk} 𝒞(C_f) ≥ β. Since {C_f : f ∈ ∏_k∈K J_k} is up-directed and ⋂_k∈K Ak=⋃f∈∏k∈KJkdCf, it follows that

   ⋁⋃t∈TdBt=⋂k∈KAk⋀t∈TC(Bt)≥⋀f∈∏k∈KJkC(Cf)≥β.

   By the arbitrariness of β, we obtain that for each {A_k : k ∈ K} ⊆ 2^X,

   α∧⋀k∈K⋁⋃j∈JkdBk,j=Ak⋀j∈JkC(Bk,j)≤⋁⋃t∈TdBt=⋂k∈KAk⋀t∈TC(Bt).

   This means that

   α≤⋀{Ak:k∈K}⊆2X(⋀k∈KC∗(Ak)→C∗(⋂k∈KAk)).

   By the arbitrariness of α, we obtain that

   ⋀{Ak:k∈K}⊆2X(⋀k∈KC∗(Ak)→C∗(⋂k∈KAk))≥⋀{Bk:k∈K}⊆2X(⋀k∈KC(Bk)→C(⋂k∈KBk)),

   as desired.

3. D_⋃^d(X, 𝒞^*) ≥ D_⋂(X, 𝒞). It suffices to show the following inequality.

   ⋀{Ak:k∈K}⊆d2X(⋀k∈K⋁⋃j∈JkdBk,j=Ak⋀j∈JkC(Bk,j)→⋁⋃t∈TdBt=⋃k∈KdAk⋀t∈TC(Bt))≥⋀{Bk:k∈K}⊆2X(⋀k∈KC(Bk)→C(⋂k∈KBk)).

   Take any α ∈ M such that

   α≤⋀{Bk:k∈K}⊆2X(⋀k∈KC(Bk)→C(⋂k∈KBk)).

   Then for each {B_k : k ∈ K} ⊆ 2^X, it follows that α ∧ ⋀_k∈K𝒞(B_k) ≤ 𝒞(⋂_k∈KB_k). Now we need only show that for each {A_k : k ∈ K} ⊆d 2^X,

   α∧⋀k∈K⋁⋃j∈JkdBk,j=Ak⋀j∈JkC(Bk,j)≤⋁⋃t∈TdBt=⋃k∈KdAk⋀t∈TC(Bt).

   Take each β ∈ M such that

   β≺α∧⋀k∈K⋁⋃j∈JkdBk,j=Ak⋀j∈JkC(Bk,j).

   Then β ≤ α and for each k ∈ K, there exists an up-directed set {B_k,j : j ∈ J_k} such that ⋃j∈Jkd B_k,j = A_k and for each j ∈ J_k, 𝒞 (B_k,j) ≥ β. Let

   A=⋃k∈KdAk=⋃k∈Kd⋃j∈JkdBk,j.

   Define a mapping σ : 2finA ⟶ 2^X by

   ∀F∈2finA,σ(F)=⋂{Bk,j|F⊆Bk,j}.

   It is easy to check that

   A=⋃F∈2finAF=⋃F∈2finAσ(F)=⋃F∈2finA⋂F⊆Bk,jBk,j.

   Obviously, σ is order-preserving. Since 2finA is up-directed, we know {σ(F)∣F ∈ 2finA } is up-directed. Now for each F ∈ 2finA , put {B_t∣ t ∈ T} = {B_k,j∣F ⊆ B_k,j}. Then

   ⋀t∈TC(Bt)=⋀F⊆Bk,jC(Bk,j)≥β.

   This implies

   C(σ(F))=C(⋂F⊆Bk,jBk,j)=C(⋂t∈TBt)≥α∧⋀t∈TC(Bt)≥β.

   By the arbitrariness of F, we obtain ⋀F∈2finA 𝒞(σ(F)) ≥ β. Since ⋃F∈2finAdσ(F)=A=⋃k∈KdAk, it follows that

   ⋁⋃t∈TdBt=⋃k∈KdAk⋀t∈TC(Bt)≥⋀F∈2finAC(σ(F))≥β.

   By the arbitrariness of β, we obtain that for each {A_k : k ∈ K} ⊆d 2^X,

   α∧⋀k∈K⋁⋃j∈JkdBk,j=Ak⋀j∈JkC(Bk,j)≤⋁⋃t∈TdBt=⋃k∈KdAk⋀t∈TC(Bt).

   This means that

   α≤⋀{Ak:k∈K}⊆d2X(⋀k∈KC∗(Ak)→C∗(⋃k∈KdAk)).

   By the arbitrariness of α, we obtain that

   ⋀{Ak:k∈K}⊆d2X(⋀k∈KC∗(Ak)→C∗(⋃k∈KdAk))≥⋀{Bk:k∈K}⊆2X(⋀k∈KC(Bk)→C(⋂k∈KBk)),

   as desired.

   By (1), (2) and (3), we have

   Dcon(X,C∗)=D⊤(X,C∗)⋀D⋂(X,C∗)⋀D⋃d(X,C∗)≥D⊤(X,C)⋀D⋂(X,C)⋀D⋂(X,C)=Dclo(X,C).

   □

   In order to introduce the approximate degree of M-fuzzifying Alexandrov topologies, we first give the following notation. For each 𝒞 ∈ P_M(2^X), define

   D∪(X,C)=⋀{Ak:k∈K}⊆2X(⋀k∈KC(Ak)→C(⋃k∈KAk)).

   Actually, this definition offers an approximate degree description to which 𝒞 is closed under arbitrary unions.

   Now let us equip each 𝒞 ∈ P_M(2^X) with some degree to which 𝒞 is an M-fuzzifying Alexandrov topology.