Sobriety of quantale-valued cotopological spaces

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Fuzzy Sets and Systems ••• (••••) •••–•••

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www.elsevier.com/locate/fss

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Sobriety of quantale-valued cotopological spaces

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Dexue Zhang

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School of Mathematics, Sichuan University, Chengdu 610064, China

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Received 18 December 2016; received in revised form 6 September 2017; accepted 11 September 2017

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Abstract

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For each commutative and integral quantale, making use of the fuzzy order between closed sets, a theory of sobriety for quantalevalued cotopological spaces is established based on irreducible closed sets. © 2017 Published by Elsevier B.V.

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Keywords: Fuzzy topology; Quantale; Quantale-valued order; Quantale-valued cotopological space; Irreducible closed set; Sobriety

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1. Introduction

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O  pt

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between the category Top of topological spaces and the opposite of the category Frm of frames [10]. Precisely, X is sober if ηX : X −→ pt(O(X)) is a bijection (hence a homeomorphism), where η denotes the unit of the adjunction O  pt. In the classical setting, a topological space can be described in terms of open sets as well as closed sets, and we can switch between open sets and closed sets by taking complements. So, it makes no difference whether we choose to work with closed sets or with open sets. In the fuzzy setting, since the table of truth-values is usually a quantale, not a Boolean algebra, there is no natural way to switch between open sets and closed sets. So, it may make a difference whether we postulate topological spaces in terms of open sets or in terms of closed sets. An example in this regard is exhibited in [3,4]. The frame approach to sobriety of topological spaces makes use of open sets; while the irreducible-closed-set approach makes use of closed sets. Extending the theory of sober spaces to the fuzzy setting is an interesting topic in fuzzy topology. Most of the existing works focus on the frame approach; that is, to find a fuzzy counterpart of the category Frm of frames, then establish an adjunction between the category of fuzzy topological spaces and that of fuzzy frames. Works in this regard include Rodabaugh [26], Zhang and Liu [33], Kotzé [13,14], Srivastava and

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A topological space X is sober if each of its irreducible closed subsets is the closure of exactly one point in X. Sobriety of topological spaces can be described via the well-known adjunction

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✩

This work is supported by National Natural Science Foundation of China (No. 11371265). E-mail address: [email protected].

http://dx.doi.org/10.1016/j.fss.2017.09.005 0165-0114/© 2017 Published by Elsevier B.V.

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Khastgir [28], Pultr and Rodabaugh [21–24], Gutiérrez García, Höhle and de Prada Vicente [6], and Yao [30,31], etc. But, the irreducible-closed-set approach to sobriety of fuzzy topological spaces is seldom touched, except in Kotzé [13,14]. In this paper, making use of the fuzzy inclusion order between closed sets, we establish a theory of sobriety for quantale-valued topological spaces based on irreducible closed sets. Actually, this theory concerns sobriety of quantale-valued cotopological spaces. By a quantale-valued cotopological space we mean a “fuzzy topological space” postulated in terms of closed sets (see Definition 2.6). The term quantale-valued topological space is reserved for “fuzzy topological space” postulated in terms of open sets (see Definition 3.16). It should be noted that in most works on fuzzy frames, the table of truth-values is assumed to be a complete Heyting algebra (or, a frame), even a completely distributive lattice sometimes. But, in this paper, the table of truth-values is only assumed to be a commutative and integral quantale. Complete Heyting algebra, BL-algebras and left continuous t-norms, are important examples of such quantales. The contents are arranged as follows. Section 2 recalls basic concepts about quantale-valued ordered sets and quantale-valued Q-cotopological spaces. Section 3, making use of the quantale-valued order between closed sets in a Q-cotopological space, establishes a theory of sober Q-cotopological spaces based on irreducible closed sets. In particular, the sobrification of a stratified Q-cotopological space is constructed. The last section, Section 4, presents some interesting examples in the case that Q is the unit interval [0, 1] coupled with a (left) continuous t-norm.

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p&q ≤ r ⇐⇒ q ≤ p → r.

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The binary operation → is called the implication, or the residuation, corresponding to &. Some basic properties of the binary operations & and → are collected below, they can be found in many places, e.g. [2,27].

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In this paper, Q = (Q, &) always denotes a commutative and integral quantale, unless otherwise specified. Precisely, Q is a complete lattice with a bottomelement 0  and a top element 1, & is a binary operation on Q such that (Q, &, 1) is a commutative monoid and p& j ∈J qj = j ∈J p&qj for all p ∈ Q and {qj }j ∈J ⊆ Q. Since the semigroup operation & distributes over arbitrary joins, it determines a binary operation → on Q via the adjoint property

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2. Quantale-valued ordered sets and quantale-valued cotopological spaces

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Proposition 2.1. Let Q be a quantale. Then

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1 → p = p. p ≤ q ⇐⇒ 1 = p → q. p → (q → r) = (p&q) → r. p&(p  ≤ q.    → q) j ∈J pj → q = j ∈J (pj → q).    (6) p → j ∈J qj = j ∈J (p → qj ).

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We often write ¬p for p → 0 and call it the negation of p. Though it is true that p ≤ ¬¬p for all p ∈ Q, the inequality ¬¬p ≤ p does not always hold. A quantale Q is said to satisfy the law of double negation if

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(1) (2) (3) (4) (5)

(p → 0) → 0 = p,

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i.e., ¬¬p = p, for all p ∈ Q. A commutative and integral quantale that satisfies the law of double negation is also called a complete regular residuated lattice sometimes.

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Proposition 2.2. ([2]) Suppose that Q is a quantale that satisfies the law of double negation. Then

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(1) p → q = ¬(p&¬q) = ¬q → ¬p. (2) p&q  = ¬(q →¬p) = ¬(p → ¬q). (3) ¬( i∈I pi ) = i∈I ¬pi .

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In the class of quantales, the quantales with the unit interval [0, 1] as underlying lattice are of particular interest in fuzzy set theory [7,12]. In this case, the semigroup operation & is called a left continuous t-norm on [0, 1] [12]. If a left continuous t-norm & on [0, 1] is a continuous function with respect to the usual topology, then it is called a continuous t-norm.

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Example 2.3. ([12]) Some basic t-norms: the first three are continuous, the fourth is left continuous but not continuous. (1) The minimum t-norm: a&b = a ∧ b = min{a, b}. The corresponding implication is given by  1, a ≤ b, a→b= b, a > b. (2) The product t-norm: a&b = a · b. The corresponding implication is given by  1, a ≤ b, a→b= b/a, a > b. (3) The Łukasiewicz t-norm: a&b = max{a + b − 1, 0}. The corresponding implication is given by a → b = min{1, 1 − a + b}.

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In this case, ([0, 1], &) satisfies the law of double negation. (4) The nilpotent minimum t-norm:  0, a + b ≤ 1, a&b = min{a, b}, a + b > 1. The corresponding implication is given by  1, a ≤ b, a→b= max{1 − a, b}, a > b. In this case, ([0, 1], &) satisfies the law of double negation. A Q-order (or an order valued in the quantale Q) [2,29] on a set X is a reflexive and transitive Q-relation on X. Explicitly, a Q-order on X is a map R : X × X −→ Q such that 1 = R(x, x) and R(y, z)&R(x, y) ≤ R(x, z) for all x, y, z ∈ X. The pair (X, R) is called a Q-ordered set. As usual, we write X for the pair (X, R) and X(x, y) for R(x, y) if no confusion would arise. If R : X × X −→ Q is a Q-order on X, then R op : X × X −→ Q, given by R op (x, y) = R(y, x), is also a Q-order on X (by commutativity of &), called the opposite of R. A map f : X −→ Y between Q-ordered sets is Q-order-preserving if X(x1 , x2 ) ≤ Y (f (x1 ), f (x2 )) for all x1 , x2 ∈ X. We write Q-Ord

subX : QX × QX −→ Q,

subX (A, B) =

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for the category of Q-ordered sets and Q-order-preserving maps. Given a set X, a map A : X −→ Q is called a fuzzy set (valued in Q), the value A(x) is interpreted as the membership degree. The map

given by

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A(x) → B(x),

x∈X

defines a Q-order on QX . The value subX (A, B) measures the degree that A is a subset of B, thus, subX is called the fuzzy inclusion order on QX [2]. In particular, if X is a singleton set then the Q-ordered set (QX , subX ) reduces to the Q-ordered set (Q, dL ), where dL (p, q) = p → q.

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For all p ∈ Q and A ∈ QX , write p&A, p → A ∈ QX for the fuzzy sets given by (p&A)(x) = p&A(x) and (p → A)(x) = p → A(x), respectively. It is easy to check that

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for all p ∈ Q and A, B ∈ QX . In particular, A ≤ B if and only if 1 = subX (A, B). Furthermore,

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p → subX (A, B) = subX (p&A, B) = subX (A, p → B)

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for all p ∈ Q and A, B ∈ QX . Given a map f : X −→ Y , as usual, define f → : QX −→ QY and f ← : QY −→ QX by  f → (A)(y) = A(x), f ← (B)(x) = B ◦ f (x).

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f (x)=y

The fuzzy set f → (A) is called the image of A under f , and f ← (B) the preimage of B. The following proposition is a special case of the enriched Kan extension in category theory [11,17]. A direct verification is easy and can be found in e.g. [2,15].

(A), B) = subX (A, f

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(B))

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A fuzzy upper set [16] in a Q-ordered set X is a map ψ : X −→ Q such that

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X(x, y)&ψ(x) ≤ ψ(y)

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for all x, y ∈ X. It is clear that ψ : X −→ Q is a fuzzy upper set if and only if ψ : X −→ (Q, dL ) is Q-orderpreserving. Dually, a fuzzy lower set in a Q-ordered set X is a map φ : X −→ Q such that φ(y)&X(x, y) ≤ φ(x) −→ (Q, dL ) is a Q-order-preserving map, where

Definition 2.5. A fuzzy lower set φ in a Q-ordered set X is irreducible if



x∈X φ(x) = 1

X op

is the opposite of the

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and

subX (φ, φ1 ∨ φ2 ) = subX (φ, φ1 ) ∨ subX (φ, φ2 ) for all fuzzy lower sets φ1 , φ2 in X.

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Irreducible fuzzy lower sets are a counterpart of directed lower sets [5] in the quantale-valued setting. In particular, the condition x∈X φ(x) = 1 is a Q-version of the requirement that a directed set should be non-empty. The following definition is taken from [3,32].

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Definition 2.6. A Q-cotopology on a set X is a subset τ of QX subject to the following conditions:

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(C1) pX ∈ τ for all p ∈ Q; (C2) A ∨ B ∈ τ for all A, B ∈ τ ; (C3) j ∈J Aj ∈ τ for each subset {Aj }j ∈J of τ .

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X op

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for all A ∈ QX and B ∈ QY .

for all x, y ∈ X. Or equivalently, φ : Q-ordered set X.

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(1) f → : (QX , subX ) −→ (QY , subY ) is Q-order-preserving; (2) f ← : (QY , subY ) −→ (QX , subX ) is Q-order-preserving; (3) f → is left adjoint to f ← , written f →  f ← , in the sense that subY (f

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Proposition 2.4. For each map f : X −→ Y ,

←

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p ≤ subX (A, B) ⇐⇒ p&A ≤ B ⇐⇒ A ≤ p → B

→

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The pair (X, τ ) is called a Q-cotopological space; elements in τ are called closed sets of (X, τ ). A Q-cotopology τ is stratified if

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(C4) p → A ∈ τ for all p ∈ Q and A ∈ τ .

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A Q-cotopology τ is co-stratified if

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Q-CTop

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for the category of Q-cotopological spaces and continuous maps; and write

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SQ-CTop

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for the category of stratified Q-cotopological spaces and continuous maps. It is easily seen that both Q-CTop and SQ-CTop are well-fibered topological categories over Set in the sense of [1]. Given a Q-cotopological space (X, τ ), its closure operator − : QX −→ QX is defined by  A = {B ∈ τ | A ≤ B} for all A ∈ QX . The closure operator of a Q-cotopological space (X, τ ) satisfies the following conditions: for all A, B ∈ QX ,

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(cl1) (cl2) (cl3) (cl4)

pX = pX for all p ∈ Q; A ≥ A; A ∨ B = A ∨ B; A = A.

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Proof. (1) ⇒ (2) Since p → p&A is closed and A ≤ p → p&A,  it follows that A ≤ p → p&A, hence p&A ≤ p&A. (2) ⇒ (3) For all A, B ∈ QX , let p = subX (A, B) = x∈X (A(x) → B(x)). Since p&A ≤ B, then p&A ≤ p&A ≤ B, hence subX (A, B) = p ≤ subX (A, B). (3) ⇒ (1) Let A be a closed set and p ∈ Q. In order to see that p → A is also closed, it suffices to check that p → A ≤ p → A, or equivalently, p ≤ subX (p → A, A). This is easy since p ≤ subX (p → A, A) ≤ subX (p → A, A) = subX (p → A, A). 2

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(1) X is stratified. (2) p&A ≤ p&A for all p ∈ Q and A ∈ QX . (3) The closure operator − : (QX , subX ) −→ (QX , subX ) is Q-order-preserving.

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Proposition 2.8. Let X be a Q-cotopological space. The following are equivalent:

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A map f : X −→ Y between Q-cotopological spaces is continuous if f ← (A) = A ◦ f is closed in X whenever A is closed in Y . We write

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Remark 2.7. If Q is the quantale obtained by endowing [0, 1] with a continuous t-norm T , then τ ⊆ [0, 1]X is a strong Q-cotopology on X if and only if (X, τ ∗ ) is a fuzzy T -neighborhood space in the sense of Morsi [20], where τ ∗ = {1 − A | A ∈ τ }. In particular, if Q is the quantale ([0, 1], min), then τ ⊆ [0, 1]X is a strong Q-cotopology on X if and only if (X, τ ∗ ) is a fuzzy neighborhood space in the sense of Lowen [19].

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As usual, we often write X, instead of (X, τ ), for a Q-cotopological space.

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A Q-cotopology τ is strong if it is both stratified and co-stratified.

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(C5) p&A ∈ τ for all p ∈ Q and A ∈ τ .

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Proposition 2.8 is a counterpart of Proposition 6.1.1 in [9] that is stated for stratified Q-topological spaces. It follows immediately from Proposition 2.8 (3) that if X is a stratified Q-cotopological space and if B is a closed set in X, then for all A ∈ QX ,

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subX (A, B) ≤ subX (A, B) ≤ subX (A, B) = subX (A, B),

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hence

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subX (A, B) = subX (A, B).

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This equation will be useful in this paper.

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Corollary 2.9. ([2]) In a stratified Q-cotopological space X, 

A= subX (A, B) → B | B is closed in X

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for all A ∈ QX .

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Given a Q-cotopological space (X, τ ), define (τ ) : X × X −→ Q by  (τ )(x, y) = (A(y) → A(x)).

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A∈τ

Then (τ ) is a Q-order on X, called the specialization Q-order of (X, τ ) [16]. It is clear that each closed set in (X, τ ) is a fuzzy lower set in the Q-ordered set (X, (τ )). As said before, we often write X, instead of (X, τ ), for a Q-cotopological space. Accordingly, we will write (X) for the Q-ordered set obtained by equipping X with its specialization Q-order. The correspondence X → (X) defines a functor  : Q-CTop −→ Q-Ord.

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In particular, if f : X −→ Y is a continuous map between Q-cotopological spaces, then f : (X) −→ (Y ) is Q-order-preserving, i.e., (X)(x, y) ≤ (Y )(f (x), f (y)) for all x, y ∈ X. Conversely, given a Q-ordered set (X, R), the set (R) of fuzzy lower sets in (X, R) forms a strong Q-cotopology on X, called the Alexandroff Q-cotopology on (X, R). The correspondence (X, R) → (X, R) = (X, (R)) defines a functor  : Q-Ord −→ Q-CTop

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Proposition 2.10. ([25]) If X is a stratified Q-cotopological space, then (X)(x, y) = 1y (x) for all x, y ∈ X.

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3. Sober Q-cotopological spaces

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Let X be a topological space. A closed set F in X is irreducible if it is non-empty and for all closed sets A, B in X, F ⊆ A ∪ B implies either F ⊆ A or F ⊆ B. A topological space is sober if every irreducible closed set in it is the closure of exactly one point. Sobriety is an interesting property in the realm of non-Hausdorff spaces and it plays an important role in domain theory [5]. In order to extend the theory of sober spaces to the fuzzy setting, the first step is to postulate irreducible closed sets in a Q-cotopological space. Fortunately, this can be done in a natural way with the help of the fuzzy inclusion order between the closed sets in a Q-cotopological space.

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Definition 3.1. A closed set F in a Q-cotopological space X is irreducible if

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x∈X F (x) = 1 and

subX (F, A ∨ B) = subX (F, A) ∨ subX (F, B) for all closed sets A, B in X.

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that is left adjoint to the functor  [3,16]. The following conclusion says that in a stratified Q-cotopological space, the specialization Q-order is determined by closure of singletons, as in the classical case.

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This definition is clearly an extension of that of irreducible closed sets in a topological space. We hasten to emphasize that,  instead of the (crisp) pointwise order, the fuzzy inclusion order between closed sets is used here. The condition x∈X F (x) = 1 is a Q-version of the requirement that F is non-empty.

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Example 3.2.

Having the notion of irreducible closed sets (in a Q-cotopological space) at hand, we are now able to formulate the central notion of this paper. Definition 3.3. A Q-cotopological space X is sober if it is stratified and each irreducible closed in X is the closure of 1x for a unique x ∈ X.

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s(A)(F ) = subX (F, A).

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Lemma 3.4. Let X be a stratified Q-cotopological space. (1) (2) (3)

s(pX )(F ) = p for all p ∈ Q and F ∈ irr(X). s(A  ∨ B) = s(A)  ∨ s(B) for all closed sets A, B in X. s Aj = s(Aj ) for each set {Aj }i∈J of closed sets in X. j ∈J

j ∈J

(4) s(p → A) = p → s(A) for all p ∈ Q and all closed sets A in X. (5) subX (A, B) = subirr(X) (s(A), s(B)) for all closed sets A, B in X. Proof. We check (5) for example. On one hand,  subX (A, B) ≤ (subX (F, A) → subX (F, B)) = subirr(X) (s(A), s(B)). F ∈irr(X)

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On the other hand, subirr(X) (s(A), s(B)) =



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(s(A)(F ) → s(B)(F ))

F ∈irr(X)

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by

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s(A) : irr(X) −→ Q

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denote the set of all irreducible closed sets in X. For each closed set A in X, define

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irr(X)

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Given a stratified Q-cotopological space X, let

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• sobriety and Hausdorff separation in a stratified Q-cotopological space; • sober topological spaces and sober Q-cotopological spaces via the Lowen functor ωQ ; • sober Q-cotopological spaces and sober Q-topological spaces in the case that Q satisfies the law of double negation.

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for the full subcategory of SQ-CTop consisting of sober Q-cotopological spaces. This section concerns basic properties of sober Q-cotopological spaces. First, we show that the subcategory SobQ-CTop is reflective in SQ-CTop and that the specialization Q-order of each sober Q-cotopological space X is directed complete in the sense that every irreducible fuzzy lower set in the Q-ordered set (X) has a supremum. Then we will discuss the relationship between

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SobQ-CTop

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Write

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(1) Let X be a stratified Q-cotopological space. For each x ∈ X, the closure 1x of 1x is irreducible. This follows from that subX (1x , A) = A(x) for each closed set A in X. (2) A fuzzy lower set φ in a Q-ordered set X is irreducible in the sense of Definition 2.5 if and only if φ is an irreducible closed set in the Alexandroff Q-cotopological space (X).

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(subX (F, A) → subX (F, B))

F ∈irr(X)

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≤

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(subX (1x , A) → subX (1x , B))

=

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(A(x) → B(x))

x∈X

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= subX (A, B).

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The proof is finished.

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{s(A) | A is a closed set of X} is a stratified Q-cotopology on irr(X). We will write s(X), instead of irr(X), for the resulting Q-cotopological space.

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F ∈s(X)

F ∈s(X)

z∈X

it follows that    A(x) ≥ F (x)&subX (F, A) x∈X F ∈s(X)

x∈X

=

 

F (x)&subX (F, A)

F ∈s(X) x∈X

=



subX (F, A)

F ∈s(X)

= 1. Step 2. subX (A, B ∨ C) = (subX (A, B)) ∨ (subX (A, C)) for all closed sets B, C in X. Since s(A) is irreducible in s(X), subX (A, B ∨ C) = subs(X) (s(A), s(B ∨ C)) = subs(X) (s(A), s(B) ∨ s(C))

43

= subX (A, B) ∨ subX (A, C).

48 49 50 51 52

14

16 17 18 19 20 21

23 24 25 26 27 28

30 31 32 33 34 35

= subs(X) (s(A), s(B)) ∨ subs(X) (s(A), s(C))

47

12

29

42

46

11

22

Since for each F ∈ s(X) and x ∈ X,  F (x)&subX (F, A) = F (x)& (F (z) → A(z)) ≤ A(x),

41

45

10

15

Proof. First of all, we note that for each irreducible closed set F in X, the closure of 1F in s(X) is given by s(F ). So, it suffices to show that for each closed set A in X, if s(A) is irreducible in s(X), then A is irreducible in X. We prove the conclusion in two steps.  Step 1. x∈X A(x) = 1. Since s(A) is an irreducible closed set in s(X), it holds that   s(A)(F ) = subX (F, A) = 1.

40

44

6

13

Proposition 3.5. s(X) is sober for each stratified Q-cotopological space X.

32 33

5

9

15 16

4

8

By the above lemma,

13 14

3

7

8 9

1 2

x∈X

3

7



Therefore, A is an irreducible closed set in X. 2 Proposition 3.6. For a stratified Q-cotopological space X, define ηX : X −→ s(X) by ηX (x) = 1x . Then (1) ηX : X −→ s(X) is continuous. (2) X is sober if and only if ηX is a homeomorphism.

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D. Zhang / Fuzzy Sets and Systems ••• (••••) •••–••• 1

Proof. (1) For all closed set A in X and x ∈ X,

2

5 6 7 8 9 10 11 12 13 14 15 16 17

1 2

← ηX (s(A))(x) = s(A)(ηX (x)) = A(x),

3 4

9

3

← (s(A)) = A. ηX

hence This shows that f is continuous. (2) If X is sober then each irreducible closed set in X is of the form 1x = ηX (x) for a unique x ∈ X, hence ηX : X −→ s(X) is a bijection. Since for each closed set A in X and x ∈ X, 

→ ηX (A)(1x ) = A(z) | ηX (z) = 1x = A(x) = s(A)(1x ),

4

→ (A) = s(A). This shows that η is a continuous closed bijection, hence a homeomorphism. The it follows that ηX X converse conclusion is trivial, since s(X) is sober by Proposition 3.5. 2

9

Theorem 3.7. Let f : X −→ Y be a continuous map between stratified Q-cotopological spaces. If Y is sober, there is a unique continuous map f ∗ : s(X) −→ Y such that f = f ∗ ◦ ηX . Lemma 3.8. Let f : X −→ Y be a continuous map between stratified Q-cotopological spaces. Then for each irreducible closed set F of X, the closure f → (F ) of the image of F under f is an irreducible closed set of Y .

18 19

22 23 24 25

hence

f → (F )

is irreducible.

28

31 32 33 34 35

42 43

46 47 48 49 50 51 52

13 14 15 16 17

(F is irreducible)

23

2

= subY

22

24 25 26 27 28

(f → (F ), B) = sub

Y (1f ∗ (F ) , B)

= B(f ∗ (F )) = B ◦ f ∗ (F ), thus, B ◦ f ∗ = s(A) and is closed in s(X). Second, for each x ∈ X, since 1f (x) ≤ f → (1x ) ≤ f → (1x ) = 1f (x) ,

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

it follows that

44 45

12

21

s(A)(F ) = subX (F, f ← (B)) = subY (f → (F ), B)

38

41

11

(f →  f ← )

Proof of Theorem 3.7. Existence. For each F ∈ s(X), since is an irreducible closed set of Y and Y is sober, there is a unique y ∈ Y such that f → (F ) equals the closure of 1y . Define f ∗ (F ) to be this y. We claim that f ∗ : s(X) −→ Y satisfies the conditions. First, we show that B ◦ f ∗ is a closed set in s(X) for each closed set B in Y , hence f ∗ is continuous. Let A be the closed set f ← (B) = B ◦ f in X. Then for all F ∈ s(X),

37

40

10

20

f → (F )

36

39

8

(A ∨ B is closed)

= subY (f → (F ), A) ∨ subY (f → (F ), B),

26

30

7

19

subY (f → (F ), A ∨ B) = subY (f → (F ), A ∨ B) = subX (F, f ← (A ∨ B)) = subX (F, f ← (A)) ∨ subX (F, f ← (B)) = subY (f → (F ), A) ∨ subY (f → (F ), B)

21

29

6

18

Proof. For all closed sets A, B in Y ,

20

27

5

44

1f (x) = f → (1x ) = f → (ηX (x)), f ∗ (η

45

f∗

hence ◦ ηX = f . X (x)) = f (x), showing that Therefore, f ∗ : s(X) −→ Y satisfies the conditions. Uniqueness. Since Y is sober, it suffices to show that if g : s(X) −→ Y is a continuous map such that f = g ◦ ηX , then 1g(F ) = f → (F ) for all F ∈ s(X). Since for all E ∈ s(X), (s(X))(E, F ) = 1F (E) = subX (E, F ),

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D. Zhang / Fuzzy Sets and Systems ••• (••••) •••–•••

10

1 2 3 4

it follows that for all x ∈ X,

1 2

F (x) = subX (1x , F ) = (s(X))(ηX (x), F )

3 4

5

≤ (Y )(g(ηX (x)), g(F ))

5

6

= (Y )(f (x), g(F ))

6

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

= 1g(F ) (f (x)),

8

f → (F ) ≤ 1

→ showing that g(F ) , hence f (F ) ≤ 1g(F ) . → (F ) = s(F ) first. On one hand, since η→ (F ) ≤ s(F ), it follows For the converse inequality, we show that ηX X → → (F ) is closed in s(X), there is some closed set A in X such that that ηX (F ) ≤ s(F ). On the other hand, since ηX → (F ) = s(A). For each x ∈ X, ηX → F (x) ≤ ηX (F )(ηX (x)) ≤ s(A)(ηX (x)) = A(x), → (F ). Therefore, hence s(F ) ≤ s(A) = ηX → (F )(g(F )) ≥ g → (s(F ))(g(F )) ≥ s(F )(F ) = 1, f → (F )(g(F )) = g → ◦ ηX

hence 1g(F ) ≤ f → (F ).

37 38 39 40 41 42 43 44 45

10 11 12 13 14 15 16 17

19

Theorem 3.7 shows that the full subcategory of sober Q-cotopological spaces is reflective in SQ-CTop. For every Q-cotopological space X, the sober space s(X) is called the sobrification of X. An important property of sober spaces is that the specialization order of a sober space is directed complete [5,10]. The following Proposition 3.10 says this is also true in the quantale-valued setting if we treat irreducible fuzzy lower sets as “directed fuzzy lower sets”.

20

Definition 3.9. ([29,15]) A supremum of a fuzzy lower set φ in a Q-ordered set X is an element sup φ in X such that X(sup φ, x) = subX (φ, X(−, x))

26

for all x ∈ X.

21 22 23 24 25

27 28 29

The notion of supremum of a fuzzy lower set in a Q-ordered set is a special case of that of weighted colimit in category theory [11].

30

Proposition 3.10. Let X be a sober Q-cotopological space. Then each irreducible fuzzy lower set in the specialization Q-order of X has a supremum.

33

31 32

34 35

Proof. Let φ be an irreducible fuzzy lower set in the Q-ordered set (X). First, we show that the closure φ of φ in X is an irreducible closed set. Let A, B be closed sets in X. By definition of the specialization Q-order, both A and B are fuzzy lower sets in (X). Hence subX (φ, A ∨ B) = subX (φ, A ∨ B) = subX (φ, A) ∨ subX (φ, B) = subX (φ, A) ∨ subX (φ, B), showing that φ is an irreducible closed set in X. Since X is sober, there is a unique a ∈ X such that φ = 1a . We claim that a is a supremum of φ in (X). That is, for all x ∈ X, (X)(a, x) = subX (φ, (X)(−, x)). Since (X)(z, x) = 1x (z) for all x, z ∈ X, it follows that subX (φ, (X)(−, x)) = subX (φ, 1x )

46 47

= subX (φ, 1x )

48

= subX (1a , 1x )

49

= subX (1a , 1x )

50

= (X)(a, x).

51 52

9

18

2

35 36

7

This completes the proof.

2

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D. Zhang / Fuzzy Sets and Systems ••• (••••) •••–••• 1 2 3 4 5 6 7 8 9 10

11

It is well-known that a Hausdorff topological space is always sober [5,10]. The following proposition says this is also true for Q-cotopological spaces if Q is linearly ordered. A Q-cotopological space X is Hausdorff if the diagonal : X × X −→ Q, given by

1, x = y, (x, y) = 0, x = y,

1

is a closed set in the product space X × X.

7

Proposition 3.11. Let Q = (Q, &) be a linearly ordered quantale. Then each stratified Hausdorff Q-cotopological space is sober.

9

11 12 13 14 15 16 17

20

Proof. Let X be a stratified Hausdorff Q-cotopological space. In order to see that X is sober, it suffices to show that if F is an irreducible closed set in X, then F (x) = 0 for at most one point x in X. Suppose on the contrary that there exist different x, y in X such that F (x) > 0 and F (y) > 0. Let b = min{F (x), F (y)}. Then b > 0 by linearity of Q. Since X is Hausdorff, there exist two sets of closed sets in X, say, {Aj }j ∈J and {Bj }j ∈J , such that  (x, y) = (Aj (x) ∨ Bj (y)). j ∈J

Since (x, y) = 0, there exists some i ∈ J such that Ai (x) ∨ Bi (y) < b. Since Ai (z) ∨ Bi (z) = 1 for all z ∈ X, we have either F ≤ Ai or F ≤ Bi , hence either F (x) < b or F (y) < b, a contradiction. 2

21 22 23 24 25

28 29 30 31 32 33 34 35 36 37

λ[p] = {x ∈ X | λ(x) ≥ p}

39

41 42 43 44 45 46 47 48 49 50 51 52

5 6

8

10

12 13 14 15 16 17 18 19 20

22 23 24 25 26

An element a in a lattice L is a coprime if for all b, c ∈ L, a ≤ b ∨ c implies that either a ≤ b or a ≤ c [10]. A complete lattice L is said to have enough coprimes if every element in L can be written as the join of a set of coprimes. It is clear that every linearly ordered quantale has enough coprimes and the complete lattice of closed sets in a topological space has enough coprimes. We say that an element in a quantale Q = (Q, &) is a coprime if it is a coprime in the underlying lattice Q; and that Q has enough coprimes if the complete lattice Q has enough coprimes. It is easily seen that if 1 ∈ Q is a coprime and if F is an irreducible closed set in a Q-cotopological space X, then for all closed sets A, B in X, F ≤ A ∨ B implies either F ≤ A or F ≤ B. Said differently, in this case, an irreducible closed set in a Q-cotopological space is a coprime in the lattice of its closed sets. Let Q be a quantale and X be a (crisp) topological space. We say that a map λ : X −→ Q is upper semicontinuous if for all p ∈ Q,

38

40

4

21

Note 3.12. The assumption that Q is linearly ordered is indispensable in the above proposition. To see this, let Q = {0, a, b, 1} be the Boolean algebra with four elements; let X be the discrete Q-cotopological space with two points x and y. It is clear that X is Hausdorff. One can verify by enumerating all possibilities that the map λ, given by λ(x) = a and λ(y) = b, is an irreducible closed set in X, but it is neither the closure of 1x nor that of 1y .

26 27

3

11

18 19

2

is a closed set in X. Lemma 3.13. Let Q be a quantale with enough coprimes and X be a topological space. (1) (2) (3) (4)

λ : X −→ Q is upper semicontinuous if and only if λ[p] is a closed set in X for each coprime p in Q. If both λ, μ : X −→ Q are upper semicontinuous then so is λ ∨ μ. The meet of every set of upper semicontinuous maps is upper semicontinuous. If λ : X −→ Q is upper semicontinuous then so is p → λ for all p ∈ Q.

Proof. (1) follows from the fact that Q has enough coprimes and (2) is an immediate consequence of (1). The verification of (3) is straightforward. And (4) follows from (p → λ)[q] = {x ∈ X | q ≤ p → λ(x)} = {x ∈ X | p&q ≤ λ(x)} = λ[p&q] for all p, q ∈ Q.

2

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D. Zhang / Fuzzy Sets and Systems ••• (••••) •••–•••

12

1 2 3 4 5 6 7 8 9 10 11 12

The above lemma shows that if Q is a quantale with enough coprimes, then for each topological space X, the set of upper semicontinuous maps X −→ Q forms a stratified Q-cotopology on X. We write ωQ (X) for the resulting stratified Q-cotopological space. For each closed set K in X, 1K : X −→ Q is obviously upper semicontinuous, hence every closed set in X is also a closed in ωQ (X). Moreover, for each A ⊆ X, the closure of 1A in ωQ (X) equals 1A , where A is the closure of A in X. The correspondence X → ωQ (X) defines an embedding functor ωQ : Top −→ SQ-CTop. This functor is one of the well-known Lowen functors in fuzzy topology [18]. The following conclusion says that for a linearly ordered quantale Q, the notion of sobriety for Q-cotopological spaces is a good extension in the sense of Lowen [18].

13 14 15

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

37 38 39 40 41 42 43 44 45

x∈K

For all coprime p ≤ subX (1K , λ ∨ μ), K is clearly a subset of (λ ∨ μ)[p] = λ[p] ∪ μ[p] , hence either K ⊆ λ[p] or K ⊆ μ[p] , and then either p ≤ subX (1K , λ) or p ≤ subX (1K , μ). Therefore, subX (1K , λ ∨ μ) ≤ subX (1K , λ) ∨ subX (1K , μ). The converse inequality subX (1K , λ ∨ μ) ≥ subX (1K , λ) ∨ subX (1K , μ) is trivial. Thus, 1K is an irreducible closed set in ωQ (X). Since ωQ (X) is sober, there is a unique x ∈ X such that 1K is the closure of 1x in ωQ (X). Because the closure of 1x in ωQ (X) equals 1{x} , one gets K = {x}. 2

46 47 48 49

52

4 5 6 7 8 9 10 11 12

14 15

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

Note 3.15. The assumption in Proposition 3.14 that Q is linearly ordered is indispensable. To see this, let Q = {0, a, b, 1} be the Boolean algebra with four elements; let X be the discrete (hence sober) topological space with two points x and y. It is clear that ωQ (X) is the discrete Q-cotopological space in Note 3.12, hence it is not sober.

50 51

3

16

Proof. Necessity. Let λ be an irreducible closed in ωQ (X). Firstly, we show that for each x ∈ X, the value λ(x) is either 0 or 1. Suppose on the contrary that there is some x ∈ X such that λ(x) is neither 0 nor 1. We proceed with two cases. If there is no element y in X such that λ(y) is strictly between λ(x) and 1, let φ = 1λ[1] and ψ be the constant map X −→ Q with value λ(x). Then both φ and ψ are closed in ωQ (X) and λ ≤ φ ∨ ψ, but neither λ ≤ φ nor λ ≤ ψ, contradictory to that λ is irreducible. If there is some y ∈ X such that λ(x) < λ(y) < 1, let φ = 1λ[λ(y)] and ψ be the constant map X −→ Q with value λ(y). Then both φ and ψ are closed in ωQ (X) and λ ≤ φ ∨ ψ , but neither λ ≤ φ nor λ ≤ ψ, contradictory to that λ is irreducible. Therefore, λ = 1K for some closed set K in X. Since λ is irreducible in ωQ (X), K must be an irreducible closed set in X. Thus, in the topological space X, K is the closure of {x} for a unique x, i.e., K = {x}. This shows that in ωQ (X), λ is the closure of 1x for a unique x. Hence ωQ (X) is sober. Sufficiency. We prove a bit more, that is, if Q has enough coprimes and ωQ (X) is sober, then X is sober. Let K be an irreducible closed set in X. Firstly, we show that 1K is an irreducible closed set in the Q-cotopological space ωQ (X). That 1K : X −→ Q is upper semicontinuous is trivial. For all closed sets λ, μ in ωQ (X), one has by definition that  subX (1K , λ ∨ μ) = (λ(x) ∨ μ(x)).

35 36

2

13

Proposition 3.14. If Q is a linearly ordered quantale, then a topological space X is sober if and only if the Q-cotopological space ωQ (X) is sober.

16 17

1

47 48 49 50

At the end of this section, we discuss the relationship between sober Q-cotopological spaces and sober Q-topological spaces in the case that the quantale Q satisfies the law of double negation.

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13

Definition 3.16. A Q-topology on a set X is a subset τ of QX subject to the following conditions:

2 3 4 5

2

(O1) pX ∈ τ for all p ∈ Q; (O2)  U ∧ V ∈ τ for all U, V ∈ τ ; (O3) j ∈J Uj ∈ τ for each subset {Uj }j ∈J of τ .

6 7

10

15

24

and

28



29

31

34 35 36

A(x)&B(x) = ¬subX (A, ¬B) = ¬subX (B, ¬A).

41 42 43

x∈X

satisfies the following conditions:

46

fF (pX ) = p. fF (U (U ) ∧ fF (V ).  ∧ V ) = fF = U fF i∈I i i∈I fF (Ui ). fF (p&U ) = p&fF (U ).

49 50 51 52

28 29

31

33 34 35 36 37 38

40 41 42 43 44

Conversely, if g : ¬(τ ) −→ Q is a map satisfying (Fr1)–(Fr4), then there is a unique irreducible closed set F in (X, τ ) such that g = fF .

47 48

24

39

(Fr1) (Fr2) (Fr3) (Fr4)

44 45

23

32

Proposition 3.17. Let Q be a quantale that satisfies the law of double negation; and let (X, τ ) be a stratified Q-cotopological space. Then for each irreducible closed set F in (X, τ ), the map  fF : ¬(τ ) −→ Q, fF (U ) = F (x)&U (x)

39 40

22

30

These equations are clearly extensions of the properties listed in Proposition 2.2.

37 38

19

27

32 33

18

26

x∈X

30

15

25

subX (A, B) = subX (¬B, ¬A)

26

14

21

is a (stratified) Q-cotopology on X. So, for a quantale Q that satisfies the law of double negation, we can switch freely between (stratified) Q-topologies and (stratified) Q-cotopologies, hence between open sets and closed sets. If Q = (Q, &) satisfies the law of double negation, then for all A, B ∈ QX ,

25

27

12

20

¬(τ ) = {¬A | A ∈ τ }

21

23

10

17

is a (stratified) Q-topology on X, where ¬A(x) = ¬(A(x)) for all x ∈ X. Conversely, if τ is a (stratified) Q-topology on X, then

20

22

9

16

¬(τ ) = {¬A | A ∈ τ }

17

19

7

13

It is clear that if Q = (Q, &) is a frame, i.e., if & = ∧, then every Q-topology is stratified. Let Q be a quantale that satisfies the law of double negation. If τ is a (stratified) Q-cotopology on a set X, then

16

18

5

11

(O4) p&U ∈ τ for all p ∈ Q and U ∈ τ .

13 14

4

8

A Q-topological space in the above definition is also called a weakly stratified Q-topological space in the literature, see e.g. [8,9]. A Q-topology τ is stratified [9] if

11 12

3

6

The pair (X, τ ) is called a Q-topological space; elements in τ are called open sets of (X, τ ).

8 9

1

45 46 47

Proof. We check (Fr2) for example. fF (U ∧ V ) = ¬(subX (U ∧ V , ¬F )) = ¬(subX (F, ¬(U ∧ V ))) = ¬(subX (F, ¬U ∨ ¬V ))

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D. Zhang / Fuzzy Sets and Systems ••• (••••) •••–•••

14

= ¬(subX (F, ¬U ) ∨ subX (F, ¬V ))

1 2

= ¬(subX (U, ¬F )) ∧ ¬(subX (V , ¬F ))

3

= fF (U ) ∧ fF (V ).

4 5 6 7 8 9 10 11 12 13 14 15 16 17

Conversely, suppose g : ¬(τ ) −→ Q is a map that satisfies (Fr1)–(Fr4). Let  F = {A ∈ τ | g(¬A) = 0}. We show that F is an irreducible closed set in (X, τ ) and g = fF . Step 1. g(¬F ) =0. This follows from (Fr3) and Proposition 2.2(3). Step 2. g(U ) = x∈X F (x)&U (x) for all U ∈ ¬(τ ). On one hand, if we let p = subX (U, ¬F ), then p&U ≤ ¬F , hence p&g(U ) = g(p&U ) ≤ g(¬F ) = 0. Therefore,

20 21 22 23 24 25 26 27 28

2 3 4 5 6 7 8 9 10 11 12 13 14 15

g(U ) ≤ ¬subX (U, ¬F ) =



16

F (x)&U (x).

x∈X

18 19

1

On the other hand, since g(¬(g(U ))&U ) = ¬(g(U ))&g(U ) = 0, it follows that ¬(g(U ))&U ≤ ¬F by definition of F . Therefore, ¬(g(U )) ≤ subX (U, ¬F ), hence  g(U ) ≥ ¬subX (U, ¬F ) = F (x)&U (x). x∈X

  Step 3. x∈X F (x) = 1. Otherwise, let x∈X ¬F (x) = p. Then p = 0 and pX ≤ ¬F . Therefore, g(¬F ) ≥ g(pX ) = p, contradictory to that g(¬F ) = 0. Step 4. subX (F, A ∨ B) = subX (F, A) ∨ subX (F, B) for all closed sets A, B in (X, τ ). In fact, subX (F, A ∨ B) = subX (¬A ∧ ¬B, ¬F )

17 18 19 20 21 22 23 24 25 26 27 28

29

= ¬(g(¬A ∧ ¬B))

29

30

= ¬(g(¬A)) ∨ ¬(g(¬B))

30

31 32

= subX (¬A, ¬F ) ∨ subX (¬B, ¬F )

33

= subX (F, A) ∨ subX (F, B).

34 35 36 37 38 39 40

The proof is completed.

2

For all stratified Q-topological space (X, τ ) and x ∈ X, the map fx : τ −→ Q,

fx (U ) = U (x)

clearly satisfies (FR1)–(FR4) in Proposition 3.17. This fact leads to the following:

41 42 43

46

49

52

34 35 36 37 38 39 40

42 43

45 46 47

Proposition 3.19. Let Q be a quantale that satisfies the law of double negation. Then a Q-topological space (X, τ ) is sober if and only if the Q-cotopological space (X, ¬(τ )) is sober.

50 51

33

44

We leave it to the reader to check that if Q = (Q, &) is a frame, i.e., & = ∧, then the above definition of sober Q-topological space coincides with that in [33].

47 48

32

41

Definition 3.18. A stratified Q-topological space (X, τ ) is sober if for each map f : τ −→ Q satisfying (FR1)–(FR4) in Proposition 3.17, there is a unique x ∈ X such that f (U ) = U (x) for all U ∈ τ .

44 45

31

48 49 50

Proof. Necessity: Let F be an irreducible closed set in the Q-cotopological space (X, ¬(τ )). By Proposition 3.17, the map

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D. Zhang / Fuzzy Sets and Systems ••• (••••) •••–••• 1

fF : τ −→ Q,

fF (U ) =

2 3 4 5 6



15

F (x)&U (x)

1 2

x∈X

satisfies (Fr1)–(Fr4), hence there is a unique a ∈ X such that fF (U ) = U (a) for all U ∈ τ . We claim that the closure of 1a in (X, ¬(τ )) is F . Since  ¬F (a) = fF (¬F ) = F (x)&(¬F (x)) = 0,

3 4 5 6

7

x∈X

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8

then F (a) = 1. Therefore, 1a ≤ F . Conversely, since  F (x)&¬(1a )(x) = fF (¬(1a )) = ¬(1a )(a) = 0,

8

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x∈X

12

for all x ∈ X, hence F ≤ 1a . Sufficiency. Let f : τ −→ Q be a map satisfying (FR1)–(FR4).  By Proposition 3.17, there is an irreducible closed set F in the Q-cotopological space (X, ¬(τ )) such that f (U ) = x∈X F (x)&U (x) for all U ∈ τ . Since (X, ¬(τ )) is sober, there is a unique a ∈ X such that 1a = F . Therefore,  f (U ) = F (x)&U (x) x∈X

23

= ¬subX (1a , ¬U )

24

= U (a),

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13

F (x) ≤ ¬(¬(1a )(x)) = 1a (x)

= ¬subX (F, ¬U )

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10 11

it follows that

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9

completing the proof.

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(1a = F, ¬U is closed)

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2

4. Examples This section discusses the sobriety of some natural Q-cotopological spaces in the case that Q is the unit interval [0, 1] endowed with a (left) continuous t-norm. In this section, we will write a, instead of a[0,1] , for the constant map [0, 1] −→ [0, 1] with value a. Let Q = ([0, 1], &) with & being a (left) continuous t-norm on [0, 1]. We consider three Q-cotopologies on [0, 1]: • τC& : the stratified Q-cotopology on [0, 1] generated by {id} as a subbasis; • τS& : the strong Q-cotopology on [0, 1] generated by {id} as a subbasis; • τA& : the Alexandroff Q-cotopology on the Q-ordered set ([0, 1], dR ), where dR (x, y) = y → x. First of all, we list some facts about these Q-cotopologies. (F1) A closed set in ([0, 1], τA& ) is, by definition, a Q-order-preserving map φ : ([0, 1], dL ) −→ ([0, 1], dL ), where dL (x, y) = x → y. So, it is easy to verify that for all φ ∈ τA& : • φ is increasing, i.e., φ(x) ≤ φ(y) whenever x ≤ y; • φ(1) = 1 ⇐⇒ φ ≥ id. (F2) For each x ∈ [0, 1], the closure of 1x in ([0, 1], τA& ) is x → id, i.e., 1x = x → id. On one hand, x → id is a closed set in ([0, 1], τA& ) and (x → id)(x) = 1, hence 1x ≤ x → id. On the other hand, for all φ ∈ τA& , if φ(x) = 1, then φ(t) = φ(x) → φ(t) ≥ x → t for all t ≤ x, hence φ ≥ x → id. (F3) Since every Alexandroff Q-cotopology is a strong Q-cotopology and id ∈ τA& , it follows that τC& ⊆ τS& ⊆ τA& . Moreover, since x → id ∈ τC& for all x ∈ X, the closure of 1x in both ([0, 1], τC& ) and ([0, 1], τS& ) is x → id.

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(F4) Since finite joins and arbitrary meets of right continuous maps [0, 1] −→ [0, 1] are right continuous, and x → id is right continuous for all x ∈ [0, 1], it follows that every closed set in ([0, 1], τC& ), as a map from [0, 1] to itself, is right continuous. (F5) The space ([0, 1], τC& ) (([0, 1], τS& ), resp.) is initially dense in the topological category of stratified (strong, resp.) Q-cotopological spaces. For instance, for each stratified Q-cotopological space (X, τ ),

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A

{(X, τ ) −→ ([0, 1], τC& )}A∈τ

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is an initial source in the category of stratified Q-cotopological spaces.

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Proposition 4.1. Let Q = ([0, 1], &) with & being a left continuous t-norm on [0, 1]. Then the stratified Q-cotopological space ([0, 1], τC& ) is sober.

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Proof. It suffices to show that if φ is an irreducible closed set in ([0, 1], τC& ), then φ = x → id = 1x for some x ∈ [0, 1]. Since φ is increasing and t∈[0,1] φ(t) = 1, one obtains that φ(1) = 1. Let x = inf{t ∈ [0, 1] | φ(t) = 1}. Since φ is right continuous by (F4), then φ(x) = 1, hence φ ≥ 1x = x → id. We claim that φ = x → id. Otherwise, there is some t < x such that φ(t) > x → t. Since x → t = y y → t. It is clear that both y → id and φ(y) belong to τC& and φ ≤ (y → id) ∨ φ(y), but neither φ ≤ (y → id) nor φ ≤ φ(y), contradictory to that φ is irreducible. 2 In the following we consider sobriety of the Q-cotopologies τS& and τA& in the case that & is the minimum t-norm, the product t-norm and the Łukasiewicz t-norm. Proposition 4.2. Let Q = ([0, 1], &) with & being the product t-norm. Then the Alexandroff Q-cotopology τAP on ([0, 1], dR ) is not sober; but the strong Q-cotopology τSP on [0, 1] generated by {id} is sober. Proof. By Example 3.11 in [16], the Alexandroff Q-cotopology τAP consists of maps φ : [0, 1] −→ [0, 1] subject to the following conditions: (i) φ is increasing; and (ii) y/x ≤ φ(y)/φ(x) whenever x > y, where we agree by convention that 0/0 = 1.

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1

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We note that each φ in τAP is continuous on (0, 1], but it may be discontinuous at 0. It is easy to verify that the map φ : [0, 1] −→ [0, 1], given by φ(0) = 0 and φ(t) = 1 for all t > 0, is an irreducible closed set in ([0, 1], τAP ), but it is not the closure of 1x for any x ∈ [0, 1]. So, ([0, 1], τAP ) is not sober. In the following we prove in three steps that τSP on [0, 1] is sober. Step 1. We show that the stratified Q-cotopology τCP on [0, 1] generated by {id} is given by τCP = {φ ∧ a | a ∈ [0, 1], φ ∈ B}, where,

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B = {φ ∈ τAP | φ ≥ id, φ is continuous}. It is routine to verify that C = {φ ∧ a | a ∈ [0, 1], φ ∈ B} is a stratified Q-cotopology on [0, 1] that contains the identity id : [0, 1] −→ [0, 1], hence τCP ⊆ C. To see that C is contained in τCP , it suffices to check that B ⊆ τCP . Let φ ∈ B. For each x ∈ (0, 1], define gx : [0, 1] −→ [0, 1] by  x  gx = φ(x) ∨ ((φ(x) → x) → t) = φ(x) ∨ → id . φ(x) Then gx ∈ τCP . We leave it to the reader to check that  x     φ= gx = → id , φ(x) ∨ φ(x) x∈(0,1]

x∈(0,1]

hence φ ∈ τCP . Therefore, C ⊆ τCP .

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Step 2. We show that

2

τSP = {φ ∈ τAP | φ is continuous}.

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17

It is easily verified that S = {φ ∈ τAP | φ is continuous} is a strong Q-cotopology on [0, 1] that contains the identity id : [0, 1] −→ [0, 1], hence τSP ⊆ S. Conversely, for each φ ∈ S with φ(1) > 0, let ψ = φ(1) → φ = φ/φ(1). Then ψ ∈ τAP , ψ(1) = 1, and ψ is continuous. Thus, ψ ∈ B ⊆ τSP . Since τSP is strong and φ = φ(1)&ψ , then φ ∈ τSP , therefore S ⊆ τSP .  Step 3. ([0, 1], τSP ) is sober. Suppose φ is an irreducible closed set in ([0, 1], τSP ). Since φ is increasing and t∈[0,1] φ(t) = 1, one has φ(1) = 1, hence φ ∈ B ⊂ τCP . Since τCP is coarser than τSP , then φ is an irreducible closed set in the sober space ([0, 1], τCP ), and consequently, φ = x → id for a unique x ∈ [0, 1]. This shows that φ is the closure of 1x for a unique x ∈ [0, 1] in ([0, 1], τSP ), hence ([0, 1], τSP ) is sober. 2

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Proposition 4.3. Let Q = ([0, 1], &) with & being the Łukasiewicz t-norm. Then the strong Q-cotopology τSL on [0, 1] generated by {id} is sober and coincides with the Alexandroff Q-cotopology τAL on the Q-ordered set ([0, 1], dR ).

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(i) φ is increasing; and (ii) φ is 1-Lipschitz, i.e., φ(x) − φ(y) ≤ x − y for all x ≥ y.

19

Firstly, we show that the stratified Q-cotopology τCL on [0, 1] generated by {id} is given by τCL = {φ ∧ a | a ∈ [0, 1], φ ∈ τAL , φ ≥ id}. It is routine to verify that C = {φ ∧ a | a ∈ [0, 1], φ ∈ τAL , φ ≥ id} is a stratified Q-cotopology on [0, 1] that contains the identity id : [0, 1] −→ [0, 1], hence τCL ⊆ C. To see that C is contained in τCL , it suffices to check that for all φ ∈ τAL , if φ ≥ id then φ ∈ τCL . For each x ∈ [0, 1], define gx : [0, 1] −→ [0, 1] by gx = φ(x) ∨ ((φ(x) → x) → id),

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i.e.,

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gx (t) = max{φ(x), min{φ(x) + t − x, 1}}. Then gx ∈ τCL . We leave it to the reader to check that   φ= gx = (φ(x) ∨ ((φ(x) → x) → id)), x∈[0,1]

x∈[0,1]

hence φ ∈ τCL . Therefore, C ⊆ τCL . Secondly, we show that τSL = τAL . For any φ ∈ τAL with φ(1) > 0, it is clear that φ(1) → φ ∈ τAL and (φ(1) → φ)(1) = 1. Thus, φ(1) → φ ∈ τCL ⊆ τSL . Since τSL is strong and φ = φ(1)&(φ(1) → φ), it follows that φ ∈ τSL , hence τAL ⊆ τSL . The converse inclusion τSL ⊆ τAL is trivial. Finally, we show that ([0, 1], τSL ) is sober. Let φ be an irreducible closed set in ([0, 1], τSL ). Since φ is increasing  and t∈[0,1] φ(t) = 1, then φ(1) = 1, hence φ ∈ τCL . Since τCL is coarser than τSL , it follows that φ is an irreducible closed set in the sober space ([0, 1], τCL ), hence φ = x → id for a unique x ∈ [0, 1]. This shows that φ is the closure of 1x for a unique x ∈ [0, 1] in ([0, 1], τSL ). Therefore, ([0, 1], τSL ) is sober. 2

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Proposition 4.4. Let Q = ([0, 1], &) with & being the t-norm min. Then the strong Q-cotopology τSM on [0, 1] generated by {id} is sober; but the Alexandroff Q-cotopology τAM on the Q-ordered set ([0, 1], dR ) is not sober.

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5

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4

Proof. By Example 3.10 in [16], the Alexandroff Q-cotopology τAL on ([0, 1], dR ) consists of maps φ : [0, 1] −→ [0, 1] that satisfy the following conditions:

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3

15

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2

12

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1

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Proof. First of all, since & = min, every stratified Q-cotopological space is obviously a strong Q-cotopological space, then the strong Q-cotopology τSM on [0, 1] generated by {id} coincides with the stratified Q-cotopology τCM on [0, 1] generated by {id}, hence it is sober by Proposition 4.1.

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With the help of Example 3.12 in [16], it can be verified that

2

τAM = {φ ∧ a | a ∈ [0, 1], φ : [0, 1] −→ [0, 1] is increasing, φ ≥ id}.

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For any a ∈ (0, 1), the map φ : [0, 1] −→ [0, 1] given by

1, t > a, φ(t) = t, t ≤ a, is an irreducible closed set in ([0, 1], τAM ), and it is not the closure of 1x for any x ∈ [0, 1], so, ([0, 1], τAM ) is not sober. We would like to record here that τCM = τSM = {φ ∈ τAM | φ is right continuous}.

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It is easily verified that C = {φ ∈ τAM | φ is right continuous} is a stratified Q-cotopology on [0, 1] that contains the identity id : [0, 1] −→ [0, 1], hence τCM ⊆ C. To see the converse inclusion, we show firstly that for all φ ∈ C, if φ(1) = 1 then φ ∈ τCM . For each x ∈ [0, 1], define gx : [0, 1] −→ [0, 1] by gx = φ(x) ∨ (x → id). Then gx ∈ τCM . Since   φ= gx = (φ(x) ∨ (x → id)),

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x∈[0,1]

then φ ∈ τCM . Secondly, for any φ ∈ C, since φ(1) → φ ∈ C, (φ(1) → φ)(1) = 1, and φ = φ(1) ∧ (φ(1) → φ), it follows that φ ∈ τCM . Therefore, C ⊆ τCM . 2 Acknowledgement The author thanks Dr. Hongliang Lai for the stimulating discussions during the preparation of this paper and thanks the reviewers for their valuable comments and helpful suggestions. References [1] [2] [3] [4] [5] [6]

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