More- over, we investigate the relations among Alexandrov topologies, non-symmetric pseudo-metrics and lower approximation operators

https://doi.org/10.7468/jksmeb.2020.27.3.125 ISSN(Online) 2287-6081 Volume 27, Number 3 (August 2020), Pages 125–135

ALEXANDROV TOPOLOGIES AND NON-SYMMETRIC PSEUDO-METRICS

Ju-mok Oh^a and Yong Chan Kim^{b, ∗}

Abstract. In this paper, we investigate the properties of Alexandrov topologies, non-symmetric pseudo-metrics and lower approximation operators on [0, ∞]. More- over, we investigate the relations among Alexandrov topologies, non-symmetric pseudo-metrics and lower approximation operators. We give their examples.

1. Introduction

H´ajek [3] introduced a complete residuated lattice which is an algebraic structure for many valued logic. Pawlak [12,13] introduced the rough set theory as a formal tool to deal with imprecision and uncertainty in the data analysis. By using the con- cepts of lower and upper approximation operators, information systems and decision rules are investigated in complete residuated lattices [1-11,14,15]. Kim [6-10] inves- tigated the properties of Alexandrov topologies, fuzzy preorders and join-preserving maps in complete residuated lattices.

In this paper, we investigate the properties of Alexandrov topologies, non-symmetric pseudo-metrics and lower approximation operators on [0, ∞]. We give their exam- ples. In fact, categories of Alexandrov topologies, non-symmetric pseudo-metrics and lower approximation operators are isomorphic.

2. Preliminaries

Let ([0, ∞], ≤, ∨, +, ∧, →, ∞, 0) be a structure where

Received by the editors January 01, 2020. Accepted March 05, 2020.

2010 Mathematics Subject Classification. 03E72, 54A40, 54B10.

Key words and phrases. Alexandrov topologies, non-symmetric pseudo-metrics and lower approx- imation operators.

This work was supported by the Research Institute of Natural Science of Gangneung-Wonju National University.

∗Corresponding author.

c

° 2020 Korean Soc. Math. Educ.

125

x → y =^

{z ∈ [0, ∞] | z + x ≥ y} = (y − x) ∨ 0,

∞ + a = a + ∞ = ∞, ∀a ∈ [0, ∞], ∞ → ∞ = 0.

Definition 2.1. Let X be a set. A function d_X : X × X → [0, ∞] is called a non-symmetric pseudo-metric if it satisfies the following conditions:

(M1) d_X(x, x) = 0 for all x ∈ X,

(M2) d_X(x, y) + d_X(y, z) ≥ d_X(x, z), for all x, y, z ∈ X.

The pair (X, d_X) is called a non-symmetric pseudo-metric space.

Remark 2.2. (1) We define a function d_[0,∞]X : [0, ∞]^X × [0, ∞]^X → [0, ∞] as d_[0,∞]X(A, B) =W

x∈X(A(x) → B(x)) =W

x∈X((B(x)−A(x))∨0). Then ([0, ∞]^X, d_[0,∞]X) is a non-symmetric pseudo-metric space.

(2) If (X, d_X) is a non-symmetric pseudo-metric space and we define a function d⁻¹_X (x, y) = d_X(y, x), then (X, d⁻¹_X ) is a non-symmetric pseudo-metric space.

(3) Let (X, d_X) be a non-symmetric pseudo-metric space and define (d_X⊕d_X)(x, z) = V

y∈X(d_X(x, y) + d_X(y, z)) for each x, z ∈ X. By (M2), (d_X ⊕ d_X)(x, z) ≥ d_X(x, z) and (d_X⊕ d_X)(x, z) ≤ d_X(x, x) + d_X(x, z) = d(x, z). Hence (d_X ⊕ d_X) = d_X.

(4) If d_X is a non-symmetric pseudo-metric and d_X(x, y) = d_X(y, x) for each x, y ∈ X, then d_X is a pseudo-metric

Example 2.3. (1) Let X = {a, b, c} be a set and define maps dⁱ_X : X × X → [0, ∞]

for i = 1, 2, 3 as follows:

d¹_X =



 0 6 5

6 0 1 15 7 0



 , d²_X =



 0 6 3 7 0 4 0 5 0



 , d³_X =



 0 3 7 6 0 9 5 4 0



 .

Since d¹_X(c, b) + d¹_X(b, a) = 13 < d¹_X(c, a) = 15, d¹_X is not a non-symmetric pseudo- metric. Since d²_X and d³_X are non-symmetric pseudo-metrics, d^k_X ⊕ d^k_X = d^k_X for k = 2, 3.

3. Alexandrov Topologies and Non-symmetric Pseudo-metrics

We define the following two definitions as a sense in [2, 5-10].

Definition 3.1. A subset τ_X ⊂ [0, ∞]^X is called an Alexandrov topology on X iff it satisfies the following conditions:

(AT1) α_X ∈ τ_X where α_X(x) = α for each x ∈ X and α ∈ [0, ∞].

(AT2) If A_i∈ τ_X for all i ∈ I, thenW

i∈IA_i,V

i∈IA_i ∈ τ_X.

(AT3) If A ∈ τ_X and α ∈ [0, ∞], then α + A, α → A ∈ τ_X where (α → A)(x) = (A(x) − α) ∨ 0.

The pair (X, τ_X) is called an Alexandrov topological space.

Definition 3.2. A map H : [0, ∞]^X → [0, ∞]^X is called a lower approximation operator if it satisfies the following conditions, for all A, A_i ∈ [0, ∞]^X, and α ∈ [0, ∞],

(H1) H(α + A) = α + H(A) where (α + A)(x) = α + A(x), (H2) H(V

i∈IA_i) =V

i∈IH(A_i), (H3) H(A) ≤ A,

(H4) H(H(A)) = H(A).

Theorem 3.3. Let d_X ∈ [0, ∞]^X×X be a non-symmetric pseudo-metric. Define H_dX(A) : [0, ∞]^X → [0, ∞]^X as follows

H_dX(A)(y) = ^

x∈X

(A(x) + d_X(x, y)).

Then H_dX is a lower approximation operator.

Proof. Since H_dX(A)(y) =V

x∈X(A(x) + d_X(x, y)), (H1) H_dX(α + A) = α + H_dX(A).

(H2) H_dX(V

i∈ΓA_i) =V

i∈ΓH_dX(A_i).

(H3) H_dX(A)(y) =V

x∈X(A(x) + d_X(x, y)) ≤ A(y) + d_X(y, y) = A(y).

(H4) For all A ∈ [0, ∞]^X, z ∈ X, H_dX(H_dX(A))(z) =V

y∈X(H_dX(A)(y) + d_X(y, z))

=V

y∈X((V

x∈X(A(x) + d_X(x, y))) + d_X(y, z))

=V

x∈X(A(x) +V

y∈X(d_X(x, y) + d_X(y, z)))

=V

x∈X(A(x) + d_X(x, z)) = H_dX(A)(z).

Hence H_dX is a lower approximation operator. ¤

Theorem 3.4. A map H : [0, ∞]^X → [0, ∞]^X is a lower approximation operator iff there exist a non-symmetric pseudo-metric d_H on X such that

H(A)(y) = ^

x∈X

(A(x) + d_H(x, y)).

Proof. (⇒) Put d_H: X × X → [0, ∞] as d_H(x, y) = H(0_x)(y) where 0_x(x) = 0 and 0_x(y) = ∞ for x 6= y ∈ X. (M1) d_H(x, x) = H(0_x)(x) ≤ 0_x(x) = 0.

(M2) Since A =V

y∈X(A(y) + 0_y) and H(0_x) =V

y∈X(H(0_x)(y) + 0_y), V

y∈X(d_H(x, y) + d_H(y, z))

=V

y∈X(H(0_x)(y) + H(0_y)(z)) ( by (H2))

= H(V

y∈X(H(0_x)(y) + 0_y)(z)) = H(H(0_x))(z)

= H(0_x)(z) = d_H(x, z).

Hence d_H is a non-symmetric pseudo-metric. Moreover, H(A)(y) = H(V

x∈X(A(x) + 0_x))(y)

=V

x∈X(A(x) + H(0_x)(y))

=V

x∈X(A(x) + d_H(x, y))).

(⇐) It follow from Theorem 3.3. ¤

Theorem 3.5. Let d_X be a non-symmetric pseudo-metric on X. Define τ_dX = {A ∈ [0, ∞]^X | A(x) + d_X(x, y) ≥ A(y)}. Then the following properties hold.

(1) τ_dX is an Alexandrov topology on X.

(2) d_X(x, −) ∈ τ_dX. Moreover, A ∈ τ_dX iff A = V

x∈X(A(x) + d_X(x, −)) = H_dX(A).

Proof. (1) (AT1) Since α_X(x) + d_X(x, y) ≥ α_X(y), we have α_X ∈ τ_dX. (AT2) If A_i∈ τ_dX for all i ∈ I, then

(V

i∈IA_i)(x) + d_X(x, y) =V

i∈I(A_i(x) + d_X(x, y))

≥V

i∈IA_i(y), (W

i∈IA_i)(x) + d_X(x, y) =W

i∈I(A_i(x) + d_X(x, y))

≥W

i∈IA_i(y).

HenceV

i∈IA_i,W

i∈IA_i ∈ τ_dX.

(AT3) If A ∈ τ_eX and α ∈ [0, ∞], then

(α + A)(x) + d_X(x, y) ≥ (α + A)(y),

(α → A)(x) + d_X(x, y) = ((A(x) − α) ∨ 0) + d(x, y)

= ((A(x) − α) + d(x, y)) ∨ d(x, y)

≥ (A(y) − α) ∨ 0 = (α → A)(y).

So, α + A, α → A ∈ τ_dX. Hence τ_dX is an Alexandrov topology on X.

(2) Since d_X(x, y) + d_X(y, z) ≥ d_X(x, z), d_X(x, −) ∈ τ_dX. Let A ∈ τ_dX. Then V

x∈X(A(x)+d_X(x, y)) ≥ A(y) andV

x∈X(A(x)+d_X(x, y)) ≤ A(y)+d_X(y, y) = A(y).

Hence A =V

x∈X(A(x) + d_X(x, −)) = H_dX(A).

Conversely, since H_dX(A)(y) + d_X(y, z) = V

x∈X(A(x) + d_X(x, y)) + d_X(y, z) ≥ V

x∈X(A(x) + d_X(x, z)) = H_dX(A)(z). So, A = H_dX(A) ∈ τ_dX. ¤

Theorem 3.6. Let H : [0, ∞]^X → [0, ∞]^X be a lower approximation operator. Then the following properties hold.

(1) τ_H = {A ∈ [0, ∞]^X | H(A) = A} is an Alexandrov topology on X such that τ_H= {H(A) | A ∈ [0, ∞]^X}.

(2) Define d_H : X × X → [0, ∞] as d_H(x, y) = H(0_x)(y) where 0_x(x) = 0 and 0_x(y) = ∞ for x 6= y ∈ X. Then d_H is a non-symmetric pseudo-metric on X such H_dH = H and τ_dH = τ_H.

Proof. (1) (AT1) Since H(α_X) = H(α + 0_X) = α + 0_X = α_X, then α_X ∈ τ_H. (AT2) For A_i ∈ τ_H for each i ∈ Γ, by (H2), V

i∈ΓA_i ∈ τ_H. Since W

i∈ΓA_i = W

i∈ΓH(A_i) ≤ H(W

i∈ΓA_i) ≤W

i∈ΓA_i, Thus,W

i∈ΓA_i∈ τ_H. (AT3) For A ∈ τ_H, by (H1), α + A ∈ τ_H.

Since α + H(α → A) = H(α + (α → A)) ≥ H(A), H(α → A) ≥ (H(A) − α) ∨ 0 = α → H(A) = α → A. Then α → A ∈ τ_H. Hence τ_H is an Alexandrov topology on X. Let A ∈ τ_H. Then A = H(A) ∈ {H(A) | A ∈ [0, ∞]^X}. Let H(A) ∈ {H(A) | A ∈ [0, ∞]^X}. Since H(H(A)) = H(A), H(A) ∈ τ_H.

(2) By Theorem 3.4, d_H is a non-symmetric pseudo-metric on X. Moreover, H_dH(A)(y) =V

x∈X(A(x) + d_H(x, y)))

=V

x∈X(A(x) + H(0_x)(y))

= H(V

x∈X(A(x) + 0_x))(y) = H(A)(y), τ_dH= τ_H from:

A ∈ τ_dHiff H_dH(A) = A iff H(A) = A iff A ∈ τ_H.

¤ Example 3.7. (1) Define maps dⁱ : [0, ∞] × [0, ∞] → [0, ∞] for i = 0, 1, 2, 3 as follows:

d⁰(x, y) =

½ 0, if x = y,

∞, if x 6= y, d¹(x, y) =

½ 0, if x ≥ y,

∞, if x < y, d²(x, y) =

½ 0, if x ≤ y,

∞, if x > y, d³(x, y) = 0.

Since H_dX(A)(y) =V

x∈X(A(x) + d_X(x, y)), we can obtain H_d0(A)(y) =V

x∈X(A(x) + d⁰_X(x, y)) = A(y), H_d¹(A) =V

x≥yA(x), H_d2(A) =V

x≤yA(x), H_d3(A) =V

x∈XA(x).

τ_d⁰ = [0, ∞]^[0,∞],

τ_d¹ = {A ∈ [0, ∞]^[0,∞]| A(x) ≤ A(y) if x ≤ y}, τ_d2 = {A ∈ [0, ∞]^[0,∞]| A(x) ≥ A(y) if x ≤ y}, τ_d³ = {α_X ∈ [0, ∞]^[0,∞]| α ∈ [0, ∞]}.

Theorem 3.8. Let τ be Alexandrov topology on X. Then the following properties hold.

(1) Define H_τ : [0, ∞]^X → [0, ∞]^X as follows:

H_τ(A) =_

{B | B ≤ A, B ∈ τ }.

Then H_τ is a lower approximation operator such that τ_Hτ = τ, H_τH = H . (2) Define d_τ : X × X → [0, ∞] as

d_τ(x, y) = _

A∈τ

(A(x) → A(y)) = _

A∈τ

((A(y) − A(x)) ∨ 0)

Then d_τ is a non-symmetric pseudo-metric such that τ = τ_Hdτ = τ_dτ. Moreover, H_τ = H_dτ and d_τ = d_Hτ.

(3) If H : [0, ∞]^X → [0, ∞]^X is a lower approximation operator, then

^

y∈X

(H(0_x)(y) + H(0_y)(z)) = H(0_x)(z) for all x, y, z ∈ X and d_τH= d_H.

Proof. (1) We show H_τ(A) = W

{B | B ≤ A, B ∈ τ } is a lower approximation operator.

(H1) For α ∈ [0, ∞], A ∈ [0, ∞]^X, α + H_τ(A)

= α +W

{B | B ≤ A, B ∈ τ }

=W

{α + B | α + B ≤ α + A, α + B ∈ τ }

= H_τ(α + A).

(H2) Since H_τ(A) ≤ H_τ(B) for A ≤ B, we have V

i∈ΓH_τ(A_i) ≥ H_τ(V

i∈ΓA_i).

Since V

i∈ΓA_i≥V

i∈ΓH_τ(A_i) ∈ τ , then H_τ(V

i∈ΓA_i) ≥V

i∈ΓH_τ(A_i).

(H3) It follows from the definition.

(H4) Since H_τ(A) ∈ τ , we have H_τ(H_τ(A)) = H_τ(A).

Let A ∈ τ_Hτ. Then A = H_τ(A) ∈ τ . Hence τ_Hτ ⊂ τ .

Let A ∈ τ . Then H_τ(A) = A. So, A ∈ τ_Hτ. Hence τ ⊂ τ_Hτ. Since H_τH(A) = W

{B | B ≤ A, B ∈ τ_H} and A ≥ H(H(A)) = H(A), we have H(A) ≤ H_τH(A). For B_i ∈ τ_H, since H(W

i∈ΓB_i) ≥ W

i∈ΓH(B_i) = W

i∈ΓB_i, then H(H_τH(A)) = H_τH(A). So, H(A) ≥ H_τH(A).

(2) We easily show that d_τ is a non-symmetric pseudo-metric.

Let A ∈ τ . Then H_dτ(A) = A because H_dτ(A)(y) =V

x∈X(A(x) + d_τ(x, y)) =V

x∈X(A(x) +W

B∈τ(B(x) → B(y))

≥V

x∈X(A(x) + (A(x) → A(y))) =V

x∈X(A(x) + ((A(y) − A(x)) ∨ 0))) ≥ A(y), So, τ ⊂ τ_Hdτ

Let A = H_dτ(A). Then

A = H_dτ(A) =V

x∈X(A(x) +W

B∈τ(B(x) → B)) ∈ τ.

So, τ_Hdτ ⊂ τ.

Let A ∈ τ . Then A ∈ τ_dτ because A(x) + d_τ(x, y) = A(x) +W

B∈τ(B(x) → B(y))

≥ A(x) + (A(x) → A(y)) = A(x) + ((A(y) − A(x)) ∨ 0))) ≥ A(y).

So, τ ⊂ τ_dτ

Let A ∈ τ_dτ. Then A ∈ τ because A = H_dτ(A) =V

x∈X(A(x) +W

B∈τ(B(x) → B)) ∈ τ.

Since A ≥ H_dτ(A) ∈ τ , then H_τ(A) ≥ H_dτ(A). Since H_dτ(A)(y) =V

x∈X(A(x) +W

B∈τ(B(x) → B(y))

≥V

x∈X(H_τ(A)(x) + (H_τ(A)(x) → H_τ(A)(y)) ≥ H_τ(A)(y), Hence H_dτ(A) = H_τ(A).

(3) Since H(0_x) = V

y∈X(H(0_x)(y) + 0_y(−)), V

y∈X(H(0_x)(y) + H(0_y)(z)) = H(0_x)(z) because

H(0_x)(z) = H(H(0_x)(z) = H(V

y∈X(H(0_x)(y) + 0_y(−)))(z)

=V

y∈X(H(0_x)(y) + H(0_y)(z)).

d_τH(x, y) =W

A∈τH((A(y) − A(x)) ∨ 0)

≥W

z∈X((H(0_z)(y) − H(0_z)(x)) ∨ 0) ≥ (H(0_x)(y) − H(0_x)(x)) ∨ 0

≥ (H(0_x)(y) − (0_x)(x)) ∨ 0 = H(0_x)(y), d_τH(x, y) =W

A∈τH((A(y) − A(x)) ∨ 0) =W

A∈[0,∞]^X((H(A)(y) − H(A)(x)) ∨ 0)

=W

A∈[0,∞]^X((V

z∈X(A(z) + H(0_z)(y)) −V

z∈X(A(z) + H(0_z)(x))) ∨ 0)

≤W

z∈X((H(0_z)(y) − H(0_z)(x)) ∨ 0) ≤ H(0_x)(y).

Hence d_τH(x, y) = H(0_x)(y) = d_H(x, y). ¤

Theorem 3.9. Let d_X ∈ [0, ∞]^X×X be a non-symmetric pseudo-metric. Then d_HdX(x, y) = H_dX(0_X)(y) = d_X(x, y) = d_τdX(x, y) for each x, y ∈ X.

Proof. Since H_dX(A)(y) = V

x∈X(A(x) + d_X(x, y)), H_dX(0_x)(y) = V

x∈X(0_x(x) + d_X(x, y)) = d_X(x, y). Since d_X(x, −) ∈ τ_dX and A = H_dX(A) for A ∈ τ_dX,

d_τdX(x, y) =W

A∈τ_dX((A(y) − A(x)) ∨ 0)

≥ (d_X(x, y) − d_X(x, x)) ∨ 0 = d_X(x, y), d_τdX(x, y) =W

A∈τ_dX((A(y) − A(x)) ∨ 0)

=W

A∈τ_dX((H_dX(A)(y) − H_dX(A)(x)) ∨ 0)

=W

A∈τ_dX((V

z∈X(A(z) + d_X(z, y)) − (V

z∈X(A(z) + d_X(z, x))) ∨ 0)

≤W

z∈X((d_X(z, y) − d_X(z, x)) ∨ 0) ≤ d_X(x, y).

Example 3.10. Let X = {a, b, c} be a set and A ∈ [0, ∞]^X as A(a) = 7, A(b) = 5, A(c) = 10 (1) Define d_A(x, y) = (A(y) − A(x)) ∨ 0 as

d_A=



 0 0 3 2 0 5 0 0 0



 , H_dA(B) =



 B(a) ∧ (2 + B(b)) ∧ B(c) B(a) ∧ B(b) ∧ B(c) (3 + B(a)) ∧ (5 + B(b)) ∧ B(c)



 .

Then d_A(a, ) = (0, 0, 3), d_A(b, ) = (2, 0, 5), d_A(c, ) = (0, 0, 0), A ∈ τ_dA. Moreover, τ_dA = {H_dA(B) | B ∈ [0, ∞]^X}.

(2) Define d¹_A(x, y) = |A(x) − A(y)| as

d¹_A=



 0 2 3 2 0 5 3 5 0



 , H_d¹

A(B) =



 B(a) ∧ (2 + B(b)) ∧ (3 + B(c)) (2 + B(a)) ∧ B(b) ∧ (5 + B(c)) (3 + B(a)) ∧ (5 + B(b)) ∧ B(c)



 .

Then d¹_A(a, ) = (0, 2, 3), d¹_A(b, ) = (2, 0, 5), d_A(c, ) = (3, 5, 0), A ∈ τ_d¹

A. Moreover, τ_d¹

A = {H_d¹

A(B) | B ∈ [0, ∞]^X}.

(3) Define d²_X and d²_X⊕ d²_X(x, z) =V

y∈X(d²_X(x, y) + d²_X(y, z) as d²_X =



 0 4 1 7 0 3 2 9 0



 , d²_X ⊕ d²_X =



 0 4 1 5 0 3 2 6 0



 .

Since d²_X(b, c)+d²_X(c, a) = 5 < d¹_X(b, a) = 7 and d²_X(c, a)+d²_X(a, b) = 6 < d¹_X(b, a) = 9, d²_X is not a non-symmetric pseudo-metric.

(4) Define d³_X = d²_X ⊕ d²_X.

d³_X =



 0 4 1 5 0 3 2 6 0



 , H_d³

X(B) =



 B(a) ∧ (5 + B(b)) ∧ (2 + B(c)) (4 + B(a)) ∧ B(b) ∧ (6 + B(c)) (1 + B(a)) ∧ (3 + B(b)) ∧ B(c)



 .

Since d³_X is a non-symmetric pseudo-metric, d³_X ⊕ d³_X = d³_X. Then d³_X(a, ) = (0, 4, 1), d³_X(b, ) = (5, 0, 3), d³_X(c, ) = (2, 6, 0) ∈ τ_d³

X. Moreover, τ_d³

X = {H_d³

X(B) | B ∈ [0, ∞]^X}.

4. Categories of Non-symmetric Pseudo-metrics, Lower Approximation Operators and Alexandrov Topologies Let LA be a category with object (X, H_X) where H_X is a lower approxima- tion operator with a morphism f : (X, H_X) → (Y, H_Y) such that f^←(H_Y(B)) ≤ H_Y(f^←(B)) for all B ∈ [0, ∞]^Y.

Let NPM be a category with object (X, d_X) where d_X is a non-symmetric pseudo-metric with a morphism f : (X, d_X) → (Y, d_Y) such that d_Y(f (x), f (y)) ≤ d_X(x, y) for all x, y ∈ X.

Theorem 4.1. Two categories LA and NPM are isomorphic.

Proof. Define H : NPM → LA as H(X, d_X) = (X, H_dX) where H_dX(A)(y) = V

x∈X(A(x) + d_X(x, y)) from Theorem 3.3. Let d_Y(f (x), f (z)) ≤ d_X(x, z). Then f^←(H_Y(B))(x) =V

w∈Y(B(w) + d_Y(w, f (x)))

≤V

z∈X(B(f (z)) + d_Y(f (z), f (x)))

≤V

z∈X(f^←(B)(z) + d_X(z, x)) = H_Y(f^←(B))(x).

Hence H is a functor.

Define a functor G : LA → NPM as G(X, H_X) = (X, d_HX) where d_HX(x, y) = H_X(0_x)(y) from Theorem 3.6(2). Let f^←(H_Y(B)) ≤ H_Y(f^→(B)). Since

d_HY(f (x), f (z)) = H_Y(0_{f (x)})(f (y)) = f^←(H_Y(0_{f (x)}))(y)

≤ H_X(f^←(0_{f (x)}))(y) ≤ H_Y(0_x)(y) = d_X(x, y).

Hence G is a functor. Moreover, by Theorem 3.9, G(H(X, d_X)) = G(X, H_dX) = (X, d_HdX) = (X, d_X) and, by Theorem 3.6(2), H(G(X, H_X)) = H(X, d_HX) = (X, H_dHX) = (X, H_X). Thus, LA and NPM are isomorphic. ¤ Let ATOP be a category with object (X, τ_X) where τ_X is an Alexandrov topology with a morphism f : (X, τ_X) → (Y, τ_Y) such that f^←(B) ∈ τ_X for all B ∈ τ_Y. Theorem 4.2. Two categories ATOP and LA are isomorphic.

Proof. Define U : ATOP → LA as U (X, τ_X) = (X, H_τX) where H_τX(A) =W {B ∈ [0, ∞]^X | B ≤ A, B ∈ τ_X} from Theorem 3.8(1). For B ∈ τ_dX,

f^←(H_τY(B)) =W

{f^←(C) | C ≤ B, C ∈ τ_Y}

≤W

{f^←(C) | f^←(C) ≤ f^←(B), f^←(C) ∈ τ_X}

≤ H_τX(f^←(B)).

Hence U is a functor.

Define W : LA → ATP as W (X, H_X) = (X, τ_HX) where τ_HX = {A ∈ [0, ∞]^X | A = H_X(A)} from Theorem 3.6(1). For B = H_Y(B), f^←(B) ∈ τ_HX because

f^←(B) = f^←(H_τY(B)) ≤ H_τX(f^←(B)) ≤ f^←(B).

Hence W is a functor. Moreover, by Theorem 3.8(1), U (W (X, H_X)) = U (X, τ_HX) = (X, H_τHX) = (X, H_X) and, by Theorem 3.8(1), W (U (X, τ_X)) = W (X, H_τX) = (X, τ_HτX) = (X, τ_X). Thus, LA and NPM are isomorphic. ¤ Theorem 4.3. Two categories ATOP and NPM are isomorphic.

Proof. Define T : NPM → ATOP as T (X, d_X) = (X, τ_dX) where τ_dX = {A ∈ [0, ∞]^X | A(x)+d_X(x, y) ≥ A(y)} from Theorem 3.5. Let d_Y(f (x), f (z)) ≤ d_X(x, z).

For B ∈ τ_dY, we have

f^←(B)(x) + d_X(x, z) ≥ B(f (x)) + d_Y(f (x), f (z))

≥ B(f (z)) = f^←(B)(z).

Hence T is a functor.

Define P : ATOP → NPM as P (X, τ_X) = (X, d_τX) where d_τX(x, y) =W

A∈τX(A(x) → A(y)) from Theorem 3.8(2). Let f^←(B) ∈ τ_X for B ∈ τ_Y. We have

d_τY(f (x), f (y)) =W

B∈τY(B(f (x)) → B(f (x)))

=W

B∈τY(f^←(B)(x) → f^←(B)(x))

≤W

A∈τX(A(x) → A(y)) = d_τX(x, y)

Hence P is a functor. Moreover, by Theorem 3.8(2),T (P (X, τ_X)) = T (X, d_τX) = (X, τ_dτX) = (X, τ_X) and, by Theorem 3.9, P (T (X, d_X)) = P (X, τ_dX) = (X, d_τdX) =

(X, d_X) ¤

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aMathematics Department, Gangneung-Wonju National University, Gangneung, 25457, Korea

Email address: [email protected]

bMathematics Department, Gangneung-Wonju National University, Gangneung, 25457, Korea

Email address: [email protected]