Seminarberichte Nr. 29

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Seminarberichte aus dem Fachbereich Mathematik der FernUniversität

29 – 1988

der Mathematik (Hrsg.)

Seminarbericht Nr. 29

A remark on ultrapower3 of incidence geometries

Giorgio Faina

2T spaces and closure operators

0

Eraldo Giuli and S. Salbany

Initial morphisms in universal algebra Part I: Equations

Dieter Kutzner

The Macneille completion of certain categories of T -spaces coincides with the bireflective huZl

0

Dieter Kutzner

Coproducts of diametric frames A. Pultr

Eilenberg-Moore Algebras revisited Dieter Pumplün

Les famiZles mesurab1es de paraboZoides equilateraux de L'espace

Salvatore Vassallo

On the range of L-decomposable measures

Aldo G.S. Ventre, Massimo Squillante and Mario Fedrizzi

- 1 -

- 11 -

- 41 -

- 75 -

- 81 -

- 97 -

- 14 5 -

- 16 3 -

A REMARK ON ULTRAPOWERS OF INCIDENCE GEOMETRIES

by

Giorgio Faina

Abstract. Non trivial extensions G* of incidence geometries G

are

constructed using proper ultra- powers with respect to non principal ultrafilters.

Such extesnsions have strong ties to the incidence geometries which generate them. but we prove that. in general, the automorphism group Aut(G*) is not trivially obtained by Aut(G).

This work was performed under the auspicies of the G.N.S.A.G.A. of the C.N.R. and of the

Italian M.P.I ..

ANS subject classification 11980): Pri

•

ary 51B10; Secondary 51D20, 03C20.

Key Mords and phrases: Nobius plane, affine plane, ultrafilter, ultraproduct,

1. The construction of ultrapowere (see [31) have proven to be ueeful tool for algebraic etructures theorists. lt seems to us that theee constructions might be useful also in studying incidence geometries (like Projective planes. Möbius planes. Sperner spaces etc.). so that, in this brief note. we want to fit this method based on ultrapowers to construct infinite incidence geometri~s with cardindlity large as we like. This leads us to consider problems about embedding incidence geometries in incidence geometries of the same class or ~ - and about their automorphism groups.

Definitions ar.d basic planes can be found in (9).

spaces can be found in (5).

facts about projective planes and Möbius Benz planes. Buekenhout ovals and Sperner (61 and [1] respectively. For background material on Model Theory and ultrapowers see (31. [41. and [7]. Our

tenninology and notation will be in accordance with this treatise.

2. For completness we recall in this nwnber only some basic notions and properties from the theory of ultrapowers w.r.t. non principal ultrafilters and from the theory of incidence geometries.

A filter r on a set Ais a non-empty set of subsets of

A.

such that 0fr. F1 ,F2Er implies F1nF2EE and FEE, F~S forces S tobe in[. A maximal filter (w.r.t. inclusion) is called ultrafilter. lt is well known that a filter Eis an ultrafilter on A iff. for each ScA either S or its complement S' belongs to r. For each cxEA. ro:•<F,A cxEF} is an ultrafilter on

A.

These ultrafilters are called principal (generated by a). Let E be a non-principal ultrafilter on A and E a set. The quotient of the cartesian product

.{I L , where E.•E , for all aEA with respect to the equivalence relation defined by

f,

gE TT

Eo. , f "'g

CmodE>

i

ff ~.

^{Q •} < aEA

-,.A f

(a)

•g

(a) }EE

is called a proper ultrapower of

E

w.r.t E and is denoted by EA/E ^(or

E*

for short). The e lements of

E* wi 11

be denoted by { f ( a) },,. ..

^{A (}

or f

for short) for

each

fEEA.

We require the following well known results (see

[3]

or

[7]):

( 2 .1)

Let

A

be

An

infinite set. ll

Bis any

infinite set. then there are non principal ultrafilters E 2n As such ~hat

IA'A ^> /El2lBl.

Let

B a

set. If Ais any infinite set and E a non principal ultrafilter on A, then there is a canonical injection (see

[3]. [4]

,or[7))

t:B

- >

B*

such that

'v

xEB, • Ac.EE Then we have:

t(x)•{x(a)

}~EA

where x(a)•x for all

aEA.:..

C 2. 2)

(1) l l Bis finite then t is surjective.

(2) l l Bis an infinite set such that lAl1lBl then tCB>iB*.

An incidence geometry is a triple S•(f,1,1), in which fand l arE disj oint (non empty) sets of objects called points and blocke, respectively, and for which l ^{is a} symmetric point-block incidencf relation. All classes of incidence geometries considered later will

bt

defined by certain conditions (incidence axioms) on 1-

Let S•(f,1,1> ^and T•(Q,t.~> be two incidence geometries. A monomo- rPhiem of Sinto T ie an injective mapping

u

of f~ into QUt such that

(1)

p

l

b

implies u(p)

^~

u(b} for all pEf and

bEB

(2) u{f)~Q and u<B>~t-

An isomorphism of S onto T is

a

monomorphism

u

with u(f)•Q and u(~)•~.

Automorphisms are isomorphisms of an incidence geometry onto itself. The set of all automorphisms is

a

group, denoted by

Aut(S).

Let i be the class of all incidence geometries. Let G1 .<hEi, with G1~G2, be two incidence geometries satisfying some incidence axioms

A1.

'Aa, . . . , A-, .

We say that G1 and

(h

are of the same class (and G(A1 . . . .

^,An)

will denote such class) of elements of i-

We shall now outline that our binary relation in

~

is not an equivalence relation. We illustrate it with an example.

Let G1 be an affine plane and

(h

a projective plane. Clearly G1 ,G2Ei{A1), where A1 is the following incidence axiom:

to any two distinct points. there exists a unique line (block) incident with both of them.

But

G:z belongs also

to the class iCA1,lb) where

A:z

is the following incidence

axiom:

to any two distinct lines. there exists a unique point incident with both of them.

Obviously, G1,i(A1,lb). This proves our statement.

3. From

i (

A1 . . . An) ,

now on assume G•(f.~-1) be an incidence geometry of class

E an

infinite set and r a non principal ultrafilter on E

with

IElilfl.

We indicate by e• ^and &• the proper ultrapowers e-,-·/E and

~';/E reepectively, lf p•{p(e) } •

.i,a.:Ef•

and b•{b(e) } •

^{.i: ..} E~• .

we write Pl•b iff {eEE :

^p(e)

l b(e) }EE.

The incidence geometry G•

•(f.•

.~•.1•> is called proper ultrapower of the incidence geometry G with respect to E,

The main purpose of the present paper is to show that

THEOREM. ( i) GEfi (

A1 , ...• An) if

f G* E~ (

A1 . . . . , An) ,

where A1 ...

An

~

n arbitrary (but compatibles) incidence axioms.

(ii) .il. Gis finite. then

G

and G* are isomorphic.

(iii) .il. Gis infinite, then there is

g

proper monomorphism J of G into G* <i-~-

J(G)

is

~

proper subgeometry of

G*).

(iv) There exist incidence geometries

G

such that (Aut(G))* is not isomorphic to Aut(G•) for every non Principal ultrafilter E on g set

E

with

IEIS.lfl.

(V)

(Aut(G))*

geometry Aut CG*)) .

G

There exist non is not isomorphic to (i.e. (Aut(G))* is

principal ultrafilters E such that Aut(G*) for every infinite incidence isomorphic to g proper subgroup of

The next Corollary now follows immediately from (2.1) and our Theorem.

COROLLARY. In every fixed class there exist incidence geometries

with cardinality large as we like.

We should mention the fruithful surge of activity in the claesification of incidence geometries by means of their authomorphism groups. We refer the reder to [91 for survey the fundamental classification of projective planes given by Lenz and Barlotti. and the classification of Mobius planes given by Hering. We outline the celebrated work of Buekenhout [6] for the classification of abstract ovals and the recent paper of Klein for the classification of Minkowski planes

[10].

In this context. with regard to (iv) and (v) of our Theorem.

it is very interesting to find

a way

of deciding whether or not

a

given incidence geometry G and a given its ultrapower G* are of the same

^~

w.r.t. their automorphism groups. This problem is very difficult to solve and will.be dealt in a following paper.

As a final remark, we mention that in [2] it is proved that if Gis a projective plane, and G* is a given its ultrapower, then G and G* are of the same Lenz-Barlotti type.

4.

Proof of the Theorem. In the light of the broad treatise

^UPOTI

Model Theory

(see

[3]), it easy to check that:

(4.1) Any class of incidence geometries can be interpreted as class of relational .sYstems satisfying a set

~

of axioms formulated in an appropriate language L within many-sorted first order logic.

We will consider languages for two familiar kinds of incidenci

- 6 -

geometries to illustrate and explain this idea.

ProJective planes T•(f,1,1> (and Mobius planes M•(f,,,l>>. One usually employs a two-sorted formalism. with

P,Q,R •...•

varying over pointe and 1,m,n .... varying over lines (circles). With

a

little hanky- panky we can regard

^{T (}

M) as the relational structure Y•Cf,i,B>

( ~-(f,,,B) ) , where B ie the relation corresponding to incidence of

T (

M). respectively.

The big

advantage

of ultrapowers is that elementary properties of models carry over to their ultraproducts. Thus. applying the fundamental theorem on ultraproducts (t6s's Theorem, see [31) and the

(4.1).

one

has

irnrnediately

(4.2) Let L be a first-order language of incidence geometries and I

a set of sentence of L. We write G

~ I

if every sentence of

I

is true in G. Then

G* iff

G ~

I.

Hence. if I is any set of (consistent) sentences of L then the class of all models for I is closed w.r.t. ultrapowers. This proves Ci).

(ii) is a simple consequence of (2.2)

In order to proof (iii). let a:f - > f* and ß:i - > i* .be the canonical injections and let J:G --> G* be a map defined as follows:

(1) V pEf

C2) Y bEI.

J(p)•a(p) J(b)•ß(b) .

Let TE(Aut(K))* such that o(T) (x)•x for all xE4'CK). Thue

YeEE1 T(e)(x(e))•x(e)

•>

-r(e)•I~"

•> o(T)•ld.<*.

Hence the proof of (iv) is completed.

We now turn on our attention to the next result Ccfr.

which follows immediately (v) of our Theorem:

[81)

f rom

(4. 3) .

Let

M

be an arbitrary infinite model and assume thE Generalized Continuum Hypothesis. Let E be a K•-good ultrafilter over

E

and assume IMliK• and K•w. Then Aut(rf</E)•Aut(M"') is not isomorphic tc (Aut(M))K/[•(Aut(M))*.

Since the quoted result is not out. a short account of its proof i1 done~ Since countably indexed ultrapowers are CJ.\.-saturated (see

[71,

n 6

.1).

we have that

M"'

is K•-saturated of power K• and (Aut CM))* is K•·

saturated. But Aut CM"') is not ~-saturated because there ex ist e lement1 gn. ni

W,

such that

{h

but for

each k< W

{ h : gwh•hg,.., }

,1,

r'-o {

k

h : gn h•hgn }

[e.

g. take the g'

s to be pennutations on a set of order indiscernibl1 (see [71. n. 3.3) and

g

automorphisms extending such permutationsl.

Thus (Aut(M))* is not isomorphic to Aut(M"'), and our Theorem is proved.

lt

Plb

then

J(p)l*J(b) .

In tact, by

definition

• E1EE: V eEE1

^p(e)•p

and • E..~EE: V eEE2 b(e)•b.

and p ( e >

lb

^< e > f

or

a 11 eEE, ,

which proves our Thus,

assertion. This fact ensures J is a monomorphism. lt remains to prove that J is proper, but this ie a simple consequence of

(2.2).

In order to prove (iv), let

1r

be an infinite projective plane over some algebraically closed field K with infinite trascendence degree. lt is well known (see

(9))

that Aut(T)•PrL:s(K) and that

PT'L:s(K)•PGL:s(K)xAut(K).

lt is

a

simple computation to verify that

(Aut(T))*•(PGL:s(K))*x(Aut(K))*

and that

Furthermore, since

(PGL:s(K))* is isomorphic to PGL:s(K*).

Aut(1r*)•PGL:s(K*)xAut(K*),

we have that the canonical homomorphism

6: (Aut(1r)*)

- >

Aut(1r*)

is surjective iff the canonical injective map of groups

a: (Aut(K))*

- >

Aut(K*)

is surj

ective ._

We recall that a is defined as follows:

Yr•{r(e) } ••• E(Aut(K))* . a(r)

(z) •z' z ' ( e) •r ( e) z ( e) f

or all

zEK*.

Since K* is a trascendental algebraically closed extension of K. ^it

admits plenty of automorphisms leaving

K

pointwise fixed (see [111). If

we prove that a((Aut(K)*) do not contains non-trivial automorphisms

which leave K pointwise fixed. the proof will be completed.

{1] A. BARL01TI. Una costruzlone di una classe dl seazl afflnl generalizzati . Boll. Un. Mat. Ital.. (3) 17 (1962). 182-187.

[21

U. BARTOCCI - G. FAINA. Sul tipo di Lenz-Barlotti di certe estensloni di un piano grafico infinite. Rend. Accad. Naz. Lincei,

58

(1975). 703-707.

[31

J.

BARWISE. Handbook of Mathematical Logic. North-Holland Publ.

Co .. Amsterdam 1977.

[41

J.L. BELL-A.B. SLOMSON. Models and ultraproducts: an introduction, North-Holland Publ. Co .. Amsterdam 1969.

[5] W. BENZ. Permutations and plane sections of a ruled guadric.

Symposia Mathematica I.N.D.A.M.,

5

(1970). 325-339.

[61

F. BUEKENHOUT. Etude intrinsegue des ovales. Rend. Mat. e Appl ..

25

(1966). 333-393.

[71

C.C. CHANG-H.J. KEISLER. Model theory, North-Holland Publ. Co ..

Amsterdam 1973.

[8] H.J. KEISLER. Private communication.

[91 P.

DEMBOWSKI. Finite geometries. Springer-Verlag, Berlin 1968.

[10]

M.

KLEIN.

A

classification of Minkowski planes. to appear.

[111 0.

ZARISKI-P. SAMUEL. Commutative algebra I and II. Springer- Verlag, Berlin 1960.

Indirizzo dell'Autore:

Giorgio Faina

Dipartimento di Matematica-Universita Via Vanvitell_i 1

06100 PERUGIA

Eingegangen am 9.2.88

2T0 SPACES AND CLOSURE OPERATORS

E.

Giuli and S. Salbany

Abstract: In this note we study the category 2T of bitopological spaces

---0

which is minimally separated and corresponds in this respect to the category of T topological spaces. We also identify a

-o

cogenerator for the category which has a role analogous to that of the Sierpinski dyad\n the category of topological spaces.

An explicit formula for the closure operator associated with 2T is obtained,from which follows a diagonal theorem

----0

characterizing 2T -spaces as well as an explicit description of

---0

the epimorphisms in 2T. The closure operator obtained leads

---0

naturally to a reflective subcategory of 2Top which is related to, but distinct from, the category of bisober spaces introduced by

B.

Banaschewski, G.C.L. Brümmer and

K.

Hardie [B.B.H] ; we also give an answer to the question raised in [B.B.H.] concerning whether or not the epireflective subcategory of bisober spaces is simple.

AMS Subject classification:

54 E 55, 54 B 30, 54

A

05, 18 B 30.

Key words Bitopological spaces. Closure operator.

Epimorphisms. Extension-closed spaces.

Bisober spaces. Epireflections.

This work was partially supported by research funds from the Italian Ministry of Public Education and a Research Grant from the University of Zimbabwe.

1. Introduction The category of topological spaces will be denoted by Top and 2Top will represent the category of bitopological spaces, whose objec ts are triples (XI P ,0), where X 1s a set and P, Q topological

structures on X and whose morphisms f:(X,P,O) - (Y,L,R) are maps

f:X-Y such that f:(X,P)-(Y,L) and f:(X,O) -(Y,R) are continuous.

2Top is a topological category .in the sense of [

ttJ :

initial and final structures, in particular, products, embeddings, coproducts and

quotients are obtained in 2Top by applying the corresponding

constructions in Top to each component topology. The indiscrete 2Top objects are exactly those (X,P,O) for which P and Q are both

{~,x_}_

The discrete 2Top objects (X,P,Ol are those for which both P and Q are all subsets of X.

If Pisa topology on X, denote by P the topology on X where each x

•

in X has c~x as its smallest P•-open neighbourhood [see [s

1

J, ^[s

₂

JJ.

The correspondence l'•F• determines two functors F. : Top ...,,.. 2Top

• • 1 - - - - -

b y F~(X,P)

=

(X,P,P) and F2(X,P)

=

(~,P ,P).

F

1,

F,

are right adjoints of the projections u

1 ,u2 :2Top - Top , where u

1 (X,P,O) = {X,P), u₂(X,P,Q) = (X,Q); u

1, u₂ also have right adjoints R

1, R_{2 ,} re3pectively, given by R (X,P)

=

(X,P,I) where I is the indiscrete topology on

1

X. The embedding F

1 :Top .... 2Top has been explored in

[s

₁

J, [s,J.

There is another natural embedding D:Top - 2Top given by D(X,P)

=

(X,P,P). D has a left adjoint: the supremum functor

S:2Top -Top given by S(X,P,O)

=

PVO. This relationship was used

effectively in [a.B.H ].

2.

z.1

0

§paces

2.1 Definition and characterizations

If a topological space (X,T) is tobe minimally separated, then any two distinct points can be distinguished by open sets; equivalently, every continuous map from the two point indiscrete space (D,i) into

(X,T)

is constant. This idea extends naturally to 2Top and we define, as in [Br]

2.1.1 Definition constant.

(X,P,O) is 2T if every f:(D,i,i) -{X,P,Q)

0 is

2. 1. 2. Lemma: For (X,P,Q) the following properties hold:

( i) cl = ^{cl XI'\ cl x, X f} X.

PVQ ^p Q

• , •

(ii) (PVQ)

=

^{p VQ}

(iii) b(PVO)

=

^{b(P) v b(Q).}

As a corollary, we have:

2.1.3 Proposition: Fora topological space (X,T), the following are equivalent:

(i) (X,T) is T

0

(ii) (iii)

(X,T) is

•

T

0

(X,b(T)) is T

0

(iv) _{(X, T) and (X, T )}

•

_{are T}

0

2.1.4 Proposition: Fora bitopological space (X,P,O) the following conditions are equivalent:

( i) (X,P,O) is 2T

0

( ii) (X,PVQ) is T

• •

0

(iii) (X,P VQ) is T

0

(iv) (X' b(PVO) is T

0

(v) If ^{X ,J}

Y,

^{then cl X}

'

^{cl y}

PVO PVO (vi) If ^{X ,J} y, then clpx

'

clpy or cl

0x -J. cl 0y.

(vii) If x -J y, then Cl X -J cl y or cl x ^,J. cl y.

b(P) b(P) b(Q) b(O)

Proof We shall only prove the equivalence of (i) and (ii). The other implications follow without difficulty from the observations in lemma 2.1.2 and proposition 2.1.3.

Assume (i) To prove (ii), consider f:(D,i)- (X,PVO). Then

f:(D,i,i)-(X,P,C), so that f is constant and hence, (X,PVO) is T.

0

Conversely assume (ii). Consider f:(D,i,i)- (X,P,O), then

f:(D,ivi)-- (X,PVO) so that

f

is constant, since ivi = i. Hence (X,P,O)

is 2T.

0

2.2. Cogenerators for 2T

0

Every topological space is initial with respect to its mappings into the

Sierpinski two point space (D,u), with

{O\

as the only non trivial u-open set.

In fact (X,T) is T iff the continuous maps f:(X,T) - (D,u) separate

0

the points of X. For T spaces (X,T), the canonical embedding e into the

0

C(X D)

product (D,u) ' provides a homeomorphism from (X,T) onto e[X1 with the relative topology.

We shall exhibit two bitopological spaces which are cogenerators for 2Top in the above sense ( (M1 ,

(H. s1 ) .

2.2.1 The triad ^(T,U,L)

rr ⁼ {a,B,Y}

u {4>, {a,B},rr}

L

{~, {B,Y},rr}

Note: UVL =

{4>,{8},{a,8} ,{B,Y}, rrl

_is a T

0 topology on 1I'.

Also: (D,u,l) «;.

ar,U,Ll

under the identification

o-a, 1- y.

2.2.2. Proposition Every bitopological space (X,P,O) is initial with respect to its maps into

Ol',U,L).

Proof: Given (X,P,0), x ^E: G and G a P-open set, let g:X

--rr

be defined by : { g

=

Y on X - G

l

^g

⁼ a

^{on G •} Then g: (X' p) -

or, u)

and g: (X' Q) - -

or'

L) are both continuous and G

=

g+(a

,B}],

^g(x)

= ß.

Similarly, if His 0-open, there is h:(X,P,O)-

Ol',U,L)

such that H = h

+[ {

B,

'Y} J ,

^{in fact, h}

⁼ l:

^{on H} _{off H.}

2.2.3 Proposition points.

If (X,P,O) is 2T, the mappings into (II',U,L) separate

0

Proof: Suppose Xi y, or cl xi cl y.

0 Q Assurne

Since (X,P,O) is 2T

0, then cl

X'

^{cl y}

c; Xi clpY. Then ^X

f

^{cl y}

p

p p

or y

f

^{cl x.} _p

Suppose x

+

^{clpY· Then x ~}

^{X -}

^clpy

^{= G,}

sog defined above is such that g: (X,P,Ol- (II',U,L) and g(x)

=

ß, g(y)

=

^y. If y f= clpx, then there is

g':(X,P,O) ('JI',U,L) such that g'(x) :Y, g'(y)

=

ß. Similarly, for the case cl xi cl y.

Cl 0

2.2.4 Proposition If the ma.ppings f:(X,P,O) - (II',U,L) separate points, then (X,P,O) is 2T.

0

Proof: Any f:(X,P,Ol- OI',U,L) gives f:(X,PVQ) - (II',UVL), hence ^PVO is

T

0, since UVL is

T

0•

In fact, no space with smaller cardinality can be cogenerator for 2T

---0

This observation can be proved by examining (D,P,O) where P,O range through all the possible topologies on D

=

lü,1}: the discrete topology, the indiscrete topology, the u-topology (with

\0~

as the only non trivial open set) and the 1-topology {with {1} as the only non trivial open set).

There is a larger cogenerator which contains (II',U,L) and is an injective object i n ~ (unlike

(n',U,Ll)

in the same way that (D,u) is an injective in

T.

-0

2.2.6

The

quad

(.,U,Ll ,.

& 't &

= {a,ß,y,6} .

0

~

u ^{= {~,{a,B},} O} .

{~,{ß,Y},O} ,. _•

L

=

{

~

, { ß} , { a, ß}, { ß , Y}, c l

^\j

^L

Note:

UVL = =

^T ₀ ^topology.

The propositions corresponding to 2.2.2, 2.2,3, 2.2.4 above remain true for (Q.,U,L) in place of OI',U,Ll.

2.2.6 Proposition ^(II' ,U,L) is not an injective object in 2T ---<>

Proof. Let i: (II', U, L) - (II', U, L) denote the identity map and j: (Il',U,L) - (<l,U,L) the inclusion map. Then there is no map

f: (O,U ,L) - m',U,L) such that f.j = 1, since f: ( Q., u)

-

^{<rr,U) implies}

f(Ö) :Y and f: (0,L) -m',L) implies f(Ö) = ^Cl.

2.2.7 Proposition (O,U,L) is an injective object in 2T ---<>

Proof Suppose (X,P,O) -(X' ,P' ,O') and f: (X,P,0) - (O,U,L).

Now v:r({a,ß}] is P open and W=r•[{ß,Y}] is 0-open, so there are sets v·, P'open, and W', O'open, such that v'n X= V,

w'"

^X

= w.

Define f' on ^X' by: f'= 8 on V ' f"\ W' ; f'

=

Y on

W'- ^V,; f'

=

^{Cl on V'-}

w,;

f': ö on X'- (V' VW'). f' is well defined and:

(i) f':(X',p') - (O,U) since r•+[{a, ß}] =

v:

(ii) 'f':(X',Q') -(O,L) since f'+[{ß,

Y}]

₌

_w'.

(ili) If X ~ X and f(x) ^{- Cl}

- '

then xeV:V'r-.

f(x)

=

Cl; similarly for the cases f(x):ß, Thus r'Jx

= r.

2.3 The 2T reflection

X and x

f w;

^{so that}

f(x)=Y, f(x) = ö

We show that the situation is analogous to that of Top, where the T

0

reflection can be obtained, as is well known, Internally - Let X

0 be the set of equivalence classes (x] with respect to the equivalence relation

induced by

q:x-[x].

xRy iff cl x

=

cly, with the quotient topology Then X is a T space with the universal property

0 0

with respect to maps into T

0 spaces. Moreover, the map q is an open map, closed map and an initial map.

Externally - Consider all maps f:(X,T) -(Y,S), where (Y,S), is a T space

0

with cardinality not exceeding that of f, then the map e:(X,T)-rt(Y,S)f induces the reflection e:(X,T) -.(e(x],S), where S is the relative product topology on e[x].

In fact, one need only consider the maps f:(X,T) -(D,u) into the Sierpinski space to obtain the reflection.

2.3.1 The internal description

Given (X,P,Q) define R by xRy iff cl ^X= cl y and cl ^X= cl y,

p p - Q Q so

that R is the equivalence relation that determines the T reflection of

0

(X ,PVQ).

X' = X/R

Let P'

,a·

be the quotient topologies on the quotient space

induced by Panda, respectively, and the quotient map q:X-X/R.

2. 3. 1. 1. Lemma

=

A.

If Ais P-closed or P-open or Q-elosed or Q-open, then

q lq [A11

Corollaries q is a elosed map with respect to both topologies q is an open map with respeet to both topologies cl q(x)

=

^{q tel x1 , cl q(x)}

=

q tel

x1

p· P

a·

o

q:(X,P,Ol-(X',P',Q') is an initial map.

2.3.1.2. Proposition (X',P',Q') is 2T.

0

Proof. If q(x) i. q(yl, then cl X f. cl y or: cl

0x i cl 0y.

p p Assume

cl x i. cl y. Then

x+

^clpy .or y

f

^clpx. Hence q(x)

•

^{q(e~ y) or}

p p

i.e.

cl X f. cl y.

Q Q

2.3.1.4 Proposition then there

is a

unique

q(x)

4

cl~(y) or q(y)

f

^{el 9(x).}

p Similarly for

If (Y,L,R)

is

2T

0 and

f': (X' ,P' ,O') - - (Y ,L,R)

f: (X,P,Ol - (Y ,L,R), such that f 'o q

=

f.

Proof Since fdX,PVQ) _. (Y,LVR) and

LVR

is

T

0, it follcws that f is constant an q(x), for all x in X. Define f'(q(x))

=

f(x). Then f' is well defined and f':(X',P', Q')-{Y,L,R), since q:(X,P) -(X',p') and q: (X,Q) - (X'

,a')

are quotient maps.

Finally, the relations between the 2T

0 reflection and the T

0 reflection can be stated explicitly as follows:

Denote the T reflection of (X,T) by (X ,T ), the reflection map by q

O O 0 0

and the reflector by

.:Eo·

2. 3. 1, 5 Proposition 2.3. 1.6 Proposition Corollary:

2.3.1.7 Proposition

_,.

Corollary:

T oS = S 02T -0 ---o

~ F = F

•!o

T

=

u •2T •·F

-0 --0

2T oD = D •T

----0 -0

T

=

So2T •D

-0 - - 0

Thus, T can be recovered from 2T ,but not conversely.

--0 - - 0

We shall only give the justification of 2.3.1.6, which depends on the lemma:

2.3.1.6 Lemma

Proof. V is a q { ₀

r*

l neighbourhood of q { ₀ x l

~

^{q+ LV} ₀

1

neighbourhood of x ^#- cl x _T <. q ,._ _o

t_v]

++ q _o

t_

cl x _T

1

^{~ V ~}

is a T

•

cl{q(x))~V ++

q {T)

V is a(q

•

0(T )) - neighbourhood of q₀ (x).

0

2.3.2 The external description

The external description of the 2T reflection depends on the following

- - 0

easily verified property which we state for completeness.

2.3.2.1 Proposition The full subcategory of 2T is productive and hereditary.

----0

The triad and the quad deterrnine the 2T reflection in the same way that

---0

the Sierpinski dyad detennines the

Io

reflection:

Let S denote either11' or~ consider all maps f:{X,P,O)- {S,U,L), which will be denoted by C(X,S). Let e be the product map

C(X,S) [ ]

e: {X,P,O) - {S,U,L) and q the map q:X - e X induced by e.

Let

e[x] =

X', ---~:, the relative topology on e[x] induced by {S,U)C(X,S) and Q' the relative topology on e[x] induced by (S,L)C(X,L). Then

(X',p',Q') is a 2-T -space such that, for every f:{X,P,O) - {Y,L,R), there

0

is f:(X',P',C')-(Y,L,R) such that f'•q

=

f.

.2.4 A diagonal characterization of 2T spaces -

The following characterization is required for the explicit description of the closure operator associated with 2T . It is in the same spirit as

~

(s,1 ;

such diagonal theorems have been studied extensively, for instance, [DGJ for some quotient reflective subcategories;

[G.HJ

for all quotient

reflective subcategories; andin a more general context in [G.M.T1. We denote the diagonal of

XxX

by ÄXxX , or simply by A.

.2.4.1 Proposition (X,P,O) is 2T

~ i f and only i f

Proof.

b.,

=

cl D. f'\ cl A b(P)xb(P) b(O)xb(Q) Assume A

=

cl ~ t"'\

b{P)xb(P)

cl A b(Q)xb(O)

Let ^X,,_ y, Then (x,y) E\:

A,

so { x , y ) (: c 1

A

b{P)xb{P)

or (x,y) ~ cl A b(O)xb(O)

Assume the first possibility

holds, th~n there is a Popen set V containing x and a P-open set W containing y such that (V t"\ cl x)x(W r-. cl y) ,-.. Ä:

4,.

Thus

p p

(V f"I clpx) t"\ (W r'\ clpy)

= + ,

^{so that}

^{x 4}

^{W or}

^{x 4}

^clpY• in which case

x and y are separated by some Popen set. If

(x,yl.

cl

Ä ,

then there b(Q)xb(O)

is a 0-open set which separates x and y. ^Thus

PVQ

is a T topology an

X.

0

Conversely, assume

PVC

is a

T

topology. Suppose

x

'1- y, then there

0

is a Popen set which does not contain both points or a 0-open set which does not contain both. Assume the latter is the case, assume the 0-open set

W

contains x and not y. Then cl

0y "W

=4' ,

so that (x,y) ': W x cl0^y and Wxcl0y f'\ A.

= J; ,

otherwise there is z E- W f\ cl

0y, which implies y

e-

W.

Thus (x,y) ~ cl 6.

b(O)xb(O)

Similarly, for the first possibility,

(x,y)

4

cl

A

^{Thus, cl 6. f'\ cl A f;. A .} The result follows.

3. 1,T

0-closure and epirnorphisrns 3.1 Definition ( S

2) For f,g:(X,P,O)-(Y,L,R), let E = E(f,g) denote the equalizer of f and g. For

M~

X, define

[M1

X as the intersection of all E

=

E(f,g) such that f,g: (X,P,Q) - (Z,U,V), UVV is T and M <ä E,

0

The object of this section is to give an explicit description of the closure operator [

1

in terms of the topologies involved. Firstly, for E sets:

3.2 Proposition For any E

=

E(f,g), f,g:(X,P,O) - (Y,L,R), LVR is T we have E = cl E t'\ cl E •

b(P) b(Q)

Proof. Functo_!'ialit~ of

r - r* gi

ves f ,g: (

x, p* ,o*) _

• • • •

Hence f,g:(X,PVP ,OVO )-(Y,LVL ,RVR ). Thus fxg: (X,b(P) ,b(O))

-frxY,

b(L)xb(L), b(R)xb(R)}

Now: E

= (

fxg/"(.~xY ]

= (

fxg)+( cl Ä ""' cl A]

b(L)xb(L) b(R)xb(R) since A

=

cl C:::,.. ,.--.

b(L}xb(L)

cl A b(R)xb{R)

by 2.2.4.1.

* *

(Y,L ,R ).

.,

A 1 so : c 1 E

<:. (

^{f xg)}

+(

c l ll ] cl E ~ (fxgl+[ cl6 ]

b(P) b(L}xb(L) b(Q) b(R)xb(R)

Hence: cl E r\ cl E ~ ( fxgl+[

b(P) b(O)

cl A /""\ clA ] b(L)xb(L) b(R)xb(R)

= E.

3.3 Proposition For M ~ (X,P,O) , cl M ^f\ cl Mt;..[ M].

b(P) b(O)

Proof: Suppose M <=.. E : E(f,g). Th en c 1 M f'\ cl •·\ "" tä cl E f'\ cl E : E.

b(P) b(O) b(P) b(O) Hence

(M]

~ cl

M

^r\ cl

M ,

by definition of

[M].

b(P) b(Q)

o'

3.4 Proposition For M c;;. (X,P,Q), we have: cl

M

2

(M],

cl M;:2

[M].

b(P) b(Q)

Proof: . We prove that ^cl

M

^'2

[M].

b(P)

Suppose x ~ cl M, b(P)

then there is

a p open set U containing x such that U nclpx f"\ M = ~ • Define f ,g: X .... 1I

Y on cl x u ( X-U) p

U f'\(X-cl x) p

on

X - U

on U.

It is easy to verify that M ~ E(f,g).

f,g:(X,P,Q) - tlr,U,L) and that

Moreover f(x) =y, g(x)

=S,

so x~ E(f,g), hence x

•

^M. Similarly, [M}~cl M •

Q

3.5 Theorem For Mt;. (X,P,Q), we have

tM1 =

cl

M

^{I"\ cl M}

b(P) b(Q) 3.6 The hereditary property

For M ~ N ~ (X,P,O) one can form [M] relative to M as a subspace of N we denote this

[M] by [M]N;

one can also form

[M]

relative to

X,

we denote this by

[M]x.

Proof: Let PN be the relative topology on N topology on N induced by Q. It is clear that:

induced by P and QN the relative cl⁰M : cl M f""\ N

b(PN) b(P) and cl M

b(ON)

=

cl M n N.

b(O)

Hence cl M "' cl M b(PN) b(QN)

: cl M "" cl MAN b(P) b(Q)

Thus, [ ] is a hereditary closure operator in the sense of [DG

3.7

Non additivity of the 2T -closure operator

0

Recall that in Top, the closure operator associated with the T

0

reflection is L. Skula's b-closure operator

([s

2 ] ) , which is a Kuratowski closure operator. There are many closure operators in Top which arise as above and are not additive; examples were provided by

A.

Misior (private communication),

E.

Giuli,

F.

Cagliari, very soon after these operators were introduced, in direct response to the request in

[s

2 ] . However, the 2T -[ ]-operator provides a more natural and much simpler instance of this

0

phenomenon.

Example On OI',U,L), let c(M)

=

cl MA cl M b(U) b(L) Observe that: c({"T'})

:{Y},

cl{a}) =

\<X) ,

Thus: c[{a} v {'.}] +_c[{a}] u

~[{Y}].

c ( { Cl ,Y·} ) : { ^{Cl ,} ß ,Y } •

3.8 Functional properties of the 2T -closure operator

---.!..----=---o

We have described how a bitopological space (X,P,O) determines a closure space (X, (]X), where, for M ~ X we have

LMJx

= cl M f"\Cl M

b(P) b(Q)

Recall that f: ( \ , Cl.) - ( X_{2 ,} C_{2 )} is continuous, where Ci are closure

r[

^C

[A]]

c;

C2[f[A]]

^{for all A (;_ X (See}

[c] ) ,

opera tors, i f

l

3.8.1. Proposition For (X,P,O), the identity i is continuous as

i: (X,[ ])

~

^{(X,P) and i:}

(X,[ ] ) -

(X,O), but not necessarily continuous as i: (X,

l ] ) -

(X, PVC) •

Proof. Let H ~ X, then: i[M] : i [ c l M r. c 1 M ] c; i [ cl M ] " i [ c l M ] ~ c 1 M

b ( p )

bO .

^{b ( p ) b ( Q ) p}

\ '

Similarl:,·, i[M] C.: cl 0M.

To justify the last statement, consider the triad OI',U,L). Then i[{a,8}]

=

({a,8,~})

=11'

i cl -

i({CL,!}) =

cl{a,Y}

={a,Y}.

UVL UVL

3.8.2 Example: For (X,P,C), the identity map i is not necessarily continuous as i:(X,PVO)- (X,[]).

Consider, again, OI',U,L) : i(clUVL{B})

=

i({a,S,y})

=

{a,S,y} i

[B] = {B}

In spite of the above, the correspondence (X,P,Q)~-(x,[ ]Xl is functorial. This is a consequence of the following:

3.8.3 Proposition If f:(X,P,O) -(Y,L,R), then f:(X,[ ]Xl

-(Y,[

]yl.

Proof Let A 4- X, f([A]Xl = f(cl A ,..._ cl Al~ f(cl Alf"\ f(cl A)t;.

b(Pl ^{b(Q) b(P)} t(:l)

~ cl f(A) ('\ cl f(A)

=

[r(A) ]y.

b(P) b(Q)

3.9 Epimorphisms

It is clear that ^f: (X,P,O) ^{- ( Y} ,L,R}

f(X) is [ ]-dense in

Y.

is an epi in 2T if and only if

0

3.9.1 Proposition f: (X,P,O) - (Y,L,Rl is an epimorphism in 2T if and

0

only i f cl

f[X]

^f'\ cl

f[X] = Y.

3.9.2.

since

b(L) b(R)

Example The inclusion map

Ö E cl 'Ir t'\ cl 'Ir , so that b(U) b(L)

i:al',U,L) - (~,U,L) is an epimorphism,

o = [rr)

It is possible to ra:over from this characterization, Baron's description of epimorphisms in T

0r

3.9.3 Proposition Y

=

cl f(XL

f:(X,T) -(Y,L) is an epimorphism in T if and only if

0

b(L)

Proof: If f:(X,Tl-(Y,L) is an epi in T, then clearly,

0

f:(X,T,T)-+ (Y,L,Ll is epi in 2T, so that

0 Y: cl f X ^f"\ cl f X= cl f X

b(L) b(L) b(L)

Conversely, suppose f:(X,Tl-(Y,Ll is such that Y

=

cl f (X).

b(Ll Then

above.

then

f:(X,T,Tl-(Y,L,L) is an ep1 in 2T , by the characterization

o f h

Moreover, if (X,T)..f... (Y,L)B.(Z,M):(X,T,) -(Y,L) .. (Z,M), (X,T,T} !...(Y,L,L)~Z,M,M)

=

(X,T,T) L.(Y,L,L)..b..(z,M,M), so that

4. Extension closed spaces

4.1 Definition [I] : An object X in a concrete category Ais called absolutely A-closed if for every embedding X-Y, ^m where Y 6 A ,

Xis A-saturated if epi, is an isomorphism.

each embedding e:

x-Y, Y

Ei: A which is also

Note that an absolutely A-closed object is always A-saturated. The converse is true if and only if [ }A is weakly-hereditary.

It is shown in [I] that saturated algebras need not be absolutely closed and it is shown in

[D.Q

3

l

that Urysohn-saturated spaces need not be

absolutely-Urysohn-closed.

In our case the two notions coincide as the [

1

-closure operator in 2T is hereditary.

0

In the topological case, a

T

topological space is absolutely

T

-closed

0 - - - - 0

if and only if it is b-closed in every T -space that contains it, i.e. if

0

and only if it is sober. Recall that a T space (X,T) is sober if every

0

non-empty irreducible closed set Fis the closure of a (unique) point x in F, where F is irreducible if F=F, U F 2 , where F i ~ F and F is closed, is only possible when F

=

^F .

,

^{or F}

=

^F2 •

In the bitopological case, (X,P,O) is absolutely 2T space {Y,L,R) that contains (X,P,Q) we have

0

2T -closed

- - 0 - - -

cl X f\cl X b(L) b(R)

if for every

= X.

We shall present an internal characterization of absolutely 2T -closed spaces

0

and also an external characterization involving the canonical product

.(0,U,L)C(X,Cl)

4.2 Theorem (X,P,Q) is absolutely 2T -closed if and only if

0 .

For every pair (Fp,F

0), where Fp is a non-empty irreducible P-closed set and F

0 is a non-empty irreducible 0-closed set, there is x in Fp/'\ F 0

such that Fp

=

cl x

p ^F = cl X

Q Q

Proof Assume the given property holds for the stated pairs ^(Fp•F

0),

Suppose (X,P,Q) is not absolutely 2T -closed. Then there is a 2T space

0 0

(X,L,R) which contains (X,P,Q) andin which Xis not [ ] -closed. We FPtaY assume, without loss of generality, that Y- X

= {

p} . Now consider

Fe

=

cl p nX.

R

Fp i !2S, since

p.

E cl X " cl X ~ b(L) b(R)

cl X

b(L) , so that cl p " X i ^!2S L

It is clear that cl ( cl

p

^r, X) c;; cl p

L L L

conversely, suppose ^Cl

e

cl p

L

and let V be an L-neighbourhood of ^et., then p ~ V, so that there is

·xe-cl p" V r. X, L

hence similarly c l ( FQ ) : c l p •

R R

Vf'\(clp "X). and,

L

Moreover,

Fp

is an irreducible P-closed set, for suppose F p = F, ^\J F 2 , F

1 P-closed. Then cl P L

= cl F₁ ticl F2 , so that cl F₁

=

cl P

L L

^{or cl F 2}

=

L

cl

P.

L

In

L

either case, ^F

p or _{F p= F z.}

Similarly ^F

0 is

a

non-empty irreducible ^Q closed set. By assumption, there is x 4: F P /"\ F

0 such tha t Fp

=

cl

x,

^Fo

=

cl ^X.

Q

cl(cl x)

L

p

: cl (cl XA X)·: cl P, so that cl X: cl P, L L cl

x =

cl

P,

R R

L L L

which is impossible since (Y,L,R) is

2T ·

0

But then Similarly,

Conversely, assume

(X,P,O)

is absolutely

2T

0-closed. Let

(FP, F 0

^{l be} a pair of non empty P-closed, 0-closed irreducible sets, respectively.

Suppose there is no x in F Pi\ F

O such that cl x

= Fp,

p

Define an extension (Y,L,R) as follows:

Y • X U {p} , where p

t

^X. L open sets are of the form

{:

where V is Popen and V

n

Fp - <P

u

{p} V is P-open and V

n

Fp 1 $.

Similarly for R-open sets. lt is straightforward to verify that the

"L open sets" do constitute a topology on Y , using the irreducibility of Fp.

Moreover, LIX • P. Similarly for R,

The space (Y,L,R) is 2T : lt is only necessary to exhibit and L-open set

0

separating x and p, x EX. lf x

t

Fp, then Y-FP is an L-neighbourhood of x which does not contain p. Similarly if x

4

^FQ. So we may assume x E Fp

n

FQ.

Now ctpx

i

Fp or ctQx

i

FQ, by assumption on (FP,FQ). lf ctpx

i

Fp, then Y - ctpx is an L-neighbourhood of p which does not contain x. Similarly, if c~Qx

i

^{FQ, then Y -} ciQx is an R-neighbourhood of p which does not contain x.

ßP,Q)

is [ J-dense in (Y,L,R) : Let

V

be an L-neighbourhood of p. Then ct1^p

n

V

n

FP = V

n

(c\p

n

FP)

=

^V

ⁿ

FP (by definition of L, we have c-½_p

=

^{Fp). Now V}

ⁿ

^{Fp # <P,} by definition of the L-open sets. Hence p E d, X.

b(L)

Similarly, ^{p €} ci X. Hence p E

[XJy.

b(R)

The above characterization suggests that every space (X,P,Q) can be extended to an absolutely 2T -closed space by adjoining certain pairs of non-empty

0

irreducible closed sets to the space. This is in fact the case and will be discussed in detail elsewhere. For the moment, the followi.ng examples illustrate the idea.

4.3 Examples

1. ffi',U,L) is not absolutely 2T -closed as it is a [ ] -dense subspace

0

of (O,U,L). The only non empty P-closed and P irreducible, 0-closed and Q-irreducible pairs in ar,U,L) are ar,{a} } llith 'Il'= cl a,

{o} :

cl ^Cl1

L

u

({y} ,1!') with cly = y,cly

=TI-,

U L

({Y} ,{a}l

for which there is no representing point in1I'. In fact, it corresponds to Ö in (l !

2. (~,U,L) is absolutely 2T -closed. The relevant pairs and representing

0

points are:

(i)

({Y,ö},{a,ö} ,)

with

clö = {y,6} ,

u

^cl

6

=

{a,6}

L ( ii}

({Y,6},0),

with clu Y

= ^{Y,ö} _'

^clLY

⁼

⁰

(iii) (O,{a,6})

'

^{with c10a}

=

^{Cl, clLa}

⁼

^{a,

^ö}.

(iv) ( 0, Cl) , with cluß

=

^{0 , clLß}

⁼

^1D

Gbserve that in both cases UVL is a sober T topology.

0

3.

4.

(D,U,L) is such that both topologies are sober and T and so is uvl.

0

However, (D,U,L) is not absolutely 2T -closed since:

0

(D,U,L)

c;

01',U,L) and

[o] =

1I' under the identification

o-a,

^{1-1' .} Moreover, the irreducible pairs and representing points are:

({1},oi : -r (D, {o}l :

a;

({1}, {o}l:

ö;

(D,Dl : ß.

so that the absolutely 2T -closed extension of (D,U,L) is (O,U,L).

0

(D,u,u) is such that both (D,u,u) is not absolutely the identification O - B

({ 1}, Dl:l;

(D, {1} )

topologies are sober and T

0 • However, 2T -closed since: (D,u,u), (O,ti,L) under

0

1 - ö, and [ D]

=

0.

a ; ({1}, {1}): 6 ; (D,D): ß

There is a characterization of absolutely 2T -closed spaces analogous to

0

that of T sober spaces in terms of join filtere (see [R~E.H] for a historical

0

survey). Recall that a filter of open sets is a join f il ter if it is the union of open neighbourhood filters and that a T space is sober if and only if every

0

join filter is the open neighbourhood filter of some point (which is necessarily unique).

4.4 Proposition. A2T space (X,P ,P) is absolutely 2T -closed if and only if

0 1 2 0

for every pair (F

1,F

2) , where fi is a join filter of Pi open sets, i = 1,2 ,

there is a point x such that

F.

is the P.-open neighbourhood filter of x ,

l l

i • 1,2.

Proof. Suppose (X,P ,P) is absolutely 2T -closed and (f

,F) a

pair as

1 2 0 1 2

specifted. Let F. denote the set of all y whose P.-open neighbourhood system

l l

N.

is contained in

f. ,

so that

F.

⁼

y1 l l U N ••

yEF. y1 l

It is straightforward to check

that Fi is a Pi-closed irreducible set, so that there is x such that clP.x = Fi.

l

Then

F.

is the P.-open neighbourhood filter ef x for i = 1,2 : it is clear that

l l

N.

^c

F. ,

^{since x}

E F. ;

^if

V E F. ,

then

V E N.

for some y in F. , so that

Xl - l l l Yl l

V must contain x , hence F. c N ••

l - Xl

Conversely, given the pair

(F ,F) ,

1 2

the family F. • U N • is

l yEF. y1 l

a join P.-open

l

where

F.

is P.-closed and irreducible,

l l

filter (since Fi is irreducible),

i = 1,2. By assumption, there is x such that

F. • N. ,

^{1 = 1,2. Then}

l Xl

F. '"'cip x

l •

l

if y

E F.

^and

V E N. ,

then

V E N. ,

so x

E V

l y1 Xl

and V E

N. ,

then V E

N. ,

so V E

F. ,

so that y E F

1 .•

Yl Xl l