Compactness in constructive analysis revisited

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deposit_hagen

Publikationsserver der Universitätsbibliothek

Mathematik und

Informatik

Informatik-Berichte 49 – 09/1984

Compactness in constructive analysis

revisited

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Christoph Kreitz and Klaus Weihrauch Fernuniversität

D-5800 Hagen West-Germany

UB Hagen

1 111

841176201

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In [ 5 ],l 3] we have developed a unified Type-2 theory of cornputability and continuity and a theory of representations.

In a third paper [ 6 J representations useful for a new kind of constructive analysis were presented. As an application of these concepts we shall now consider constructive cornpactness. We

introduce "reasonable" representations of closed and cornpact sets and prove two different versions of the Heine-Borel theorern.

Theorems concerning functions continuous on cornpact sets are investigated relative to constructivity properties and i t is shown that usually topological properties (i.e. discontinuity) are the true reasons for nonconstructivity.

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2. Representation of closed and compact sets

In [ 6] we have introduced several admissible representations of the open subsets of 1R . Since a set Ac 1R is closed if and only if 1R \ A is open, representations of the closed sets can immediatly be derived.

1. Definition (c.f.

l

6] Definitions 3.1/3.4/3.6)

Define representations a,a ,a of the closed subsets of

C -

( 1 ) a ( p) : = n { 1R \ Ik \ k ^E JMP } for every p ^E JF

,

1R by

( 2) p ^Edom ( a ) : ^~JM = {k ¹Ik n A =!:

0}

for some closed AC JR

C p

( 3)

a (p) := n

C n

a := an a

C

Representations u

j {I<. _J_,n> <j I

n>

^E JM _p}

.

(i.e. the standard infimum of a and and of the closed nonempty subsets of JR are defined by

(4) dom(a<) := {p ^E JF 1 (3A'.::_JR ,A =!: i;zj) (Vq ^Edom(p))

ex ) •

C

'

d(p(q),A) =p<~p(q)}

a<(p) := {x ^E1R ¹ (Vq ^E p-1

{x}) p<~p(q) = O}

(5), (6) and o:

1 are defined similar to a< using p> resp.

p instead of p<

In Bishop ^Is [ 2 ] constructi ve analysis a set Ac 1R for which dA (i.e. the distance function x - d(x,A)) "exists" is called

"located". Since every a

1-name of a closed set A is also a name for the distance function dA, in our theory locatedness

corresponds to the representation o:

1 .

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The following Lemma shows that representations of closed sets have essentially the same properties as the corresponding representations of open sets.

2. Lemma

Let M be a farnily of subsets of a set N and ö be an adrnissible representations of (M,T) . Define

ö c : JF - Mc :

=

{X 2:_ N j N \ X ^E:M} by Ö ^C(p) := N\o(p).

Then öc is an adrnissible representation of (Mc,Tc) where and O c

= {

X ^E: Mc j N \ X ^E: 0}

Proof

•

Since 6 =töu ~ öc =tö~ assurne ö

=

öu where U is a nurnbering of sorne basis of T . Define Uc(i) := (U(i))c Then Uc is a nurnbering of sorne basis of T ^C

corresponding representation ö c u satisfies:

and the

JM = { i \ X i:: Uc ( i) } = { i \ N \ X E: U ( i) } p

N\X=öu(p) = ö(p) for every Therefore öc = ö c is adrnissible.

u

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3. Corollary ( c. f. [ 6 ] Theorem 3. 2/ 3. 5)

( 1 ) Let

u~

:= {Y C JR ₁ y is closed and K. nY=~} where

J J

K. =

u {r

¹^NED.} and let T be the topology induced by

J n _J

the basis {U~ ¹ j E JN }

.

J

Then T is the final topology of C(, and C(, is admissible.

( 2) Let V~ := {Y ^CJR IY is closed and ( Vk E D.) IknY=t:~}

J J

Then C(, is admissible and its final topology T is

C C

induced by the basis {V~ j E JN } • J

( 3) a is admissible and { U

c. n vc.

¹

< . . >

_{]_, J} E JN }

]_ J is a basis of

the final topology of a .

(4) a and ac are incomparable w.r.t ~t .

.

Furthermore a is essentially equivalent to a< , ac is

essentially equivalent to a> and a is essentially equivalent to ( c. f. l 6 ] Theorem 3 • 7) •

As we know, a subset K c JR is compact if i t is closed and bounded. Therefore the restricitons of a, a , a (resp.

C -

C(,<, C(,>, al) to the bounded subsets of JR yield canonical representations Cl ' , Cl' Cl' (Cl~, ¹ Cl ' of the set K (JR)

c'

-

^Cl>, ₁

of all the compact subsets of JR (resp. of K ( JR) \ {~})

.

Unfortunalely these representations do not contain enough

finitely accessible information about compact sets. For example ist is not possible to abtain a P-name of the supremum of an nonempty compact set continously from its name w.r.t

Cl' c' ^~¹⁾ ^•

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4. Lemma

The operator sup K (IR)\{~} - IR is not (a.

1,

^{p>) -} continuous

Proof

Assume there is some continuous

r:

JF ---JF wi th

sup a.

1

^(p)

⁼

p>r (p) whenever p E dom(a.

1) .

^Choose ^p^E ^dom(a.

1)

with vD(O) ~ sup a

1

(p) . Then by continuity of T there is some n E JN such that ₁_E

]Mr

^(q) holds for every q E [ p [ n ] ] n dom ( r) •

On the ohter hand for every n there is some

q E [pln]]

n

dom(a.

1)

with vD(O) < sup a.

1

(q) . Because of 1 ^E JMr (q) ~ p>r(q) ::; vD(O) there is a contradiction.

•

An immediate consequence is that sup is not continous w.r.t (a

1

,p) , (a..;, p>) , (a.<, p>) etc. Similar i t can be

shown that sup is not (a.<, p<) continous. However the following holds.

5. Theorem

There are computable functions E, !::., r : JF --- JF such p<:E (p)

=

sup a.>(p) if a.> (p) is bounded above,

p>f<i,j>

=

sup a.<(p) if i is an upper bound of a.<(p) p!::.<i,p>

=

sup a.l (p) if i is an upper bound of al(p)

that

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A similar result holds for the infimum of compact sets.

Proof

Since a> is essentially equivalent to a there is some

C

computable

n :

IF --- IF with

JMn

(p) = {<j ,n> ¹I<j ,n>

n

a> (p)

*

^12)}.

Therefore v0 ( j) - 2-n (p) ~ <j ,n> ^E

JMr;i

(p) holds whenever sup a>(p) exists; Now i t is easy to construct some computable ^I: with JMI: (p) = {i

I

v

0 (i) (p)}

(i.e. p<I:(p) = sup a>(p)) whenever a>(p) is bounded above.

Nc..iv,1 let p ^Edom a< and i ^E JN be an upper bound of a< (p).

Then v

0 (j) > sup a<(p) ^~^(v0 (j) >i or [v

0(j);i]~JR \a<(p)), Since there is some computabl

c

with

JR \ a<(p) = U{Ij \ j E MJ\.(p)} and [v

0^(j);i] satisfies the Heine-Borel property, there is some computable

r :

IF --- IF.

with JM,-r<, >= {j

I

VD(j) >sup Cl<(p)}

1' .1-'p ^{( i .}^e.

whenever i is an upper bound of a<(p)

The existence of ~ follows immediately from a

1 ^EInfc{a< ,a>}.

•

As we have seen, for determining continuously the supremum of a compact set i t is not sufficient to know the fact that i t is bounded but information about an explicit bound is neces-

sary (c.f. Bishop's [ 2] concept of constructive compactness).

By tupling functions <, > : JN x IF -- IF this information can be inserted into name of a compact set.

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o. Definition

Define a representation Cl ^b

E dom a b ab<i,p>

: ~ p E dom a := a(p)

JF---K(IR) by

and a(p) c [-i;i]

^Edom a b Representations

whenever

and are defined correspondingly.

Clearly sup is and (al,p) ^b computable.

Another method to characterize compact subsets of JR is to describe them by their Heine-Borel property. This will lead us to (at least) two different representations.

Let C :

= {

I . j E JM } and

p J p

C :={I. (::liED) p ( i ) = j + 1 } .

p ,n J n

Let us say ^IIf'4 E [ JF - JN] proves compactness of Sc JR ¹¹

i ff [ (

s

^CuC <z ;> p ^E dom f'4) and (

s

^Cu^C ^;>

s

^CuC ^{r, (} ^{) ) ]}

- p - p - p ,~. p

holds for every p ^E JF

It is easy to see, that for every compact set S E JR there is some rl ^E

l

JF - JN] proving i ts compactness. Therefore compact sets in JR can be represented in the following way.

7. Definition

The weak Heine-Borel representation ^K_w ^JF^---+^K ^(JR) is defined by

p ^Edom ( Kw) : <z ;>

XP

prove·s compactness of some ^SC^JR

.-

The unique set S whose compactness is proved by ( i . e.

s =

n{uc

P,xP(q)

q E dom X } ) p

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From real analysis we know, that a set Sc JR satisf ies

the Heine-Borel property if and only if i t is closed and bounded.

In our theory we can describe a constructive version of this theorem by a simple equivalence of representations.

8. Theorem

Proof a b

C K w

(weak Heine-Borel theorem)

Let E dom a ^b . Then ab<i,p> = [-i;i] \U{I.

J j E JM }

p and [ -i; i] is compact in JR • Therefore

ab<i,p>CUC ~ ^;> (3n) [-i·i] CUC LJUC and

- q ' - p ,n ^q,n

[-i·iJCUC LJUC ;> ab<i,p>CUC

' - p.n q,n - q,n

Since [-i·i] cuc u uc is decidable in <i,n,p,q>

' - p,n q,n

there is some computable

I :

^{JF - JF} ^with

ab<i,p> = K (I<i,p>) for every w

E dom a b Conversely for p ^Edom ^K ^, ^K (p) c uc. d holds (where

w w - J. id(n) =n)

and because UC. C [ -i · i]

1.d,n - ' is decidable, there is some computable ^!:::,. ^: JF - - - :N wi th ^K ^(p)^C [-t,(p) ;ß(p)]

w - if

p E dom K • w

Furthermore K (p) =JR\U {r.\Y.nK (p) =0}

w J J w ^and

Y. n

^K (p) = ~ <=> ( 3q E dom

x )

I.

n

uc = ~

J w p J q,xP(q)

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Using the projection-theorem (see

l

5 ]) one can construct some computable

r :

^{JF - - -} ^IF with

JMr

(p) = {j \ Ij

n

Kw(p) = ~}

and hence Kw(p)

=

af(p) . Therefore K ::; ab .

W C

•

Because of theorem 8 sup is not (Kw,p) continuous. Therefore K is called the weak Heine-Borel representation of K(JR)

w

As we already know, the names w.r.t. a b contain more finitely

accessible information about compact sets than that w.r.t. ab It can be shown that this information corresponds with Bishop's

[ 2 ] notion "totally bounded", since from a name p for any covering C of ^~b (p) a finite minimal subcovering C

q q,n

can be obtained (where "minimal" means: C

q,n contains no superfluous elements I.

J with Ij

n

^a^b₁^(p)

= ~) .

This leads to a strong Heine-Borel representation of K(JR)

9. Definition

Define K : JF - - - K ( JR) by dom K : = { p i=: dorn

x

w ( Vq ^i=: dom XP) C ( )

q,xP q is minimal covering

K(p) ^:=^K_w^(p)

of Kw(p)}

whenever p ^i=: dom K .

The strong version of the Heine-Borel theorem (also expressing Bishop' s version: "A set S c JR is compact - i. e. totally bounded and complete - if i t is closed, bounded and located")

can also be formulated by a simple equivalence.

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10. Theorem (weak Heine-Borel theorem)

b

K - Ci.

C

The proof is similar that of theorem 8.

3. Functions continuous on compact sets

Because of admissibility of the representation p: JF ---JR continuous functions on JR can be described by weak (p,p)- continuity (c.f. [6] Theorem 2.3, [3] Theorem 23).

Furthermore from topology i€ is well known, that continuous real functions have G

0-subsets of JR as natural domains, i.e. every continuous function can be extended to an continuous function having a G

0-set as domain. It turns out that these

"natural" continuous functions are exactly the (p,p) - continuous ones.

11. Theorem

A function f : JR - - - JR is (p,p) - continuous iff f is topologically continuous and dom(f) is a G

0-subset of

]R •

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Proof

By [ 3 ] Theorem 2 3 we only have to show

" f (p,p) - continuous ;> dom(f) is a G

0-set."

Our proof uses the construction of continuous functions in

•

[ IF - - - IF ] ( see [ 5 ] ) •

Let f be (p,p)- continuous, i.e. there is some y: W(JN) -W(JN) that fp(p)

=

py(p) whenever p dom(y) and

p (p) E dom(f) ~ p dom(y) n dom(p) ,

where y(p)

=

sup{y[p[i] J i E JN} (if sup{ . . . } exists) . Now let

s =

{i EJN ¹ [vJN (i)] ndom(p) =l=~} and for i ES

define Ui := (v

0 ^(j)-2(k+ 1 ) ; -v

0 ^(j)+2-(k+ 1 )) where

k

=

lg(vJN (i)) and j

=

vJN (i) (k) (note that

-k . -k

P [ \J JN ( i ) ]

= [

^\JD ( j ) - 2 ; v D ( J ) + 2 ] ) . Let

o

:= u{u. ¹ i ES and (Vj ES) (U. = U. ;::,. lg(y-vJN (j)) ~ n)}

n i i J

Then by continuity of p and since for every XE JR one can show that

p -1 {x} is compact dom(f)

= no

n n is a

By

l

3] Definition 4 p induces a canonical representation [p-p] of all the (p,p) - continous functions. Because of Theorem 11 the set represented by [p-p] is the set of all the continuous real functions having a G0-set as domain.

In the following investigation of continuous real functions defined on compact sets we shall use shortly p instead of [ p - p] .

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It is well known in real analysis, that the continuous image of a compact set is compact again. In our theory this property is shown tobe effective.

12. Theorem

There is some computable

r :

^{JF - JF} such that

p(p) (K(q))

=

Kf<p,q> holds whenever K(q) ::,dom(p(p)) .

Proof

Since p -c

o

1 ^{by [} 3] Theorem 25 there is some computable such that for p E dom p , r E JF ,

(p ) - 1

ucr

=

ucZ<p,r>

n

dom(p(p)) and for every j

- -1

(p(p)) Ir(j)-₁

=

^U{Ik¹ (3i)Z<p,r><j,i>

=

^k+1}.

Now assume K (q) ::, dom(p (p)) and p(p) (K (q)) C uc • - r Then K ( q) c uc'"' . Hence Z <p, r> ^Edom ( Xq)

- L, and

C is a minimal covering of K(q)

Z<p, r>, X Z<p, r> q

There is some computable

r :

^{JF - JF} such that

=

{j ^J {3i)<j,i> ED Z< r>}

Xq p,

Then is a minimal subcovering of p(p) (K(q)).

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•

Furthermore and therefore

r E dom ( x

r

<p ,

q)

i ff p(p) (K(q))

=

Kf<p,q>

p(pl <x<ql, cuc r

Theorem 12 also holds for the weak Heine-Borel representation K _w . Effective versions of two theorerns of real analysis are an immediate consequence.

13. Corollary (A function continuous on a compact set

s

is bounded on S . )

There is some cornputable 6. : IF --- JN such that p (p) is bounded on K (q) by 6.<p,q>

w whenever K (p) c dorn(p (p)).

w -

14. Corollary (A function f continuous on a compact set

s

assumes the values sup f(S) and inf f(S) .)

There are computable operators

I,

^{r2 :} ^{IF - IF} such that

PI<p,q> = sup p(p) (K(q)) and p~<p,q>

=

inf p(p) (K(q)) whenever K (q)

:=,

dom(p (p)) .

Since sup and inf are not (Kw,p) - continuous, Corollary 14 only holds for the strong Heine-Borel representation. Furthermore there is no continuous way to obtain a point x ·where a continuous function f assumes its suprernum on a compact set S even not

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for fixed S and a fixed K-name of S .

15. Theorem

There is no continuous

r :

JF - - - JF such that for every

p E dom(p). p (p) assumes its supremum on [-1;1] at pf(p)

Pröof

whenever [-1; 1] ::_ dom(p (p)) .

Define f ,g :JR-JR

n n by f ( x)

n

• - X • 2

-n

and

g (x)

n

: = -x • 2 -n • Then ( f ) and are convergent lim f

=

lim g . n

sequences of continuous functions and

n n

Furthermore f assumes its supremum on [·-1; 1]

n at ^X=

and assumes its supremum on [ - 1 ; 1 ] at. X= -1 .

g1 (x) = -x/2

-4

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Now assume p (p) ( pf (p)) = sup{p (p) (x) x E [ -1; 1]}

whenever [ -1; 1] ~ dom(p (p)) for some

r

IF - - - IF • Using the definition of

-

^p ^(resp. ^p ^{[6] and}

^~

l/J [ 5]) one can easily construct two convergent sequences (pn) and

(qn) ⁱⁿ ^IF such that

-

p(pn) ⁼ f

-

n' p(qn) ⁼ gn and lim p _n = lim qn

.

^Since for every n pf(pn)

=

¹ ^and

pf(qn) ⁼ ^-1

. r

cannot be continuous.

•

A continuous function f defined on a cornpact set S is

uniforrnly continuous, i. e. there exists a function m: IN - IN such that ¹x-y 1 :s; 2 -rn ( n) -;> -n

lf(x) - f(y) 1 :s; 2 holds for every

x ÊS, y Êdorn ( f) and n ÊJN • rn is called a modulus of continui ty for f on s . The next theorern shows, thät frorn a

-

^p-narne

of f and a K-narne for S a rnodulus of continuity for f on S can be cornputed.

16. Theorem

There is sorne cornputable

r :

IF ---- IF such that f is a rnodulus of continuity for p(p) on K (q) whenever

w q ^Edorn ( K ) , p ^E dorn ( p)

w and ^K ⁽q ) C dorn ( p ( p ) ) . w -

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Proof

•

Let S and U. be defined as in the proof of theorem 11.

l

Then for every p ^Edom(p) , n ^EJN we have dom(p (p)) c O ,

- n ,p where O . - U{U. 1 i ^E:

s /\

(VjES) (U.=U. >lg(-v]Np(i)) 2:n}

n,p ^l ^l J

Since s is recursive and U. = I

l r(i) for some recursive r there is some computable 6 : JF - - - JF such that

0 = U{I.

1 j E JM t<n,p>} = uct<n,p> for every p E dom (p)

n+1 ,p _J

q E dom ( K ) ,

- -

Now let w p E dom ( p) such that Kw ( q) ~ dom ( p ( p) ) Then K (q) c uc and therefore

w - t<n,p> ^Kw ^(q)^C- ^UCt<n,p>,x ti<n,p> _q ^• There is some computable

r :

JF - - - JF such that

f<p,q>(n) := max{k \ (3i) (3j ED A< >) <i,k> + 1 = t<n,p>(j)}

XqL..I n ,p

r

has the desired properties.

From real analysis we know, that a function f defined on an intervall [a;b] with f(a) *f(b) assumes every value between f(a) and f(b) . A contructive version of this fact cannot hold in general. A counterexample is given in the following theorem.

17. Theorem

There is no continuous

r :

^{JF - - -} 1F such that for every pEdom(p) p(p) vanishes at pf(p) ^E [-2;2] whenever

[-2;2]~dom(p(p)) and p(p)(-2)<0<p(p)(2).

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Proof ( c. f. Aberth [ 1 ] § 7. 4)

•

Define functions f ,g : JR - JR by n n

f (x) .- min{x + 1 ,max{x - 1 ,2 -n } } and n

g (x) n

:= min{x + 1 ,max{x - 1 ,-2 -n } } . Then ( f )

n ^and are convergent sequences of continuous functions with lim f _n = lim g _n . Furthermore f (-2) _n

=

^{g (-2)}_n

=

^{-1 ,}

2

y =x-1 y=2 -n

- -

/ f o go

-4 -3 -1 2 3

/

y= -2 -n

-1

y= x+l

/

-2

Therefore

r

cannot exist.(Use arguments similar to that of Theorem 15 for formalization).

An immediate consequence of theorem 17 is, that the intermediate value theorem and the mean-value theorems for derivates and

integrals cannot be formulated effectively. However there are restricted versions excluding functions that are constant on an intervall [ c; d] ~

l

a; b] . See Aberth [ 1 ] for further information.

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In this contribution we have investigated representations of compact sets and "cl3.ssical" theorems concerning compactness relative to the constructivity. It turned out that at least two different characterizations of compact sets can be given corresponding to the finitely accessible information contained in the representation. Furthermore an interesting observation can be made: whenever an effective version of a mathematical theorem does not hold this is so for topological reasons

(discontinuity) indepentently of Church's thesis and if something is continuous then i t is even easily computable. The fundamental role of topology in contructive analysis has already been pointed out by Nerode [ 4] and others. Seemingly there is - except for artificial combinatorial counterexamples - no natural correspon- dence which is continuously effective but not computably w.r.t.

the (effective) standard-representations.

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[2] Bishop, E.: Foundations of constructive analysis, McGraw-Hill, New York, 1967.

[3] Kreitz, C. & Weihrauch, K.: Theory of representations, to appear.

[4] Nerode, A.: Lectures on "Constructive analysis", 1982 Summer institute on recursion theory, Cornell university, 1982.

[5] Weihrauch, K.: Type 2 recursion theory, to appear.

[6] Weihrauch, K. & Kreitz, C.: Representations of real numbers and open subsets of the set of real numbers, Informatik- Berichte 43, Fernuniversität Hagen, 1984.