Embedding metric spaces into CPO-S

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Embedding metric spaces into CPO-S

EMBEDDING METRIC SPACES INTO CPO-S

by Klaus Weihrauch and Ulrich Schreiber*

Abstract

We show that metric spaces with Polish topology (MP) can be represented very \vell by bounded complete w-algebraic cpo-s with continuous weight and distance (CWD). An operator r: CWD^• MP is defined such that (1) and (2) hold.

(1)

rö

can be embedded topologically into

ö

such that a (partial) function on

rö

is (metrically) continuous, iff it can be extended to a (total) continuous function on Ö. (2) Every MEMP is isometric to some r(Ö). Therefore the continuous cpo-functions are exactly adequate for . describing the continuous functions on the corresponding metric spaces.

The approach could be refined in order to study computability on metric spaces with Polish topology. Examples of application are the Euclidean space, the space of total functions (Baire space) and spaces of infinite trees.

* Die Arbeit wurde teilweise durch Mittel der Deutschen Forschungs- gemeinschaft gefördert.

1. Introduction

In one of his early papers on complete partial orders (cpo-s) Scott [1]

suggests that an appropriate cpo of intervals may be useful to study continuous and computable functions on the real numbers. Indeed, 20 years before, Lacombe [2] (see also Martin-Löf [3]) already showed that

computability on certain metric spaces (e.g. the Euclidean space, Cantor space, or Baire space) can be defined by embedding them into appropriate spaces of "approximations" on which computability can be defined in a natural way. These "spaces of approximations" are similar to "countably based complete partial orders^11, ccp-s, which have been studied recently by

Smyth [4]. In this paper we a~ply_the easier theory of w-algebraic cpo-s.

As a fir?t step we

consider only continuity, computability will be studied in a forthcoming paper. The metric spaces which we are able to embed are exactly the metric spaces with Polish topology (Bourbaki [5]), MP. The structures into which we embed are w-algebraic cpo-s (see e.g. Egli/Constable [6] or

Markowsky/Rosen [7]) enriched by continuous weight and continuous distance, CWD.

We define an operator

r: cwo

^• MP such that (a) and (b) hold.

(a) For every OE0/0,

ro

can be embedded topologically into Ö such that a (partial) function on

rö

is (metrically) continuous, iff it can be extended to a (total cpo-) continuous function on Ö. (See Theorem 21 for a more precise formulation) Therefore, the continuous cpo-functions are exactiy adequate for describing the continuous functions on the corresponding metric spaces.

( b) For every NE MP there i s some D such that

rö

i s i sometri c to N. There- fore, a metric space has Polish topology, iff it is in the range of

r

(up to i somorphi sm).

In Section 2 basic definitions and concepts will be summarized, in Section 3 weight and distance are introduced for w-algebraic cpo-s and their interrelation is studied. In Section 4 the operator

r

is defined, and it is proved in Section 5 that Ö is adequate for studying the continuous functions on

rö.

In Section 6 range

(r)

is characterized.

Finally, in Section 7 examples illustrate the abstract constructions of the previous sections.

2. Basic Definitions and Concepts

Let

ö

⁼ (D, ~) be a partial order. A subset X~D is directed in Ei, iff (vx,yEX)(3zEX)(xEz and ySz). i5 is d-complete (b-complete, fb-complete), iff every non empty subset X.s;:D which is directed (bounded, finite and bounded) has a join (= least upper bound) in

ö.

If a partial order with minimum is b-complete, then every non empty subset has an infimum. A chain in Ö is a sequence (x.).E with (Vi)(x.EO and

J. J. JN l

x. ~ x.

1 ) • LJ x . 'l'li 11 a l ways

1. l + l denote the join of a chain. If no confusion is possible we shall use the same symbols and ¹¹.L" for order and minimum on different spaces.

A function f:

ö

1 ^•

ö

2 between partial orders is monotone (d-continuous, fb-continuous), iff it preserves order (joins of directed sets, joins of finite bounded sets). f is an embedding, iff

x c.y.:,, fx '.:fy.

ö

^:= (D,~,J.) is a cpo, iff (D,~) is a d-complete partial order with minimum .L.

Ö := (D,B,S,J.) is an w-algebraic cpo, iff (D,i;,J.) is a cpo and B.s;:D is denumerab l e with ( 1) and ( 2) •

- F o r a 11 x E D , x = U { b . \ i E lN} fo r s ome c h a i n ( b . ) . E i n 8 . ( 1 )

l J. J. JN

Whenever b!= Li{x. ¹ i E :N} where bE B and (x.). E is a chain in D,

l ' J.J. N

then b ~ xj f or some j E l'J. ( 2)

Bis called the basis of Ö=(D,B,s,.1). It is determined uniquely by

(1) and (2).

In our extension theorem we shall need b-complete w-algebraic cpo-s.

Such a cpo can be obtained as the chain-completion of an fb-complete partial order. We shall now study fb-completion and chain-completion separately (see Wright/Wagner/Thatcher [10]).

Lemma 1: (fb-completion)

Let Ä = (A,~) be a partial order. Then there are an fb-complete partial order (C,E) and an embedding K: Ä^• C such that the following holds.

For every fb-co~plete partial order E = (E,S) and for every monotone f : Ä • E there i s a uni que fb-conti nuous mappi ng g : C ^• E with f = g ^K.

Cis uniuely determined by (3) up to isomorphism.

Proof: (outline)

Let (B^{1 , ~ )} be defined by B' := {Xs;A

l

X finite, X:J: 0, X bounded}, b' ~ c' :.;> (VbE b' )(:lcE c') b Sc. (ß' ,=) is a pre-order. Define

b'=c' :<c>(b'sc'Ac'~b'), C:=(B¹,S)/=, andK(b):={b}/=. Finally define g(X /=) := sup{f(x) / xE X}. The desired properties can now be verified.

'-<. E. D.

Lemma 2: (chain-completion)

Let Ä = (A,c,1.A) be a partial order with minimum ..LA and denumerable A.

Then there are an w-algebraic cpo Ö =(D,B,!::,..L) and an embedding 1 : Ä• Ö with l(A) =Bsuch that the following holds.

For every d-complete partial order E = (E,.'.:) and every monotone f: Ä^• E there i s a uni que d-conti nuous g : Ö ^• E with f = 91.

ö

is defined uniquely up to isomorphism by (4). Furthermore, if Ä is fb- complete, then ö is b-complete and sup XEB for every finite bounded non empty X s;.B.

( 3)

(4)

Proof (outline)

Let O':={(a.).c:JN ¹ (Yi)(a.EA and a.sa.

1)} and (a.). c' (b.). :~(vi)(3j)a.i:::b.

J. J.~ J. J. l + J. l - l l l J

Then (D', ~') is a pre-order. Oefine a'=b' :~(a'sib' and b'~'a') for a',b'ED' Let D : = D ' / = , 1 ( a ) : = ( a , a , ... ) / = , ~ : = ~' / = , B : = { 1 ( a)

l

^{a E A} , J. : =} l ( J.A) .

Then Ö := (D,B,~,J.) has the desired properties. For given E and f define g((a.). /=) :==Uf(a.).

J. l l

The desired properties can now be verified.

Q.E.D.

The cpo obtained by chain-completion can be embedded into that one obtained by first applying fb-completion and then chain-completion.

Lemma 3:

Let Ä be a partial order as in Lemma 2. Let B be its fb-completion with embedding K, let C its chain-completion with embedding 1, let

ö

be the chain completion of B with embedding l ¹•

Then there is a unique d-continuous embedding µ:C ^•

ö

with 1 ¹ K=µ•1.

K

c ö

]J

Proof: 1 'K is monotone (Lemma 1,2). By Lemma 2 there is a unique continuous u with 1¹K ==µ1.. It remains to prove: xi:::y~µxsµy. "•" holds by continuity of µ. "~": Suppose µXEµy. Since l(A) is the basis of C, x= L,J 1a. and

l l

y = L,11b. for chains (a.) and (b.). Continuity of µ implies

1. 1. 1. 1.

u 1 'K a = uu 1 a =uu1 a. c µu 1 b. =u u 1 b. = u 1 'Kb .. Since 1_{i . i . 1.- 1. . 1. 1.} ¹KA s;.basis of i5, (vi)(3j)1'K aii; 1'Kbj by (2). Since 1¹ and Kare embeddings,

(Vi)(3j) a. c b., thus 1 a. ~ 1 b., thus x~y.

1. - J 1. J

Q.E.0.

We shall study the embedding ~ once ~ore in Theorem 17.

In Section 3 we also want to extend a binary monotone function on the base to a continuous function on the completion.

Let Ö. _1. =(D.,!::.) be partial orders (i=l,2). The (cartesian) Product _{1. 1.}

i5 =

E\

^{x i5}₂ = (O,~) is defined by O := 0

1 x 0

2 and

(x,y) E (x',y') :~ (x~x• and y~y'). A binary function ist a function from a cartesian product. Continuity etc. are defined accordingly.

By the following Lemma we can apply our extension lemmas also to binary functi ons.

Lemma 4: Let c be the completion operator from Lemma 1 or Lemma 2.

Then c(i\ Xi\) is isomorphic to c(Bl) X cn\).

(proof straightforward)

Our aim is, to embed metric spaces into cpo-s, where embeddin; seans compatibility of topologies. Therefore, we also have to consider the natural topology (Scott [8]) of a cpo.

Let i5 = (0,8,f;,j_) be an w-algebraic cpo. For bEB let Ob:= {xE O I bSx}.

By (1) and (2) Obnoc =U{Oa

!

b~a, c~a}. Thus {Ob [ bEB} is a base of a topology TD on 0. The following lemma is fundamental.

Lemma 5

Let Ti be the topology induced by the w-algebraic cpo ö. on D ..

l l

(Fora proof see Scott [8]).

For simplicity we shall say complete instead of d-complete and continuous instead of d-continuous.

3. CPO-s with Weight and Distance

It is our aim to embed metric spaces into cpo-s. The cpo-elements are _ approximations, e.g. intervals on the reals. We need a criterion by which the elements of the metric space can be selected. Apparently, order alone (maximal elements) is not sufficient. We therefore introduce a new concept weight. Weight is a monotone function from a cpo into the real nurnbers.

The weight of an element indicates its precision. Computability theory on cpo-s is a qualitative theory until today. We claim that a quantitative theory of computability can be developped for weighted cpo-s. Especially computational complexity theory on the real numbers and on other metric spaces can be based on weighted cpo-s. But this is not the objective of this paper.

We shall see that a weighted cpo determines a metric space in a natural way. But there seem tobe metric spaces which cannot be generated this

way. We therefore introduce as a second concept distance on the cpo-s which satisfies certain axioms.

We shall now consider partial orders with weight and distance.

Definition 6:

Let i5 := (D,i;;;, 11,d). i5 is a partial order with weight and distance, iff (D,i;;;) is a partial order and II :D^• lR :=[0,^{00 ]} and d:DxD^• lR are

+ +

functions with:

( 1) ^Xi; y • lxl~lyl~O ( 2) ^{Xi;;; y •} d(x,z)::; d(y,z) ( 3) d(x,y) = d(y,x)

~o

( 4) d(x,x) = 0

( 5) d ( x ,y) S: d ( x, z) + d (y, z) + I z I for a 11 x ,y, z E D.

Two simple consequences are: xr;y • d(x,y) =0 and (3z)(xr; z and yr;_ z) • d(x,y) =

Thus 11 and d satisfy properties which also hold for diameter and distance on subsets of a metric space. In case of cpo-s we also require continuity.

We also require that no basis element is most precise.

Definition 7:

Let i5 := (D,ß,i;;;,.L,11,d). Ö is an w-algebraic cpo with weight and distance, iff (D,B,r;,.1.) is an w-algebraic cpo and (D,r;, 11,d) is a partial order v1ith weight and distance with

( 1 ) ( Vb E B ) I b I > 0 ,

(2) 11 : (D,r;) ^• (JR+' ~) is continuous >

(3) d: (D,r;) x (D,~) ^• (JR+, ::;) is continuous.

The structures from Definition 7 are rather complex.They can, however, be obtained by completion from simpler ones.

Lemma 8:

Let Ä=(A,i;,11,d) be a partial order with weight and distance. Let

Ö:=(C,!; ,II ,d) where (C,~) is the fb-completion of (A,i;) and II and

C C C C C

d are the fb-continuous ex tens i ons of 1 1 and d ( see Lemmas 1, 4) . Then C

C

is a partial order with weight and distance. (The corresponding statement hords for chain-completion.)

For the proof the properties of Definition 6 have tobe verified, which is a straightforward calculation.

Therefore_, an w-algebraic cpo with weight and distance can be constructed from a denumerable partial order with weight, distance,and minimum by completion.

We will have to consider weight and distance indepentently. However, for every partial order with weight there is a distringuished distance ct

11 .

Definition 9:

Let (O,t;;;) be a partial order and 11 be a weight satisfying (1) from Definition 6. A path from b

0 to bk is a finite sequence on D p:=(b.)._ ^k with (Yi)(3c)(b.i; C/\b.

1cc). Define its lenght by

l l - 0 , . . . 1 l l + -

1 g p : = I:{ ¹ b. 1 ! 1::;; i < k}. Let

l

ct11 (b,b¹⁾ :=inf{lgp I p is path from b to b^{1} .} Then the following lemma says that ct

11 is a distance which is maximal among all distances for 11.

Lemma 10:

(D,~, l l ,d11 ) is a partial order with weight and distance. Whenever (0,!;, 11,d) is a partial order with weight and distance, then

d(x,y)s:d

11 (x,y) for all x,yED.

The proof is a straightforward calculation again.

If no confusion is possible, we shall use the same symbols 11 and d for weights and distances on different spaces.

4. The Construction of the Metric Space

For every w-algebraic cpo with weight and distance,

D,

we shall now construct a metric space

rö

such that the following properties hold:

(a) The metric space

rö

can be embedded topologically into the cpo.

(b) A (partial) function on

ro

is continuous iff it can be extended to a (total) continuous function on

ö.

From Definition 6 it follows immediately, that the elements with weight 0 forma pseudometric subspace. This space, however, is not metric in general.

We shall define a method for selecting at most one member from every distance- 0-equivalence class. These points from a metric space. For an intuitive

justification of the approach consider B as a set of open subsets of a metric space, das metric distance, 11 as metric dia1neter, and r; stands for :2.

Completion yields a cpo. We are interested in those chains of open sets, the intersection of which contains exactly one point from the metric space. This point will be identified with the cpo-element. The following definition

expresses intuitively that the open set y and a surrounding of it of width c

is contained in the open set x.

Definition 11:

Let ö = (D,B,c:,.1, 11,d) be an w-algebraic cpo with weight and distance.

For cS>O and x,yED let

(1) x c

8 ^{y :-.>} (xi;y and (VeEB)(d(y,e) + lel <o=>x ~e)) (2) xr;:.y :-.> (38>0) x c

8 ^y ("strong approximation").

The crucial point is that Definition 11 does not require that we have sets of points. The points will be defined using Definition 11.

The following lemma summarizes some properties of the new relations. We shall use these properties later without explicit mentioning.

Lemma 12:

( 1) x~y c o y' c x' - ^=> x c:,._ o x'

(2) Suppose, sup X exists for Xf;.D. Suppose (VxEX) x c6 ^y. Then sup X c6 ^y.

( 3) (O<cS.ss and x cs y)

=

x c0 ^y.

(Proof straightforward)

We shall be interested in strongly increasing sequences of D which lead us to the definition of points.

Definition 13:

D := {xED i x =L,J b. for some chain (b.) in B with b. ^c::: b.

1}.

S · l l l l l+

(Strang elements of D)

The next Lemma will be used later in proofs.

Lemma 14:

( 1)

( 2)

Let xED and (c.).E _{S l l J N} be a chain in B with x=uc _{i '} Then x=uch(i') with ch(i) iz ch(i+l) for some monotone h.

Let x =½l x. for some chain (x.) in D with x. ^G: x.

1, let lxl = O.

l l l l l+

Then x ED .

s

Proof:

( 1)

( 2)

Since xED, x=ub. with b.iz b.

1, b. ES. From (2) we know that for all

S l l l+ l

i there are j,k with cii:; bi izbj+l~ck, thus CitI ck. Thus h can be defined.

fo r b, ck E 8, ck i; ck + l :

First we show ( b c

6 u ck and there is some

luckl

<%)

^{=? (3k) b c}₀ ck. Since 11 is continuous,

o ·

²

o

k with lckl <

2.

Suppose d(ck,c) + lcl <

2

for some cE B.

Then d ( u c. , c) + 1 c ^{1 ::;} d ( u c. , ck) + d ( c, , c) + 1 ck ¹ + 1 c ¹

l l K

< 0

+! + 1

^{= 0}

by Definition 6 and assumption for c. b c

0uck implies b;;c, therefore b c..Q. ck.

2

Suppose x. = u b ...

l l ]

A:={b .. ¹ i,jE:N}

l ] .

Using (2) one can easily show that

is directed with x = sup A. Let b .. be given. Then

l ]

for some o and m,b .. ~x. c.r x. 8

1cx =u b. and lxml <-₂. As we have

l ] l U l+ - m J ill]

shown above b .. c, b for some n. With thi s property and with

l ] _Q_ mn

2

X=supA one can easily construct a sequence (c.) in B with c.izc.

1

l l l+

and x = u c ..

l

Q.E.D.

By the following Theorem the strong elements with weight 0 forma metric space.

Theorem 15:

( a) (xED and lyl =0 and d(x,y) =0) ^=> x~y.

s

(b) (x,yE D and lxl = \yl = 0 and d(x,y) = 0) => x =y.

s

Proof:

( a) Let x=ub., y=uc., b.e::b.

1, c.cc. ₁• Definition 6 yields

l l l l.+ l - l + -

d(b.,c.) =0 for all i,j. Let i be given. There are o>0 and j with

l J

b. es b.

1 and \c.\<o. Then d(b.

1,c.)+\c.l<o which implies

l. u i+ J i+ J J

b. c c. by Definition 11. Therefore X!;;;y.

i - J

(b) (follows immediately)

Definition 16:

Let Ö be an w-algebraic cpo with weight and distance.

Let M:={xED [ lx\ =0}. Call the restriction of dto MxM also d.

s

r(Ö) := (M,d)

By Theorem 15, (M,d) is metric space.

0ur next aim is to show that r(E) is isometric to r(B) where E and Ö are as in Lemma 3. This means, we may assume wlg. that our cpo is b-complete.

The result can be used for the application of the extension theorem (Theorem 21).

Theorem 17:

Let A = (A,C,l., 11,d) be a denumerable partial order with minimum, weight, and distance. Let

B, C,

^and

D

be the completions of Ä from Lemma 3 where the unique extensions of 11 and d on B, C, or iS are also cal led 11 and d.

Let l, K, l ' , µ be the embeddings from Lemma 3.

Then r(C) is isometric to r(Ö).

Proof:

We use the same symbo l s 11 and d for wei ghts and di stances on all spaces.

Let f(C) = (Mc,d), r(Ö) = (MD, d). We show that µMc =MD and d(x,y) = d(1.1x,µy) forx,yEC.

By the extension property of d and 11 we have

Us i ng the representa tions x = u l a. , y = u 1 a. ' ( ( 1) and Lemma 2) we obta in

l l

immediately d(µx, µy) = d(x,y) by continuity of d.

Let XEMc. Then x = u la. with la. iz la.

1. If we show

· l l l+

1 a iz l a ' ( i n C ) ^• l ' K a e:: 1 ' K a ' ( i n D ) , t h e n JJ x = u 1 ' Kai E MD . Let 1.a~ 1.a'. We have

1a c₈ 1a¹•[ 1ai;1a'and(va₁ EA)(d(1a', 1aJ + l1a

11 <8=>1ai;1a 1)]

• [as;a' and (va

1 EA)(d(a', a

1) + lal <6 • ai;a 1)].

Let b

1EB and d(1¹Ka¹, 1.'b

1)+11'b

1I <6. We have to show 1¹K ai;1'b 1. By definition b

1 = sup KA

1 for some finite (bounded) A 1 ,;;,_A.

ib11 =min!KA

1i= 1Ka

01 for some a 0 EA

1 .Therefore d(a' ,a

0 ) + la

0 ¹ = d(Ka' ,Ka

0) + 1Ka

0 ¹ sd(Ka' ,b

1) + ib

1^I <o, therefore ai; a

0 ,

Ka~Ka 0 ~b

1, 1¹Ka~l ¹b

1. This proves µMc,;;,_M 0. Let xEM

0. We want to show x =µy for some yEMc. xEMa implies x = ~, 'bi

with ¹¹ b. rz ^1' b.

1, b. E B ( i E lN) . Let i E lN. Then 1¹ b. es 1 'b and

l l + l l U ffi

Ibm\ <cS for some cS>O and mE:N. bm=sup KA

1 for some finite A

1r;;_A, thus

1 bm ¹ = m i n 1K A

1 \ . Let a E A w i t h \ b ¹ = ¹ K a ^{1 •} Th e n d ( 1¹ b , 1¹ K a ) + 1 1 ' K a \

1 m m

= 0 + lal = lb ¹ <cS. 1¹ b. c._ 1¹ b implies 1¹ b. c1¹ Ka. Thus for all i there

m i 6 m i -

are some aEA and mE :N with 1¹ b. c 1¹ Ka i:;1' b rz 1¹ b

1. This implies that

i - m m+

there is some sequence (a.), a EAwith x=u1'Ka. and 1¹ Ka.c;:1'Ka.

1.

l i l l l +

Let y=u1a .. Then 1a. c;:1a.

1 (easy argument), thus yEMc and µy=x.

l l l +

Q.E.D.

From Lemma 10 we know, tha t d ~ d

11 for a gi ven wei ght II . Subs ti tuti ng d in Definition 11 yields a new kind of strong approximation, c;: . ¹ Obviously xc;:y => x c;:' y. Therefore the set D in Definition 13 and the

s

set M in Definition 16 is increased. Roughly speaking, d

11 yields the maximal metric space for a given weight II.

We have nowdefined r, but we do not yet know whether r(Ö) is trivial or not and whether the cpo ö is reasonably related to the metric space r(Ö).

5. Connections between ö and r(Ö)

We shall demonstrate now, that the cpo ö is adequate for

rö

in a very streng sense.

Theorem 18:

Let Ö be an w-algebraic cpo with weight and distance. Let (M,d) =f(Ö).

Let ,

0 be the canonical topology on

ö,

let, be the topology induced by

M , 0 on M ~ D. Let ,

0 be the topo l ogy of the open subsets of the metri c space (M,d). Then

Proof:

(a) Let bEB. We prove, that b :=ObnM is open in (M,d). ...

Suppose XE b. Then x =Ubi for some strong sequence (bi) of elements in B. From (2) and Definition 13 we conclude bb bk c

0 bk+l for some kE:N and o>O. We prove that the ball B(x, 1) is subset of b. Let yEB(x,

2).

8 Then y=uci for some strong sequence (cJ in Band

lyl = 0. By continuity of weight, ¹ ci ¹ <1 for some i > k, especially b c~ b .. We have d(b. ,c.) + lc. l ~d(b. ,x) +d(x,y) +d(y,c.) + lc. l <o

U l l l l l l l

(by Definition 6), and b c~ b. yields bf:c.cy. Therefore yEb, and

u l l

cS -

B(x,

2

^)f.b.

(b) Let xEM, o>O. Let yEB(x,o). We show: yEb~B(x,o) for some bEB.

y =Uc. for some chain (c.) in B. Then lc. l <o -d(x,y) for some 'i by

l l l

continuity of weight. Let zEc .. Then d(x,z) ~d(x,y) +d(y,c.) +d(c. ,z)

l l l

+ lcil <o. Therefore yEcif.B(x,o), and B(x,o) is open in ^TM.

Q.E. D.

Every functi on f : ö

1 ^• ö

2 has a ^II trace ^II from rEi

1 into rö

2.

Definition 19

Let ö. be w-algebraic cpo-s with weight and distance, let (M. ,d) := r(O.)

l l l

the induced metric spaces(i = 1,2). Let f: 0

1 ^• 0

2. Define Tr(f): Def(f)f.M

1 ^• M

2

1

by Oef(f) := {xEM -1

1 fxEM) =M

1nf M

2 and Tr(f)(x) :=fx for all xEDef(f).

The following theorem asserts that our construction of the metric space is sound wrt. continuity.

Theorem 20: ( Res tri cti on Theorem)

Let f:

ö

₁ ^•

ö

₂ be continuous on cpo-s. Then Tr(f) is continuous on the induced metric spaces.

The proof is straightforward from Theroem 18.

Continuous functions on cpo-s have only continuous functions on the metric spaces as traces. Here immediately the natural question arises, whether every partial continuous function on the metric spaces can be obtained as the trace of a contunuous cpo-function. We shall answer this negatively below. However, a somewhat weaker, but nevertheless as useful Theorem can be proved. lt says that every (partial) continuous function on the metric spaces can be extended to a (total) cpo-continuous functions.

Theorem 21 (Extension Theorem)

Let

ö

₁

,ö

₂ be w-algebraic cpo-s with weight and distance. Let

ö

2 be b-complete.

Let (1\,d) =f(i\) (i =1,2), let g: X.s;M 1 ^• M

2 be continuous on the metric spaces. Then there is a cpo-continuous (total) function f: 0

1 ^• o

2 i.-iith

( VX E X) f X = g X •

Proof:

If X = 0 choose f = >.x . ..L. Suppose X t 0.

For bEB. let b :=ObnM ..

1. l

For cEB₁ let q(c) := inf(lbl / bEB

1, bbC, bnXt 0}.

Xt 0 ^• .lE { ... }, thus q(c) exists and lcl >0 implies { b E B1 1 b b C , b () X

*

^{0 , 1 b 1 ::_:; q ( C) + 1 C 1 } t 0 •}

Let r(c) be an arbitrary fixed element of this set. Define

h(a) := inf u {gr(c) ~ ¹ a bc}.

/'-,

For every c, gr(c)

*

0,

ö

₂ is closed under inf of non empty sets, therefore, h(a) exists. 0bviously h is monotone on B

1. By Lemma 2, h can be extended uniquely to a continuous function f:0

1^• o

2. We have to show: xEX ^• fx=gx.

Let x=ub., b.EB₁, b.C!:b. A

1, xEX. We have b.bX • r(b.) ₆ x""XEr(bi.)

l l l l+ l l

,,,,.,--...,

• gxE gr(bJ • h(bJ i;; gx, therefore fx =Uh(bJ k gx. Using continuity of g we prove lh(b.)i ^• 0, thus lfxJ =0. With Definition 6 and Lemma 15a we can

l

conclude gx!;fx, and therefore gx=fx. Let s>0 be given. Since gxEM

2,

c₀ !;gx for some c

0 EB

2 with ic

0 J <s. c

0 is open in M

2 (Theorem 18), the_refore gxEB(gx,s₁)cc₀ for some s

1>0. By continuity of g, gß(x,6)s;,B(gx,s

1) for some 8>0. Esy Theorem 18, 0 ~B(x,o) and xE0 for some eEB

1,

e e

ekX=Ubi ^• ebbki;x for some k (by (2)), therefore gt\s;gB(x,ö)~co which implies c

-

0 kinf gbk.

We are fi ni shed if we show i nf gbk !; h( bj) for some j. For k there are ö>0 and j with bk es b. and fb. l <ö /3. Suppose, b. i; c. Then lcl <o/3,

u J J J

q(c) <ö/3, lr(c)I <28/3 and r(c)!; c. Using these properties, we obtain

d(b.,r(c)) + lr(c)I ::.:;d(b.,c) +d(c,r(c)) + lcl + lr(c)I <8, and bk c:s b. implies

J J u J

---..

- -

_,,,,,,-..,

bk 6 r(c). We conclude r(c)s;bk and inf gbki;inf u{gr(c) ¹ bji;c}=h(bj).

Therefore, for all s>0 there is some j 1tlith lh(b.)I <s.

J

Q.E. D.

The Restriction Theorem and the Extension Theorem are the fundamental theorems:whi eh j ustify the use of the cpo Ö to study the metri c space

rö.

As we have already mentioned, not every continuous function on

rö

is the trace of a continuous function on

ö.

We shall characterize now Def(f) for continuous functions f on the cpo-s. We do this only for "well behaved"

funct i ons.

Definition 22:

By the following theorem considering only strong functions does not restrict the class of continuous functions on the metric spaces.

Theorem 23:

Let f : Ö

1 ^•

ö

₂ be conti nuous. Let

ö

₂ be b-comp l ete. Then there i s some continuous strong function g:

ö

1 ^•

ö

2 such that Tr(g) extends Tr(f).

Proof:

Oefine a continuous "stretch" function cr:

ö

1 ^•

ö

2 by

for eEB

2. a is monotone on B

2, th2refore welldefined as a continuous function ( Lemma 2) . We prove tha t g : ⁼ 0f has the des i red property.

Let yEM₂, y='.Jb., b.czb.

1. For every i there are s>O and n with

l l l+

b.c b andlbl<s

i E n n

Then b. c

1b

I b and b. ~ ob ~b . This implies oy =y and Tr(g) extends

l n n i n n

T r( f).

Suppose I y 1 = 0, la y 1 = 0. Then for every i there i s some k with obi c

1b.J bi!;bk where labkl<lbil /2=:o (use Lemma 12 (2)). Suppose

l

d(obk,c) + lcl <8. Then

d(bk,c) + ICI ::;d(bk,obk) +d(obk,c) + labkl + lcl < lbil,

therefore ab. c c, therefore ab. es abk. This implies ay =uy. with

l - l u l

yiczyi+l where the yi are certain obk.

Lemma 14 (2) yields ayEM

2•

This proves our Theorem.

Q.E.D.

We shall now characterize the sets Def(f) for continuous strong functions.f.

Theorem 24:

Let M

2~D

2, M

2^:t= 0. Then {Def(f) ¹ f: 0

1 ^•

o

2, strong, continuous}

= {

A ~ M

1 ¹ A

=

n { U i I i E lN , U i o p e n i n ( M

1 , d )} }

= :

G O .

Proof:

( a) Suppose, A = n {U.

i

ⁱ E JIJ}, U. ~M

1 open. We may assume U.

1 f_U ..

l l l+ l

Let x =uci EM

2. Define f: 0 1 ^• o

2, continuous, by its values on B 1

as foll ows:

f(b)

ck where k = µ i [ not b f. U i] otherwi se,

where b : = Ob n M

1. I t i s easy to prove tha t f i s s trong, conti nuous and that Def(f) = A.

(b) (see [3]) Let f:ö

1^•

1\

be continuous and strong.

Let A : = n {u {tp( b) \ I b I < 2-n, b E B) \ n E f-1} with

<.P(b) :=U {c ! b J;fc, cEBl}. Then AEG5 (Theorem 18).

Def(f)f.A can be shown easily. Suppose xEA. Then for all n there is some b E B,,, 1 b \ < 2-n, and some c E B wi th c ^c x and b ~ f c .

n ,._ n n 1 n- n n

Therefore, b i;;;fx and \fx\ =0.

n

Since f is strong, fxEM

2. With xEAf.M

1 we conclude xEDef(f).

Q.E.D.

Using the results from Section 6 we obtain an extension theorem for metric spaces as a corollary of Theorem 20 and Theorem 21.

Extension Theorem for Polish Metric Spaces

Let (M. ,d.) (i = 1,2) be metric spaces with Polish topology. Let

l l

X f. M

1, f : X^• M

2 conti nuous. Then f can be extended to a conti nuous function g: Y^• M

2 where Xf.Y and Y is a G

0-subset of M

1.

Proof:

Apply Theorem 26 to both of the metric spaces, apply Theorem 21, apply Theorem 20.

Q.E.D.

Therefore, G0 subsets turn out tobe the "natural" domains of continuous functi ans between Po 1 i sh metri c spaces.

6. Embedding Polish Metric Spaces

We shall now characterize the metric spaces in the range of our opera tor r.

We use the following characterization of metric spaces with Polish

topology (see Bourbaki [5]). A metric space M=(M,d) has Polish topology, iff it has a denumerable dense subset and it is a G0-subspace of a complete metric space. A G

0-subset of M is a countable intersection of open subsets of M.

Theorem 25:

Let D be an w-algebraic cpo with weight and distance. Then

rö

is a Polish metric space.

Proof:

Let b:=ObnM. B

-

0 :={bEB \b=F0}. Let n:B

0-M arbitrary such that n(b)Eb.

Let T=range(n)f.M. (T,d) is a denumerable metric space. Consider its completion, (f ,d). T := {(y.) 'E /= 1 (y.) Cauchy sequence in T} where

l l JN l

(x.).~ =(y.).E :<e> lim d(x.,y.)=0 and d((x.)/=, (y.)/=) :=lim d(x.,y.)

l lc:JN l l JN l l l l l l

The mapping x-x := (x,x, ... )/= embeds T into T. For i E lN let

oi :=U {B(z,r) ⁱ r>O, ZET, (:la,b,cEB)

- - i

[a ee:bG:c, B(z,r)sc, \a\ <2 ]}.

By the above characterization, (T,d) restricted to ~ Oi is a Polish metric space. We prove now, that

µ: ub.-(nb.). /=, whereub.EM,

l l l l

is an isometric bijection from

rö

onto (n 0. ,d). It is sufficient to show

l

(a) ^µ is welldefined into T,

and (d) d(x,y) = d(uX,JJY).

(b) µ is into n 0., (c) µ is onto n 0.,

l l

(a) Let x= ub.EM. Let s>O be given. Then \b.\ <s/2 for i>n

l l S

With Definitibn 6 we obtain

d(nb. ,nb.) ~d(nb. ,b.) +d(b. ,b.) +d(b.,nb.) + \b. [+ib. \ <s

l J l l l ] ] ] l J

for i ,j > n . Therefore (nb.) 'E is a Cauchy-sequence. Suppose, also

E l l JN

X= uc .. Let s>O be given. Then \b.l<s/3 and \c.\<s/3 for i>ns.

l l l

By (2), ci ^!; bj for some j 2 i. Then

d ( nb. , nc' ) ~ d ( nb' , b' ) + d ( b' , b' ) + d ( b. , ^{C. )} + d ( ^{C. ,} nc' )

l l l l l ] J l l l

+ lb.l +lb.l +\b.l

l J l

<s

for i>n. Therefore (nb.).=(nc.)., and µ is welldefined.

S l l l l

(b) Let x =Ub. EM, let j be given. We prove JJXEO .. For some k,

l J

lbkl <2-j and bke:: b e:: b ~x for some m>l >k. By Theorem 18,

. 1 m

B(x,2r)~bm for some r>O. Since d(nb. ,x) .::;d(nb. ,b.) + lb. l +d(b. ,x).::; lb.

l l l l l l

and since (nb.) is a Cauchy-sequence, there is some s E: lN with

l

d(nb ,x) <r and d(nb. ,nb) <r/2 for all i >s. Choose z :=nb .

S l S S

Then B(z,r)~B(x,2r)~b and µxEB(z,r), therefore µxEO ..

m J

(c) Let y = (y. )/=E n 0 .. For all

l l there are

a. , b. , c. , z. , r. wi th

l l l l l

yEB(z.,r.), ß(z.,r.)Cc., \a.1<2-i, a.e::b.t:::c ..

l 1. l l - l l l l l

Prop.l:

c. ~Y. for almost all j.

l J

Proof:

yEB(zi,ri) implies d(zi,y)<ri -Ei for some Ei>O. There is jo such that d(y, y.) <s. for all j ~j . For j ~j we have

J l O 0

d ( z. ,y. ) = d (z. ,y. ) .::; d (z. ,y) + d (y

,y. )

^{< r. ,}

l J l J l J l

therefore y.EB(z.,r.)~c., i.e., c. cY.•

] l l l 1.- ]

Prop 2:

d ( c i, ak) = 0 for all i , k.

Proof:

By Prop. 1 c. cy. and a, ~Y- for some j, therefore d(c1.. ,ak) =0.

1 . - J K J

Prop 3:

(vi) ( 3k > i)

Proof:

We have bi ~

a. _l ^e:: ak. _,

C. l for some 6 > 0. Choose k wi th 2-k < 6. Then

a. e:: b. ca, , i. e. ai. e:: ak.

l l - K

Therefore, there is an increasing f with af(il ^c;: af(i+l).

Let v := uaf(k). We prove: d(µ(v) ,y) = 0. By definition d(µv,y) = lim d(riaf(k) ,y

j ,k

and d(naf(k) ,yj) .:s;d(naf(k) ,af(k)) +d(af(kl ,yj) + \af(k) \ <2-f(k) for sufficient large j by Prop. 1. Therefore d(µv,y) = 0 and µv =y.

(d) Let x =Ub., y =uc., x,yEM. We have

l l

d(nb. ,nc.) .:s;d(nb. ,b.) +d(b. ,c.) +d(c. ,nc.) + \b. l + \c. l

l l l l l l l l l l

.:s;\b.l +\c.\ +d(b.,c.),

l l l l

therefore

d(µx,µy) = lim d(rib. ,nc.) ⁼ lim d(b. ,c.) = d(x,y).

i l l i l l

Q.E.D.

We shall prove now, that every metric space with Polish topology is isometric to some rö.

Theorem 26:

Let (A,n) be a Polish metric space. Then (A,n) is isometric to rö for some w-algebraic cpo ö with weight and distance.

Proof:

Since (A,n) has Polish topology, there is a complete metric space N = (N,p) with dense countable subset X= {xi [ i E JN} and a G

0-subset G =n{Oi I i E lN} , 0. open in N, such that (G,P) :="N restricted to G" is isometric to (A,n).

l

We may assume O =N and 0.

1

co ..

We define a partial order of formal balls

0 l+_ - l

with formal inclusion (modified using the 0.) as order and formal distance

l

as distance.

Define B := (B,r;,1.,1 l ,d) as follows.

where

8 : = { ( i , q) [ i E fi, q E ~, q > 0} U {.l.}, 1.c(i,q),

(i,r)c(j,s) :~ [P(x.,x.)+s<r and h(i,r)+l::;h(j,s)J,

l J

h ( i , r) : = max { n [ B ( x. , r) f. 0 } E :N u { ^{00} ,}

l n

l..i.l : ^{= 00,}

l(i,r)I :=2r, d(.l.,(i,r)) :=0,

d((i,r), (j,s)) :=max(O,P(xi,xj)-r-s).

The following proposition can be proved straightforward.

Proposition 1:

Bis a partial order with weight and distance and minimum .l..

Let Ö == (D,B,t;;,J., l l ,d) be the chain-completion of 3. For simpl icity ¹11e

identify B with ,B. In the following, we consider balls in the metric space N==(N,p).

Proposition 2:

( 1) ( 2) ( 3)

(i ,p) !; (j ,q)

=

B(x. ,p) .2B(x. ,q)

l. J

( i , p ) C" ( j , q ) ^=> ß ( X . , p ) ;2 ß ( X . , q + o )

Ö l J

(i,p)c(j,q+o) = (i,p) c

0 (j,q)

The proofs are easy exercises.

Let (M,d) :==f(Ö) be the metric space obtained from Ö by r. (Definition 16).

Define t:.: M• N als follows:

where (n. ,r.) c:: (n.

1,r.

1) and r. ^•

o.

l. l 1.+ l+ l

We sha11 prove successively (a) t:. is wlldefined, (b) t:. is into G, (c) ^t:. is onto G, (d) t:. is isometric. These properties imply that (M,d) is isometric to (G,p), i .e. isometric to the given space (A,n).

(a) Let (n.,r.)e:: (n.

1,r.

1) and r. ^•

o.

By Proposition 2'(2)

l. l. 1.+ 1.+ l.

B(xn ,r.

1 +o.)cB(xn ,r.) for some o. >0.

i+l l+ 1. - i 1. l.

Sine N is complete, ~ B(xni,ri) has exactly one element.

Let (m.,s.)e::(m

1,s.

1) with u(m.,s.)=u(n.,r.).

l l 1.+ 1.+ 1. 1. l. 1.

Then (vi)(:lj) [(n.,r.)e::(m.,s.) and (m.,s.)e::(n.,r.)]. With Prop. 2 v,Je

l l J J l l J J

obtain n B(xni'ri) =n B(xmi•sJ.

(b) Let x= u(n.,r.), (n.,r.) e::(n.

1,r.

1). By definition and induction

l 1. l l l+ 1.+

h(n.,r.)2::i, therefore t:. XEB(xn ,r.)~O. for all i, and t:,. xEG.

l. l i ^l. l