Acta Math. Hung. 44 (3---4) (1984), 299--310

SYNTOPOGENOUS SPACES WITH PREORDER. III (SEPARATION)

K. MATOLCSY (Debrecen)

After the study of the (weakly) convex [7] and continuous [8] spaces, in the third part of our series the notion of a Ti-preordered syntopogenous space will be introduced for i=0, 2, and the definition of Burgess--Fitzpatrick [2] will be adopted for i= 1.

In Section 1 we shall examine the most important properties of these spaces, and in Section 2 a generalization of the quotient space corresponding to the separative partition of a syntopogenous space will be given. Using this construction, we shall show that the category of the T0-ordered syntopogenous spaces is epi-reflective in the category of all preordered syntopogenous spaces and continuous preorder pre- serving mappings.

O. Introduction

A preordered space is a pair (E, <=) consisting of a set E, and a reflexive and transitive relation ~= (so-called preorder) on E. An order <= is a preorder such that x<=y, y<=x imply x=y. A preorder <= is linear iff, for x, yEE, x<=y or y~x . The graph of the preorder =< is

G ( - <-) = {(x, y)EEXE: x <-_ y}.

A set X c E is called increasing (decreasing) iff xEX, x<-y (y<=x) imply yEX. X is convex iff x, zEE, x<-y<=z imply yEX. For an arbitrary X c E , the sets i(X)={yEE: x<-y, xEX}, d(X):{yEE: y<=xEX} and c(X)=i(X)fqd(X) are the smallest of all increasing, decreasing and convex sets respectively, which contain X. A mapping f of the preordered space (E, <=) into a preordered space (E', <=') is said to be preorder preserving (inversing), if x, yEE, x<=y imply f(x)<='f(y) (f(y)<-'f(x)). The product of the preordered spaces (Ej, ~-.) ( j E J r is a pre- ordered space (E, =~), where E = X E j, and (xj)<=(yj) i f f - J . < . . xj =jyj for every index

JEJ jE].

As a common generalization of topological, proximity and uniform spaces, the notion of a syntopogenous space was introduced by A. Csdsz~r. In respect of the ter- minology and the notations this paper follows his monograph [4].

A syntopogenous space [E, 5 a] equipped with a preorder <- is called a preordered syntopogenous space, and it is denoted by (E, S e, -<_) (see [1]--[3], [7]--[8]). For a semi-topogenous order <, and, for an order family ~r on a preordered space (E, ~), define

G(<) = {(x, y)EEXE: x < E - y is false}

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300 K. MATOLCSY

and G(~:) = :~ {G(<): <Ed} .

It is obvious that; for ~r {<0}, we have

(0.1) (x, y)EG(d) iff x <oE-y is false.

If ~r is a syntopogenous structure, then G(ff) is the graph of a preorder on E (cf. [1], 3.1).

The preordered syntopogenous space (E, 6 e, ~ ) (or shortly Se) is said to be increasing (decreasing) iff

G(~) c a(S') (G(_~)-~ ~ G(~))

(see [71, cf. [1]--[2]). The upper (lower) syntopogenous structure of a preordered syntopogenous space

(E, SZ, =<) will be denoted by ~ a (6~); it is the finest of all increasing (decreasing) syntopogenous structures coarser than 6 e on (E, -<-) (see [2]).

Suppose r {<x,c: X c C c E , C is convex} (cf. [4], p. 42), and let ~ be an elementary operation (see [4], p. 69). (E, ~ , <=) is called weakly a-convex iff, for every < E 6 a, there exists r such that { < } < C t u < S e (see [7]). (E, SZ, -<_) is said to be U-convex iff 6e',~(6euY~et)a (see [7], cf. [1]--[2]). (E, :T, ~ ) is symmetrizable iff there exists a symmetrical *-convex syntopogenous structure ~0 on (E, <=) such that 6e0 <6e <6a0~ (see [7]). We say that (E, 6:, ~ ) is continuous iff, for each < E 6 e, there is <~ESZ, for which A<B implies i(A)<fi(B) and d(A)<qd(B) (see [8]).

1. To-, T1- and T:preordered syntopogenous spaces

If <1, <2 are semi-topogenous orders, and ~r .~ are order families on the set E, then we define, for A, BcE ,

A(<I<2)B iff A < I X < . ~ B for some X c E , and

a r = {<1 <3: < : d , <zE~}.

Then ~r is also an order family coarser than ,~r and & (cf. [5], (2.19)). Let (E, 6e, <_-) be a preordered syntopogenous space, and put G=G(<=). Then

(t.0)

because

(1.1.1)

and

(1.1.2)

G c G(:U) ("i G(:~),

G c G(Y 'u)

G c G(:t0

by G=(G-1)-IcG(~)-I=G(~ to) (see [1], 3.2). The order family coarser than 6 au, therefore G(SeU)cG(SeuSPl0, hence

(1.2) G c G(Yu~'0.

5r is

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SYNTOPOGENOUS SPACES WITH PREORDER. n I 301

By prescribing the inclusion in the conversed direction in formulas (1.0), (1.1.1)~ (1.1.2) and (1.2), we obtain three conditions, namely

G(9 ~") ~ 6 ( ~ ' 0 ~ G;

G(Se")~G and G(5r ~ G

(To)

(7"1)

(see [2]), and

(T~) G(~"sP~O c G.

It is obvious that (T~)::~(T1)=~(To) by [1], 3.30). The (pre)ordered syntopogenous space (E, 5 ~, ~ ) will be called Ti-(pre)ordered iff it satisfies axiom (T 0 (i=0, 1, 2).

(1.3) REMARKS. (1.3.1) By 0.0)--0.2) it is clear that in the axioms (T0)--(T2) the sign-= can be written instead of c .

(1.3.2) It is also obvious that

(To),;=~x, yCE, x~:y imply x < ' E - y for some <='E5 r or y<'E-:-x for some <'C5~ t.

(T~)~x, yCE, x~:y imply x < ' E - y and y < ' E - x for some <'E5 r and <'E5 pt.

(T2),~x, yEE, x N y imply x < ' X < ' C E - y for some <'C5 r <"C5 ~ and X c E .

(1.3.3) Let (E, J , <=) be a preordered topological space, and z be the associated "classical" topology. Then (E, J , <=) is To-preordered iff, for x, y~E, x ~ y , there exists a z-open set V such that V is increasing, xE V and y r V, or V is decreasing, yEV and x~V. Further (E , J , _<-) is T~- or Tz-preordered iff (E,v, <=) satisfies the corresponding separation axiom in McCartan's "stronger" sense [9].

(1.3.4) A discretely ordered syntopogenous space (E, 5 ~, =) is T~-ordered iff [E, 5 P] is a Ti-space in the sense of [4], Ch. 14. []

Further On some theorems of [4], Ch. 14 will be generalized (see (1.4)--(1.12)). Let us consider the biperfect topologies oge and s e on a preordered space (E, ~),

which are defined by

~//e= {<'}, s {<'}, where A<'B (A<"B) iff

A c X c B for a suitable increasing (decreasing) X c E .

(see [1], 3.5 and [7], (1.2)).

(1.4) THEOREM. For any preordered syntopogenous space (E, ~ , ~), we have

(To) ~ ( 5 " VSt'z~) 'b = o?/~

(7"1) ~=~ 5 r = ql~ and 5 "ub = LP~

(T~) ~ (y,yl~),b = ~1~.

PROOF. Recalling G(q/E)=G(_<-) and G(.oq'E)=G(~) -~ (see [1], 3.5; [7], (1.2)), these statements are immediate consequences of (1.3.1), (0.1) and [1], 3.3. []

(1.5) THEOREM. Any symmetrical To-preordered syntopogenous space is T~- preordered.

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302 K. MATOLCSY

PROOF. By [7], (1.8) 6r and 5eu<Sg~2<re~Sau<Sa", provided 5" is symmetrical. Therefore (6~uSt't*)tb=(re"Seu)tb=seutb=(5~VSe"y~=(reuVralc)tb= =0gr. []

(1.6) LEMMa. l f (E, 5e, <-) is a Ti-preordered syntopogenous space and 6e <re ' , then (E, re', <-) is also Ti-preordered (i=0, 1, 2).

PROOF. S ~ < 6 p'~, 5"~<6 a'z, thus e.g. (1.3.2) can be applied. []

(1.7) THEOREM. (E, Se, <-) is Ti-preordered iff so is also (E, Set, <-) (i=0, I, 2).

PROOF. Because of 5~ etu and SPlt--~Sea (see [7], (1.9)) we have ( ~ u v,~Ic)tb = ( ~u t voQ~|ct)tb :(JO~ V,gotlc)tb. ,-,Gt~utb---r-~uttb---'---~9~tutb and 6e~b = relttb-: =5eatb. Finally (5*~Sel~)tb=(reuS/'t~)ttb=(sP~trel*t)tb=(Set~rett~)tb. In view of these equalities (1.4) can be used. []

(1.8) EXAMPLES. (1.8.1)The well-known decreasing ordered syntopological space (R, J , <-) of [7], (1.4) in T0-ordered, but it is not Twordered. A large number of examples for Tx-, but non-T~-ordered spaces can be obtained using the interval topology, in which the sets of the form i(x)Od(y) (x<-y) are the subbase of the closed sets (see e.g. [9], ex. 2).

(1.8.2) (1.7) shows that the T~-preorderedness of a space (E, 90, =) depends only on (E, 6 at, <-). This cannot be stated, if we write p instead of t.

Let (R, <-) be the naturally ordered real line, and Y-= {< } be the symmetrical topogenous structure of [7], (1.10), which is defined with the help of the sets H~= = ( _ o~, n]U[n, ~) for all natural numbers n so that A < B is equivalent to A c B , ANH,=O or H, c B for somen. Then 3-P"'--q/R, JP~-,,LeR imply that ( R , J -p, <-) is T~-ordered, however (R, ~, <-) is not To-ordered, because $ - " = ~ is the indis- crete syntopogenous structure of R (see [7], (1.10)). []

(1.9) THEOREM. Let j5 be a mapping of the set E into the Ti-preordered syntopo- genous space (E i, 5e~, <-i) for every jEJ~O. Suppose SP= V fT~(rej), and

jca x<-y iff f j ( x ) ~ j f i ( y ) for each jEJ (x, yCE). Then the preordered syntopogenous space (E, S/', =) is also T~-preordered (i=0, 1, 2).

PROOF. It is clear that 5 a is a syntopogenous structure, and <- is a preorder on E. J} is preorder preserving for every .~C J, therefore j)-l(ra~) is increasing on (E, <-) ([7], (1.1.4)~. j)-l(~,)<35-x(Spj)<5" implies J)-~(~")<:T" (jCJ). In the same way f j - ~ ( ~ ) < ~ t (j~J). If x,y~E, x~_y, ,then f~(x)N~f~(y) for at least one j~J. ffor < j ~ " ( < j ' ~ ' ) there exists < ~6 e" (< ~re~) such that fj--l(<j)~<" (f71(-<~')~-<"). If f~(x)<~E~-f~(y) (f~(y)<j"Ej-f~(x)), then

x~fi- x(fJ (x)) <'f7 ~ (Ej--fj (Y)) = E-J~-~ (~ (y)) c E-- y.

Similarly f~(y)<j"Ej-fj(x ) implies y < " E - x . Finally, if f~(x)<jC<~'~Ei-fi(y), then x<~- t (C)<"~E-y . This completes the proof by (1.3.2). []

(1.10) COROLLARY. I f { ~ : jr is a family of syntopogenous structures on the preordered space (E, <-) such that (E,S~j, <-) is T~-preordered for jr then the space (E, Y 5a~, -<) is also T,-preordered (i= 0, 1, 2).

j~a

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SYNTOPOGENOUS SPACES wn"l-I PREORDER. I I I 303

PROOF. Let fs be the identity of E for any jCJ. []

(1.11) COROLLARY. The product o f an arbitrary non-empty family of Ti-preorder- ed syntopogenous spaces is also T~-preordered (i=0, 1, 2).

PROOF. I .etfj be the projection of the product onto itsj-th component, and <= be the prOduct preorder (cf. [4], (11.4)). []

(1.12) THEOREM. 1tl order that (E, 5 ~, <=) be To-preordered, it is necessary and sufficient that one of the following conditions be satisfied for any x, yCE, x~=y:

(1.12.1) there exists a bounded preorder preserving (5 e, J)-continuous .function f on E such that f ( y )<f (x ) ;

(1.12.2) there exists a bounded preorder inversing (5 e, J)-continuous function f on E such that f (x )<f (y) .

(E, 5 p, <=) is Tl-preordered iff both (1.12.1) and (1.12.2) are satisfied for every x, yE E, x~_y.

PROOF. It will be verified that (1.12.1)r and (1.12.2)~(x,y)~ r

In fact, if (x,y)r then x<~E-y , that is y < E - x for some <E5 al. From [4], (12.41) it follows that f (E)c [0 , 1], f (y )=0 , f ( x ) = l for an (5a~,.,r - continuous function f, which is always preorder preserving by [7], (1.6). Conversely, if (1.12.1) is satisfied, then f (y)-<~R-f(x) for some e>0 is obvious, f-1(or is decreasing (see [7], (1.4), (1.1.4)), and f - l ( j ) < o w implies f - 1 ( o r thus there is .<ES~l, for which y < E - x , i.e. (y, x)~G(s or equivalently (x,y)r The other equivalence is analogous (instead of [7], (1.1.4) we need to use [7], (1.1.5)). []

(1.13) COROLLARY (cf. [10], p. 52--53, [7], (4.8)). Let (E, 5e, <-) be a symmetri- zable preordered syntopogenous space. Then the following conditions are equivalent:

(1.13.1) For x, yEE, x ~ y , there exists a bounded preorder preserving (Sa, J~)- continuous function f such that f ( y )<f (x ) .

(1.13.2) There exists a symmetrical i-convex syntopogenous structure 50o on (E, <-) such that 6ao<Sg<6eoP and (E,6eo, <=) is To-preordered.

l f 5 r is a syntopology, then one can write 5aoP~5 e, and . ~ = d ~b instead of ,,~.

PROOF. (1.13.1)=*(1.13.2). Let us denote by �9 the set of all (6 ~ JS)-continuous ordering families of preorder preserving functions of (E, <_-). Then with 5e0--~$, we have 5a0<Sa<SaoP, and 5ao is the finest one among those symmetrical i-convex syntopogenous structures, which are coarser than 5 ~ (see [7], (4.8); [4], Ch. 12). If f is a function described in (1.13.1), then f - l ( J~)<SP, and, because of that f - l ( J~ ) is symmetrical and i-convex ([4], (9.7); [7], (2.9)), f-l(J~)<5~o. In view of that in this way f i s (Se0, Js)-, and afortiori (5e 0, J)-continuous, (1.12.1) can be used for the proof of the T0-preorderedness of (E, 5a0, -<_).

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304 K. MATOLCSY

(1.13.2)=*(1.13.1). (E, SO0, -<-) is Tl-preordered (see (1.5)), hence x ~ y implies f (y )<f(x) for a bounded preorder preserving (Se0, J)-continuous function f by (1A2). f is (So0, or and (50, J~)-continuous.

If 5 a is a syntopology, then 6eoP<5~ p gives 5:f~Se, and f is (te, J~)-continuous iff it is (so,~f)-continuous by ,r (cf. [4], (8.57)). []

We can complete [7], (4.8) as follows:

(1.14) THEOREM (cf. [10], Th. 9; [1], 8.2). Let (E, J , <=) be a preordered topolo- gical space, and �9 the classical topology associated with f . Then the following statements are equivalent:

(1.14.1) (E, v, <-) is uniformizable in the sense of Nachbin.

(1.14.2) j-=sotp for a symmetrical syntopology 5" such that (E, 5#, <-) is P-convex and To-preordered.

(1.14.3) (E, ~, <=) is symmetrizable, and x ~ y implies f (y )<f(x) for a suitable bounded preorder preserving ( ~, ~r function f .

PROOf. (1.14.2)**(1.14.3). I f (1.14.2) is satisfied, then 5~ =(5:uYsoC')P,,-(S:uVSaU0P,,,SausP (see [7], (1.8)). Suppose 5Oo=5O "~t. Then 5: o is symmetrical and i-convex by [6], (4.1.2). 5r " (see [4], (8.50)), therefore Y" issymmetrizable. 5:u<5oo implies 5O"<5Oo", hence G(Seo")= cG(S:")cG (cf. (1.5), and [1], 3.3), i.e. (E, 5e0, <=) is To-preordered, and we get (1.14.3) by (1.13).

Conversely, assume (1.14.3). Then 5o0<~"<5oo p for a symmetrical i-convex syntopogenous structure 5ao such that (E, Sao, <=) is To-preordered. SCot=sot for a totally bounded symmetrical syntopology 5O on E by [4], (19.38). Owing to [7], (5.4) ow is i-convex (and afortiori P-convex). Because of (1.7) (E, 5 e, _-<) is To-preor- dered, finally 5otp =sootp < ~tp = 3r< 5,,optp = 5:otp = 5otp, i.e. ~-= 5 :tp.

(1.14.1)r Nachbin's definition of a uniformizable preordered topolo- gical space (E, ~, <=) can be transformed into the following equivalent form (cf. [1{3],. pp. 52--53):

1) If V is a neighbourhood of xEE, then there are continuous real-valued functions f, g such that f(g) is preorder preserving (inversing), f (E), g(E)=[0, 11, f(x)=g(x)=O, and max {f(y),g(y)}=l for y C E - V . (In fact, consider f ' = l - g and g '= 1 - f l )

2) If x, yCE, x ~ y , then there exists a bounded, continuous and preorder preserving functionf with f (y)<f(x) . (Indeed, if/" is an arbitrary continuous, pre- order preserving function such that f ' (y )<f ' (x ) , then putting

f (z) = max {min {f' (x), f ' (z)}, f ' (y)},

we get f'(y)<=f(z)<=f'(x) for any zEE, and f satisfies Condition 2).) Thus the proof is clear by [7], (4.8), and (1.13). []

Finally we study the connection existing between the separation properties of (E, 5O, <=) and [E, 5~

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SY'NTOPOGENOUS SPACES WITH PREORDER. III 305

(1.15) THEOREM. Let [E, 5O] be a syntopogenous space, and <- be an order on E.

(1.15.1) I f(E, 5O, ~=) is Tj-ordered, then [E, 5O] is a Tj-space ( j = 0 , 1,2).

(1.15.2) If[E, 5O] is a Tfspace, "-<_- is linear, and 5O is ~- or P-convex, then (E, 5O, <=) is Tj-ordered ( j = 0 , 1 , 2 ) .

The statement of (1.15.2) was proved by Burgess and Fitzpatrick for j = l (see [2], 5.5).

PROOF. (1.15.1) is clear by 5O"<5O, 5Oz<SO and (1.3.2), because in our case x ~ y implies x~.y or y ~ x .

(1.15.2). Suppose [E, SO] is To, and (x,y)EG(SO")f)G(SOtc). If x=y, then (x,y)EG. If xr then there exists <E5O such that x < E - y or y < E - x . If 5O is either t_ or P-convex, we have 5O<(SO~VSOt) b. For <'ESO", <"E5O Z, < C (<'13 < " ) q b = ( < ' U <,,)b, we have at least one of the following inequalities:

(1) x < ' E - y ,

(2) x < " E - y ,

(3) y-< ' E - x ,

(4) y <" E--x.

Because of the choice of (x, y), (1) and (4) are impossible. Thus d ( x ) c E - y or i ( y ) c E - x issues from (2) or (3), therefore y ~ x . On the basis of the linearity of <_-, we get (x, y)EG, that is (E, 5O, _<-) is T0-ordered.

Let now [E, 5 ~ be T2, and assume (x, y)EG(SO"SO lc) is such that x ~ y . Then y ~ x and x#y . There exists <ESO with x<C<CE-y. In view of 5O<(SO'V VSOz) p, one can find <lESO ~, <2E5O t such that <C(<xU<~)qv , and <'ESO", <"ESO t, for which < 1 C < "~, < , , C ~ "2. We can select sets CI,C2, C~,C'2 such that x<xCx, x<2C2, y-<~C~, y<2C~, CINC~cC and C~(qC~cE-C. Suppose x<'X~<'C~, x<"X~<"C~, y<'YI<'C~ and y<"Y2<:"C~. From [7], (1.2) it follows that X1cIcCa, X2cDcC2, Y~cl 'cC~ and YzcD'cC~, where I and I ' are increasing, D and D" are decreasing. We have x-<'l, x<"D, y<'I', y<"D', I O D c C and I ' f 3 D ' c E - C . Now lfqD'=O is impossible, because in this case x< I c E , D < )E-y, which contradicts the choice of (x,y). Therefore there exists zEIOD'. Moreover we get xEl" and yED from y<=x. Owing to the line- arity of _<-, we have to distinguish three different cases only, namely:

( 1 ) z < < = y = x ,

(2) y -<_ z ~_ x,

(3) y N x =-< z.

It will be seen that each one of the conditions (1)--(3) contains a contradiction. In fact; (1): z~-y implies yEI, therefore y E I A D c C c E - y . (2): y~=z implies zEI', and z~=x gives zED, thus zEIf~DAI' f~D'cCA(E-C)=f~. (3): x~-z implies xED', consequently x E I ' N D ' c E - C c E - x . Hence our initial condition cannot be satisfied, i.e. G(5Ouso~C)cG, so that (E, 50, ~=) is Tz-ordered. []

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2. To-ordered epi-reflection

Let (E, 6 r <_-) be a preordered syntopogenous space. For the sake of simpli- city, let us introduce the notation H=G(6 e~) AG(6e~ 0. It is clear that H is the graph of a preorder, and H (-1H-1 is that of an equivalence ~ on E. Let us denote by x, the .~-class containing x, i.e. x , = {yEE: x~y}i If <= is the relation of the equality (the so-called discrete order of E), then 6e"~6e ~Se, therefore H=G(6 a) ('1G(:70, thus H = H - l , and x, is a cellule of the separative partition of the space in question (see [4], ch. 14).

Returning to the general case, put E , = {x,i xEE}. Let us observe that x~u, y,~v, (x, y)E H imply (u, v)E H. (In fact, (u, x)E H, (x, y)E H, (y, v)E H=~(u, v)E H. ) Hence a relation ~ , c a n be defined on E, as follows: x , ~ , y , iff (x,y)EH. Fur- ther on let us consider the canonical mapping R of E onto E, , which is determined by R(x)=x, (xEE). Among those syntopogenous structures Se' on E, , for which R is (6 a, 6e')-continuous, there exists a finest one, and it will be denoted by 6a,. We have

se, = v

-(see [4], (9.24)).

In the rest of this article we assume that the reader is familiar with the notion of a mapping compatible with a syntopogenous structure and the definition of the image of a syntopogenous structure (see [4], pp. 105--110).

(2.1). LEMMA. The relation <=, is an order on E,. The mapping R is preorder preserving and compatible with S r and 6 r In the ordered syntopogenous space (E,, 5e,, <=,) the equivalences (6e,)",--R(6 e") and (6e,)t~R(6 et) hold.

PROOF. H is the graph of a preorder, thus owing to the definition, -<_, is also a preorder. If x,<=,y, and y,<=,x,, then (x,y), (y,x)EH, thus x~y, i.e. x , = y , .

If x<=y, then we have (x,y)EH by (1.0), so that R(x)=x,<=,y,=R(y). Now suppose <E6 eu or <E6 ez, and A<B. Then xER-I(R(A)) implies

R(a)=R(x) for some aEA. This means a..~x, hence (a,x)EHcG(6 au) and (x, a)EHcG(SezO. This gives that a < E - x is false, therefore xEB. Summing up, R-I(R(A))cB, so that R is compatible with 6 e" and Set.

R is (6~,6a,)-continuous, thus R - I ( ( 6 a , ) " ) < R - I ( ~ ) < ~ By [7], (1.1.4) R-I((6a,) ") is increasing on (E,-<_), hence R-I((S~,))<6 a . Thus (6a,)"< <R(Se,) issues from [4], (9.29). In the same way, we obtain (Sa,)~<R(6~l). On the other hand, R-~(R(Se"))~Se"<Sr R-l(R(Se'))~6~ by [4], (9.29),

u l u therefore R(6 a ) < Sa, and R(Sa)<6P,. We show that R(Sa ) is increasing, R(6 ~ is decreasing on (E,, -<_,), andthese yield R(6~u) <(Se,)" and R(SP~)<(S~,) ~. In fact, suppose < E Y u and x , R ( < ) E , - y , . By definition the relations

xER-l(x,) < R - I ( E , - y , ) = E -R- I ( y , ) c E--y

imply x.<E-y, hence (x,y)r162 consequently (x,y)r thus (x., y,)~ CG(<=,). We got G(<=,)cG(R(60")), which wasto be proved. The verification of G(<=,)-lcG(R(6~t)) is analogous. []

A c t a 1Vlathematica Hungarioa 44, 1984

SYNTOPOGENOUS SPACES WITI-I PREORDER. IH 307

(2.2) THEOREM. For every preordered syntopogenous space (E, 5 e, <=),

(2.2.1)

(2.2.2)

(2.2.3)

(E,, 5e., <=.) is To-ordered.

R is a continuous preorder preserving surjection of (E, 5 a, <=) onto (E,, coP,, ~,).

I f f is a continuous preorder preserving mapping of (E, ~ , <_=) into a To- ordered syntopogenous space (E', 6 a', <-'), then there exists a unique continu- ous order preserving mapping f , of (E,, Se,, <=,) into (E', 6e', <=') such that the diagram

E,Rx,,N,.-~*..* E' /

E

is commutative O.e. f=j .oR).

PROOF. (2.2.1) The space is ordered by (2.1). Suppose x, , y ,EE, , x, ~ . y . . Then (x,y)~H, i.e. (x,y)~G(Sr or (x,y)~G(SQIO (or equivalently (y,x)~ ~G(6el)). In the first case we have x < E - y for some <E6 a". Assume <IESe', < C < ~ , then x<aA<IB<IE-y for suitable A, BcE. R is compatible with 5 eu, therefore R-l(x,)=R-l(R(x))c.4<1~cR-l(R(~))cE--y, thus x ,R(<I )R(B)~E . -y , . Since R(Se")~(SP,) ~, we have (x,, y.)~[G((6a.)"). In the other case (y,x)~G(5r implies similarly y .R(<l)E,- -x , for some <IESPl, and because of R(6at)~(Sa,) l, we obtain (x,,y.)~G((SP.)t~).

(2.2.2) is clear. (2.2.3) Under the conditions of this statement, x, yEE, x ~ y implies f(x)=.f(y).

Indeed, if f (x)#f (y) , then f(x)~_'f(y), or f(y)~'f(x). From the T0-orderedness of (E', 5e', ~ ' ) it follows simply that one of the inequalities x f -~ (-<')E-y, yf-1 (<,,) ( < " ) E - x , y f - l (< ' )E-x , or x f - l (<" )E-y is true for a suitable <'E6 p'" o r <"E 5 ~ By [7], (1.1.4) from f - 1 (Sa,,) < f - 1 (S~,) < 50, and f -~ (6en) < f - ~ (6a,) < 5o we get f-I(Se'~)<Se" and f-~(Se't)<6al, thus (x,y)~H or (y,x)~H, which means x ~y.

This shows that it is possible to define a mapping f . : E. ~E" with f , ( x . ) =f(x) for xEE, so that f , oR=f. Since R is a surjection, this property determines f . uniquely.f, is order preserving, because x, yEE, f . (x . ) N~f.(y,) means f(x) ~_'f(y), and in this case, as we already showed, (x,y)f~H, i . e . x . ~ = , y , . Finally f . is (6~ because R-l( f , l (6~' ) )=f- l (Se ' )<6 r implies f ,1 (5~ ' )< <6e , . []

In the terminology of the categories (2.2) means that the category of the T 0- ordered syntopogenous spaces is an epi-reflective subcategory of the category of all preordered syntopogenous spaces, therefore (E.,Se., ~ , ) will be called the To- ordered epi-reflection of (E, 5 r =<) (cf. [6]).

REMARKS. Every preordered space (E, <_-) can be regarded as a preordered syn- topogenous space (E, 9 , ~_), where ~ is the discrete syntopogenous structure {c}

9 A c t s Mat l~emat ica H u n g a r i e a ~4, I984

308 K. MATOLCSY

of E. Then ~"=~'E, ~ = s and H=G(qIE)NG(L#~)=G(qIE) (see [7], (1.2)). From this x ~ y iff x<=y and y<=x. It is obvious that ~ , is equivalent to the dis- crete syntopogenous structure of E , , thus (E,, ~ , , <_-,) is an ordered space topo- logized similarly to (E, ~, <=). As a corollary of (2.2), we can easily verify that the category of the ordered spaces (with order preserving mappings) is epi-reflective in the category of the preordered spaces and preorder preserving mappings.

On the other hand, considering a syntopogenous space (E, 9 ~, =) preordered by the relation of the equality, the reader can easily prove that = , is also the equality relation on E , , and [E,, S:,] is isomorphic to the quotient space corresponding to the separative partition of [E, 6:]. This implies that the category of the separated syntopogenous spaces (with continuous mappings) is epi-reflective in the category of all syntopogenous spaces and continuous mappings (see, for topological spaces, [6], p. 105 (2)).

Two preordered syntopogenous spaces (E, :T, <_-) and (E', Se', <=') will be said to be isomorphic, if there is a one-to-one mapping f: E-+E" such that both f and g=f-1 are continuous and preorder preserving.

(2.3) THEOREM. A preordered syntopogenous space (E, 6:, <=) is To-ordered iff it is isomorphic to (E,, S:,, <=,).

PROOF. To-orderedness is invariant obviously with respect to isomorphisms. Conversely, if (E, :T, -<=) is T0-ordered, then there exists a continuous order preserving mapping id, of (E,,:T,, <=,) into (E, Se, <=) such that id= id ,oR, where id denotes the identity of E (see (2.2)). Since R is a surjection, id, is the inverse of R. []

Further on we shall study properties hereditary for the T0-ordered epi-reflection.

(2.4) L~MMA. Suppose S:,,,5 e* for an arbitrary ordinary operation * ([4], p. 74). Then 5e,,.,(6a,) k. Disregarding isomorphisms, the To-ordered epi-reflection of a preordered topological, topogenous, perfect, biperfect or symmetrical syntopogenous space is also of this kind.

PROOF. With the above notations R-l((S:,)k)=R-l(6a,)k<6:k"6:, there- fore (6e,)k<Se,, i.e. Sa,~(Se,)k. It is evident that the space (E,,6~,, <=,) is isomorphic to (E,, (60,) k, <-,), provided Se,~(6a,) k. []

(2.5) REMARK. It is worth constructing the T0-ordered epi-reflecfion of a pre- ordered topological space (E, ~,, <_--) so that this construction be applicable for "classical" spaces, too. For that purpose let us denote by U(x) (L (x)) the intersection of all increasing (decreasing) 3--open sets containing xCE. It is easy to see that (x, y)C H is equivalent to U(y)~ U(x) and L(x)cL(y) . Consequently, for x, yE E, x , = y , iff U(x)=U(y) and L(x)=L(y), moreover x,~=,y, iff U(y)cU(x) and L(x)cL(y) . We have ~ ,~(y , ) tp , and the latter is the finest of all topologies 5-' on E , , for which R is (~,, ~-')-continuous. Hence V, c E , is (~,)tP-open iff R-I(V,) is ~-open, in particular, V, is increasing (decreasing)and open in the space (E,, (j,)tp, <__,) if and only if R-I(V,) is of the same type in (E, J , _-<). []

Ac ta Mathemat~ca Hungar';ca ~4, 1934

SYNTOPOGENOUS SPACES WITH PREORDER. I I I 309

(2.6) THEOREM. Let (E, 50, <=) be a preordered syntopogenous space, and a be an elementary operation. The following conditions are equivalent:

(2.6.1) (E, 6O, <-) is ~

(2.6.2) R is compatible with 5", and (E,, 6O,, <=,) is ~

PROOf. If(E, 50, <=) is O-convex, then 6O~(6O"V6Ol) ", therefore R is compatible with 6O, and 6O,~R(6O)~R((6O"V6O~)~ by (2.1), and [4], (9.34), (9.33). Conversely, (2.6.2)=*(2.6.1) on the basis of [7], (2.9), because R-1(6O,)~6O (see [4], (9.29)). []

(2.7) THEOREM. The following conditions are equivalent for every preordered syntopogenous space (E, 6O, -<=):

(2.7.1) (E, 6O, -~) is symmetrizable.

(2.7.2) R is compatible with 6O, and (E,, 6O,, <=,) is symmetrizable.

PRoof. If (E, 50, <=) is symmetrizable, then ~ <6O <500 p for a suitable S-convex symmetrical syntopogenous structure 6o0 on (E, <=). 6ooU<6o ", 6ooI<6o l, therefore R is compatible with 6O0~6O0"V6Oo l by (2.1) and [4], (9.32), (9.34). Thus R is compa- tible with 6o0 p, and afortiori with 5~ ([41, (9.33), (9.32)). We have 6O'=R(6O0)'= =R(6Oo)~R(6Oo"V6Ool)~R(6Oo")VR(6Oot), where R(6O0 ~) is increasing, R(6O0 l) is decreasing by [4], (9.32), [7], (1.1.1), and lemma (2.1) of the present paper. Since in this way 50' is symmetrical and S-convex, (E,, 6~ <-,) is symmetrizable, namely 6O'=R(5o) <R(6O) ~6O, and 6O,~R(6O) <R(6Oo') ~R(6Oo)P=6O 'p.

Conversely, if (E,,6O,, <=,) is symmetrizable, and R is compatible with 6O, then 6O'<6O,<6O'~ for a symmetrical S-convex syntopogenous structure 9~ on (E,, <=,), thus R-l(6O ') < R-l(6O,) < R-I(Se 'p) = R-~(6O') p. R-1(6O ') is symmetrical and ~-convex, and R-1(6O,)~-,6O (see [7], (2.9), [4], (9.7) and (9.29)). Hence (E, 6O, =<) is symmetrizable. []

(2.8) THEOREM. If(E, 6O, ~) is continuous, then so is (E,,6O,, <-,), too.

PROOF. R-1(6O,)<6O, therefore if <'E6O,, we can select <E6O such that R-I ( -< ' )C<. Suppose <lE6O ", i(-<)C<1 (see [8], (2.4)), <2E6O", <~C<~. R is compatible with 5 a", and R(<~)ER(6O"),-~(6O,)"<6O, by (2.1), hence there is -<"E6O,, for which R(<2)C<". We proveA', B'=E,, A'<'B'=*i(A')<"i(B'). In fact, A'-<'B" implies R-I(A')<R-I(B'). Then

R-1 (A') = (R-1 (A'))i (<) i (R-1 (B')) c R-1 (i (B')),

because R is preorder preserving. Owing to R-X(A')<IR-I(i(B')), a set D ~ E can be found with R-I(A')<2D<2R-I(i(B')). We show R-I(i(A'))=D. Indeed, R(x)Ei(A') implies that R(a)EA', R(a)<-,R(x) for some aEE. Then (a,x)EH, therefore aER-I(A')<2D=E-x fails to be true, hence xED. Now R-I(i(A'))= =D<2R-I(i(B')) implies i(A')R(<2)i(B'), thus i(A')<"i(B').

A similar argument can be applied to prove that, for <'E6O,, there exists <"E6O, with A'<'B'=*d(A')<"d(B'). []

9* A c t a Matherna t i ca Hungar i ca 44, 1984

310 K. MATOLCSY: SYNTOPOGENOUS SPACES WITH PREORDER. III

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Ac~a M a t h e m a t i c a H u n g a r i c a 44, 1984