Separation axioms in fuzzy topological ordered spaces

FUZZ'Y

sets and systems ELSEVIER

Fuzzy Sets and Systems 98 (1998) 211-215

Separation axioms in fuzzy topological ordered spaces K . E 1 - S a a d y a'*, M . Y . B a k e i r b aMathematics Department, Faculty of Science, South Valley University, Qena 83511, Egypt bMathematics Department, Faculty of Science, Assiut University, Assiut, Egypt

Received March 1996; accepted October 1997

Abstract Since the fuzzy topological space (X, z) may be considered as a fuzzy topological ordered space when it is realised that the non-empty set X is partially ordered by agreeing that x ~< y in X if and only ifx = y. Then the study of the fuzzy topological ordered spaces not only includes the study of the abstract fuzzy topological spaces but also reveals many generalizations of well-known results concerning the abstract fuzzy topological spaces. This paper provides a certain number of separation axioms for fuzzy topological ordered spaces, which we label FTi-order separation axioms (for i = 1,2, 3, 4). The relationships between some of the FTi-order separation axioms are studied. (~) 1998 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy Hausdorffness; Fuzzy normality and fuzzy topological ordered spaces

O. Introduction The study of the relationship between fuzzy topology and order was initiated by Katsaras in [4]. In the area of the separation axiom he gave the fuzzification of the Nachbin's normality [5] and he proved a result analogous to the Urysohn's Lemma. In this paper, we introduce and study a number of separation properties in the area of fuzzy topology on order, which we label FTi-order separation axioms (for i -----1,2, 3, 4) and we show that each FTi-order axiom is successively stronger than the FTi_ l -order axiom for i = 2, 3, 4. 1. Preliminaries Let X be a non-empty set. A preorder on X is a relation ~< on X which is reflexive and transitive. * Corresponding author.

A preorder o n X which is also anti-symmetric is called a partial order or simply an order. By a preordered (resp. an ordered) set we mean a s e t X with a preorder (resp. an order) on it and we denote it by (X, ~< ). A preordered (resp. an ordered) set on which there is given a fuzzy topology is called a fuzzy topological preordered (resp. ordered) space [4] and we denote it by (X, ~<,r). Definition I . I (Katsaras [4]), A fuzzy set #, in apreordered set (X, ~< ), is called (i) increasing i f x ~< y implies #(x) ~< #(y); (ii) decreasing i f x ~< y implies #(x)/> #(y); (iii) order-convex if x ~< y ~< z implies #(y)j> min{p(x), #(y)}. The smallest increasing fuzzy set, in (X, ~< ), which contains # is called the increasing hull denoted by i(#). The decreasing (resp. order-convex) hull d(#) (resp. c(#)) is defined analogously.

0165-0114/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved PH S01 65-01 14(97)00359-X

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Proposition 1.2 (Katsaras [4]). Let # be a fuzzy set in a preordered set (X, <~ ). Then, for each x EX, we have (i) i(p)(x) = sup{p(y): y ~< x}; (ii) d(p)(x) =- sup{p(y): y>~x}; (iii) c ( # ) ( y ) = sup{min{/ffx),~ffz)}: x ~< y ~< z}. Definition 1.3 (Katsaras [4]). Let p be a fuzzy set, in a preordered set (X, ~< ) with a fuzzy topology z. Then we have (i) D(p) = inf{p: p t> p, p is closed and decreasing}; (ii) I(p) -- inf{p: p ~>/~,p is closed and increasing}. Clearly, D(/~) (resp. I(/Q) is the smallest closed decreasing (resp. increasing) fuzzy set in (X, ~< ) which contains #. A fuzzy set #, in a fuzzy topological space (X, z), is called a neighborhood of a point x E X (see [7]) if there exists a r-open fuzzy set p with p ~

0. It is shown in [7] that a fuzzy set p is z-open iff p is a neighborhood of each x E X for which

t~(x)>0. A net {x~}~cj [4] in a fuzzy topological space (X, z) is said to be converge to x E X, written as x = lira x~ or x~ ~ x, if for each neighborhood/~ o f x there exists an index ~0 such that p is a neighborhood of x~ for each ~ >/~o.

2. FTl-ordered spaces By x ~ y we mean that the relation x ~< y is false. Definition 2.1. A fuzzy topological ordered space (X, ~<, z) is said to be an upper (resp. lower) FT1ordered space iff for each pair of elements x, y E X with x ~ y , there exists a decreasing (resp. increasing) z-open neighborhood p of y (resp. x) such that ~t(x)=0 (resp. / l ( y ) = 0 ) . (X, ~<,z) is said to be an FTl-ordered space iff if it is both an upper and a lower FTl-ordered space. Theorem 2.2. Let (X, <~,z) be a fuzzy topological ordered space; then the following conditions are equivalent: (i) (X, <~,z) is upper (resp. lower) FTj-ordered

(ii) For each pair x ~ y in X, there exists a z-open neighborhood # of y (resp. x) such that d(p)(x) = 0 (resp. d(p)(y) = 0). (iii) I f x~ ~ x, andx~ >~y (resp. x~ <~y), implies that x>~ y (resp. x <~y). Proof. (i)¢=~ (ii) is immediate. (ii)¢¢,(iii) (see [4, Proposition 4.5]).

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By lx and 1A we denote the characteristic function of {x} and A C X, respectively. Proposition 2.3. A fuzzy topological ordered space (X, <<.,z) is FTl-ordered iff, for each x E X , d(lx) and/(Ix) are r-closed Proof. First, suppose that (X, ~<, z) is an upper FTIordered space. Then for a pair x g y in X, there exists a decreasing z-open neighborhood /~ of y such that /~(x)=0. For each y E X \ { x } , we have that [1 - i ( l x ) ] ( y ) > 0 and ~ < ~ [ 1 - / ( I x ) ] i.e. [ 1 - / ( l x ) ] is a neighborhood of each y so that/(Ix) is z-closed. Similarly d(lx) is z-closed. Conversely, let d(lx) and /(Ix) be z-closed for each x E X . Then, for x g y , [1 - /(lx)] is a decreasing z-open neighborhood of y where [1 - i(lx)](x) -- 0, which means that (X, ~<, z) is upper FTl-ordered. []

3. FT2-ordered spaces Definition 3.1. A fuzzy topological ordered space (X, ~<,z) is said to be FT2-ordered space iff for each pair x , y E X with xq~y, there is an increasing z-open neighborhood/.t of x and a decreasing z-open neighborhood 2 of y such that p A 2 = 0. Theorem 3.2. Let (X, <~,z) be a fuzzy topological ordered space, then the following conditions are equivalent: (i) (X, ~<, r) is FT2-ordered. (ii) For each pair of elements x, y E X with x ~ y , there are z-open neighborhoods Pi and P2 of x and y, respectively, such that i(pl ) A d(p2) -- O. (iii) x~--* x, y~---~y and x~ <~y~ imply x <<.y (for each ~ E J).

K. El-Saady, M. E Bakeirl Fuzzy Sets and Systems 98 (1998) 211~15 Proof. (i) ¢* (ii) is immediate. (ii)¢:>(iii) (see [4, Proposition 4.5]).

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Definition 3.3 (Ahsanulla [1]). If (X,z) is a fuzzy topological space, then A C X is called hyperclosed iff 1A = 1A.

Proposition 3.4. A fuzzy topological ordered space (X, <~, z) is FT2-ordered iff the graph G( ~ ) = { ( x , y ) : x <~ y } C X x X of the order <<. is hyperclosed Proof. The graph G(~<) is hyperclosed ¢=~ 1G(~<)= 1G(~<)¢e~ 1 - 1G(~<) is z-open. For p ~ q in {X × X G( ~<)} let7 E X × X such that ly(p, q) = max{ lp(p), lq(q)} then 7 E {X × X - G( ~<)} and for the z-open fuzzy set [1 - la(~<)], there exists a z-open fuzzy set # × 2 in {X × X - G( ~<)} such that # × 2 ~< 1 - 1~(~<) and p ×2(7)= min{p(p), 2(q)} >0. Thus, it follows that p ( p ) > 0 and 2(q)>0. Also, for x E X we have 0 = ~ × 2(x,x)=min{p(x),2(x)} i.e. # A 2 ~< i(p) A d(p) = 0 which means that (X, ~<, z) is FT2-0rdered. Conversely, let (X, ~<, z) be FT2-0rdered space and let 7 = (a, b) E {X × X - G( ~<)}, then for a ~ b , there exists a decreasing z-open neighborhood # of a and an increasing z-open neighborhood 2 of b such that #A)~=0. Consider the fuzzy set p × 2 such that p × 2 ( 7 ) = inf{p(a),A(b)}>0. Take any point 7' = (a',b') E X × X, then/~ × 2(~/) = inf{/~(a'), 2(b')}. Since # A 2 : 0 then/z × )o(a', a') = 0, i.e. # x 2 ~ < 1-1a(~<)

(V(a',b')EXxX).

Thus, for any point 7 E X x X - G( ~<), one can say that there exists a z-open fuzzy p x 2 such that /~ x )~ ~< 1 - 1~( ~ ) and therefore 1 - 1c(~ ) is z-open and this completes the proof. []

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the subset Y c X, the induced relative order ~
Proposition 3.6. Each z-compatibly fuzzy topological ordered subspace of FT2-0rdered space is again FT2-0rdered Proof. Straightforward.

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4. Fuzzy regularly ordered spaces Definition 4.1. A fuzzy topological ordered space (X, ~<,z) is said to be upper (resp. lower) regularly ordered space iff, the following condition holds: Given a point x E X and decreasing (resp. increasing) z-open fuzzy neighborhood 2 of x, there exists a decreasing (resp. increasing) z-closed fuzzy set v and a z-open decreasing (resp. increasing) fuzzy set p such that p ( x ) > 0 and p ~< v ~< 2. (X, ~<,z) is said to be a regularly ordered space (simply FR-ordered space) iff it is both lower and upper regularly ordered.

Proposition 4.2. A fuzzy topological ordered space (X, <~, z) is lower (resp. upper) FR-ordered iff the following condition holds: For each point x E X and an increasing (resp. decreasing) z-open fuzzy neighborhood I1 of x, there exists an increasing (resp. decreasing) z-open fuzzy set 2 such that 2(x) > 0 and )~ ~< 1(2) ~< # (resp. 2 ~< D(2) ~< #). Proof. Straightforward.

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Proposition 3.5. An FT2-ordered space is an FT1ordered space.

Lemma 4.3. I f (X, ~<, z) is FR-ordered space, then so is every z-compatible ordered subspace.

Proof. Straightforward.

Proof. Let (Y, ~y, Zy) be a z-compatible ordered subspace of the upper FR-ordered space (X, ~<, z) and let x E Y and p be a ry-open decreasing fuzzy neighborhood ofx in Y. Then, there exists a z-open decreasing

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Let Y C X, where (X, ~<, z) is a fuzzy topological ordered space. Then the triple (Y, ~
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fuzzy set 2* such that 2 = 2*/Y with 2*(x)>0. Since (X, ~<,z) is upper FR-ordered then there exists a z-open decreasing fuzzy set #* such that p*(x)>0 and g*~< D(p*)~< 2*. By the restrictions of p* and D(p*) by Y we have that # ~
Definition 4.4. An FR-ordered space which is, also, FT,-ordered is called an FT3-ordered space.

(iii) Given a decreasing (resp. increasing) z-closed fuzzy set # and a decreasing (resp. increasing) z-open fuzzy set p with p <~p, there exists an order preserving continuous function f : X--~ [0, 116, such that, for each x EX, p(x) ~< 1 - f(x)(O+) ~< 1 - f(x)(1-)<~p(x).

Proposition 4.5. An FT3-ordered space is an FT2ordered space.

Proposition 5.3. An FN-ordered space & an FRordered space.

Proof. Let x ~ y in X and suppose that (X, ~<,z) is upper FT3-ordered, then i(ly) and d(ly) are z-closed fuzzy sets for each y E X. For a point x E X and a z-open decreasing neighborhood [1 - i(ly)] of x, by the hypothesis of upper FR-ordered, there exists a decreasing z-open fuzzy set g and a decreasing z-closed fuzzy set v such that #(x)> 0 and/l ~< v ~< [ 1 - i(ly)]. Thus (1 - v ) is an increasing z-open fuzzy set containing y and # A (1 -- v ) = 0 which means that (X, ~<, z) is FTE-ordered. []

Proofi Let x E X be a point, p be a decreasing z-closed fuzzy set and p be a decreasing z-open neighborhood ofx with/~ ~< p. Fuzzy normality of(X, ~<, z) implies that there exists a decreasing z-open fuzzy set 2 such that

5. Fuzzy normally ordered spaces Definition 5.1 (Katsaras [4]). A fuzzy topological ordered space (X, ~<,z) is called a fuzzy normally ordered (FN-ordered) space iff the following condition is satisfied: Given a decreasing (resp. increasing) z-closed fuzzy set p and a decreasing (resp. increasing) z-open fuzzy set p such that p ~< p, there are a decreasing (resp. increasing) z-open fuzzy set 2 and a decreasing (resp. increasing) z-closed fuzzy set v such that # ~< 2 ~< v ~< p. Theorem 5.2 (Katsaras [4]). Let (X, <~,z) be a fuzzy topological ordered space, then the following conditions are equivalent: (i) (X, <<.,z) is FN-ordered (ii) Given a decreasing (resp. increasing) z-closed fuzzy set lZ and a decreasing (resp. increasing) z-open fuzzy set p with #<<.p, there exists a decreasing (resp. increasing) z-open fuzzy set 2 such that p ~< 2 ~
~< 2 ~< D(2) ~< p. So that (X, ~<, z) is FR-ordered.

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Definition 5.4. An FN-ordered space which is, also, FTl-ordered is called an FT4-ordered space. Lemma 5.5. An FTa-ordered space is an FT3ordered space.

Proof. Immediate by Proposition 5.3.

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6. Fuzzy perfectly normal ordered spaces Definition 6.1. A fuzzy topological ordered space (X, ~<, z) is said to be fuzzy perfectly normal (FPN-) ordered space iff it is fuzzy normal and every decreasing (resp. increasing) z-closed fuzzy set is a countable intersection of decreasing (resp. increasing) z-open fuzzy sets. Proposition 6.2. A fuzzy topological ordered space (X, ~<,z) is called an FPN-ordered space iff for every decreasing (resp. increasing) z-closed fuzzy set ~ and a decreasing (resp. increasing) z-open fuzzy set p such that # <~p, there exists an orderpreserving continuous function f : X---, [0, 1]~,

K. EI-Saady, M. E Bakeir/Fuzzy Sets and Systems 98 (1998) 211-215

such that, for each x E X, #(x) = 1 - f(x)(0÷)

~< 1 - f ( x ) ( 1 - ) =

p(x).

P r o o f . The p r o o f is trivial as a c o n s e q u e n c e T h e o r e m 5.2. []

of

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[3] B. Hutton, 1. Reilly, Separation axioms in fuzzy topological spaces, Fuzzy Sets and Systems 3 (1980) 93-104. [4] A.K. Katsaras, Ordered fuzzy topological spaces, J. Math. Anal. Appl. 84 (1981) 44-58. [5] L. Nachbin, Topology and Order, Van Nostrand, New York, 1965. [6] M. Sarkar, On fuzzy topological spaces, J. Math. Anal. Appl. 79 (1981) 384-394. [7] R.H. Warren, Neighborhood, bases and continuity in fuzzy topological spaces, Rocky Mountain J. Math. 8 (1978) 459 -470. [8] Wuyts, R. Lowen, On separation axioms in fuzzy topological spaces, fuzzy neighborhood spaces and fuzzy uniform spaces, J. Math. Anal. Appl. 93 (1983) 27-41.