• Nie Znaleziono Wyników

which play a similar role. The aim of this note is to study some relation»

N/A
N/A
Protected

Academic year: 2021

Share "which play a similar role. The aim of this note is to study some relation» "

Copied!
10
0
0

Pełen tekst

(1)

ROCZNIKI POLSKIEGO TO W AR ZYSTW A MATEMATYCZNEGO Séria I : PRACE MATEMATYCZNE X I Y (1970)

Ka r o l Bo r s u k

(Warszawa)

A note on the shape oî quasi-homeomorphic compacta

The concept of the shape (see [1]) has been introduced in order to compare compacta from the point of view of their global topological properties. There exist some other notions (as the notion of the X-likeness, [4], p. 146, and the notion of the quasi-homeomorphism [3], p. 252}

which play a similar role. The aim of this note is to study some relation»

between those notions.

§ 1. Basic definitions. We denote by Q the Hilbert cube, that is the subset of the real Hilbert space E m consisting of all points (aq, x 2, ...) satisfying the condition

0

< xn < — for every n =

1

,

2

, . . . , n

En denote the subset of E “ consisting of all points of the form (aq, a?a, ...

• • •, %n,

0

, — ) •

Let X , Y be two compacta lying in the Hilbert cube Q. A sequence of maps f k: Q ->Q is said to be a fundamental sequence (compare [

2

], p. 225) from X to Y (notation: / = { f k, X , Y}, or / : X ->Y), if for every neighborhood F of Г there is a neighborhood U of X such that

fk/U ~ fk+ilU in V for almost all Tc.

Two fundamental sequences f = { f k, X , Y} and f ' = { f k, X , Y } are said to be Tiomotopic (notation: / - / ' ) if for every neighborhood V of Y there is a neighborhood U of X such that

fk/U - f i J U in V for almost all Tc.

If X , Y, Z are compacta lying in Q and / = { f k, X , Y } , g = {gk, Y , Z }

are fundamental sequences, then { ^ , / fc, X , Z} is a fundamental sequence

which one denotes by g f . The fundamental sequence { f k, X, X} , where

f k = i (i denotes the identity map of Q onto itself) for every Tc —

1

,

2

, . . . ,

is said to be the fundamental identity sequence for X\ one denotes it by ix .

(2)

If there exist two fundamental sequences

/ = { Д , Х , Y} and g = {gk, Y , X }

such that f g — ir , then we say that X fundamentally dominates Y (com­

pare [2], p. 233). Then we write X > Y. If there exist two fundamental

F

sequences / : X - » Y and g: Y -> X such that both homotopies f g - i y and g f — ix hold, then X and Y are said to be fundamentally equivalent (notation: X - Y).

F

Since both relations > and - are topological (compare [2], p. 234),

F F

one can extend them onto arbitrary compacta as follows:

X > Y means that there exist two compacta X ', Y' <= Q homeomorphic

F

to X and Y respectively and such that X ' > Y '. On the same way one extends the relation - onto arbitrary compacta.

F

Now one defines the shape Sh(X) of a compactum X as the collection of all compacta Y satisfying the condition X — Y. The relation Sh(X)

F

> Sh( Y) means that X > Y.

F

We say that a space Y is X-liJce (compare [4], p. 146) if for every e > 0 there is a map д of Y onto X such that

(

1

.

1

) 0[ д ~ Цх ) ] < e for every point x e X .

One sees easily that if X , Y, Z are compacta and if Y is X-like and Z is Y-like, then Z is X-like.

Two compacta X and Y are said to be quasi-homeomorphic (see [3], p. 252) if Y is X-like and X is Y-like. It is clear that quasi-homeomorphism is an equivalence relation.

§ 2. X-likeness and the shape. Let us prove the following

(2.1)

Th e o r e m.

Let X , Y be two compacta. I f Y e AÎSTE and Y is X-li7ce, then S h ( X ) > S h ( Y ) .

P r o o f . Since the notions of the X-likeness and of the shape are topological, we may assume that X and Y are subsets of the Hilbert cube Q. Since Y e AXB, there is a neighborhood F of Y (in Q) and a re­

traction

r : F -> Y . Let a be a positive number so small that

(

2

.

2

)

q

(y, Y) < 3a implies y e V for every point ye Q.

It is clear, that we can assign to this number a and to the given re­

traction r a positive number e < a so small that

(2.3) g(y, Y ) < e implies g(y, r(y)) < a for every point ye Q.

(3)

Since Y is X-like, there exists a map g: Y - + X

such that g{ Y) = X and that condition (1.1) is satisfied. To the number e we can assign a number ry >

0

such that

(2.4) If A <= X and ô(A) < ?y, then <

3

[<

7

_

1

(X)] < £.

Now let us consider a finite cover of X by open (in X ) sets G0, G X, ...

..., Gm with diameters less than | r\. We may assume that none of those sets is contained in the union of the others. Thus we can select a point

a* e Gt— U Щ for every i =

0

,

1

, ..., m . ]фг

Let us assign to every point щ a point bi €g~1(ai) and let us set F i = X —Gj for i =

0

,

1

, . .., m .

Consider the function

/ : X ^ Q given by the formula

f(x) = Я

0

) - & 0

+ Д

1

(ж)-&1+ . . . + Ато(а?)-&то for every point x e X , where

U x ) = _____________ i t

l

EA_____________ _ Q (# ? -^o) + {? (#, F x) + . . . +

q

(x , Fm)

One sees easily that / is a map (that it is continuous) and if x.eGit then f(x) belongs to the convex hull of the set consisting of all points biv such that Giv n Gt Ф 0 . Since ô(Gi) < |-?y, we infer that

6

((J Gi ) < rj

v

and consequently the diameter of the set is less than s < 3 a. It follows by (

2

.

2

) that Hi cz V. Thus the formula

f = rf defines a map / : X -> Y such that

fgiPi) = / K ) = rf(ai) = r{bi) = bt.

If ye g- ^Gi ) , then

Q( f g( y) , y)

=

e ( r f g ( y ) , y ) < Q(rfg(y),fg(y)) + e ( f g { y ) , bi ) +Q{ bi , y) . But (2.3) implies that

q

(rfg(y), fg(y)) < a. Moreover, g(y)eGi, hence fg(y) belongs to the set 1Ц. Since and 0(Н{) <

e,

we infer that

Q ( f g ( y ) , k ) < e. Finally

q

(bi, y) < £, because g{bi) = a^Gi and g(y)eGi, hence both points Ъг and у belong to g~1(Gi), which implies that

$ ( bi, У) < e.

(4)

Thus we have shown that

Q { f g ( y ) , y ) < 3e < 3a for every point ye Y . It follows by (2.2) that all points of the form

t ' f g ( y ) + ( l — t)-y . with

0

< i <

1

, belong to V. Consequently, setting

<p(yf t) = r[ t - f g( y) + (l — t)-y] for (y, t)e l x <

0

,

1

>,

we get a homotopy <p: l x <

0

, l ) Г joining the identity map iY with the map fg. Hence f g - i Y .

The result, we have obtained, may be formulated as follows :

(2.6) I f X is a compactum and if a compact AN B-set Y is X-lïke, then there exist two maps f : X Y and g: Y -> X sucti that f g is homotopic to the identity map i Y .

It follows that the fundamental sequences / : X -> Y and g: Y -> X , generated (compare [2], p. 227) by the maps / and g respectively, satisfy the condition fg - iY. Hence Sh(X) > Sh( Y) and the proof of Theorem (

2

.

1

) is finished.

Let us observe that proposition (2.5) fails if we omit the hypothesis Y e Aï f B . In fact, one sees easily that if X denotes the interval

< 0 , 1>, then the closure Y of the diagram of the function y = sin — with x 0 < x < 1 is X-like. However, no map of the form fg, where X Y and g: Y X is homotopic to the identity map iY.

§ 3. Fundamental domination and components. Let □ {X) denote the set of all components of a space X . Let us prove the following

(3.1)

Th e o r e m.

I f X , Y are two compacta lying in the Hilbert cube Q and if f = { f k, X , Y} and g — {gk, Y , X ) are two fundamental sequences such that gf — ix , then there exist two functions

A : D ( X ) - > D ( Y ) and Л': □ ( Y) -> □ (X) satisfying the following conditions:

Л ' Л ( Х 0) = X 0 for every X 0e D(X).

2 I f X 0e □ (X), then f — {fk, X 0, Л ( Х 0)} and g = Л ( Х 0), X 0}

are fundamental sequences such that gf — iX().

3° I f Y 0, Y x, ...e □ (Y) andïïm Y n c Y 0, then limЛ'( Yn) c A ' ( Y 0).

71 — 0 0 7 1 = 0 0

P r o o f . Let a be a point of a component X

0

of X . Then lim g ( f k(a), Y) = 0,

k=oo

(5)

and we infer that there is an increasing sequence {Jcn} of indices such that the sequence f kl( a) , f k2{a),.>. converges to a point beY. Let У

0

denote the component of У containing b. Then for every neighborhood V0 of Y

0

there is a neighborhood F of У such that the component of V containing Y

0

lies in V0. Since / is a fundamental sequence, there exists a neighborhood U of X such that

f J U - f k+ilU in V for almost all Jc.

If U0 denotes the component of U containing -ЗГо, one infers easily that f J V 0 * f k+ll U0 in V0 for almost all Jc.

Thus

(3.2) { f k, X 0, У0} is a fundamental sequence.

It is clear that for every component X 0 of X there is only one com

1

ponent У

0

of У satisfying (3.2). Setting Л( Х0) = У0, we get a function A : □ (X) -> □ (У).

By an analogous argument one infers that there exists a function A' assigning to every component У

0

of У a component X'0 = Л ' ( У 0) of X such that

(3.3) {gk, Y 0,X'0} is a fundamental sequence.

Consider now the component X'0 = Л ' ( У 0) = A ' A ( X Q) of the eom- pactum X. Then {gk, У0, X'0] is a fundamental sequence and we infer that for every neighborhood U'0 of X'0 there is a neighborhood V0 of У

0

such that

gk( VQ) c U'0 for almost all Jc.

Moreover, (3.2) implies that f k( X 0) <= F

0

for almost all Jc. Hence (3.4) gkf k( X 0) c U'0 for almost ah Jc.

On the other hand, for every neighborhood U0 of X 0 there is a neigh­

borhood Z7 of X such that the component Û0 of Û containing X 0 lies in U0.

Since g f - ix and i{Xf) = X 0 a UQ, we infer that (3.5) gkf k( X 0) c TJ0 c UQ for almost all Jc.

It follows by (3.4) and (3.5) that every neighborhood JJ'0 of the compo­

nent X'0 intersects every neighborhood U0 of the component X 0. Hence X 0 = X'0 = A'(Y'0) = A ’ A ( X 0), that is condition 1° is satisfied.

Moreover, the relations У

0

= A ( X 0) and X 0 = A ' A ( X 0) imply that (3.2) and (3.3) may be rewritten in the following form:

/ = i f и, Л (X„)} and g = { g k, A ( Z„ ) , X , }

are fundamental sequences.

(6)

The homotopy gf - ix implies that there exists a neighborhood U' of X such that

(3.6) gJ J U ' - i/U' in Û for almost all 7c.

Let Û'0 denote the component of U' containing X 0. It follows by (3.6) that gkf k(X) c Û for almost all 7c and we infer by (3.5) that -

gjcfklÛ'o - i/Û'o in V0 for almost all 7c.

Thns we have shown that

gf — {gJk, X 0} cz iXo, i.e. condition

2

° is satisfied.

In order to prove 3°, let ns consider a sequence Y 0, Y t , ... of com­

ponents of the set Y with lim Yn cz Y 0. Let U be a neighborhood of the

71 = 0 0

component A ' ( Y 0). Then there exists an open neighborhood U0 of Л' ( Y 0) such that U0 a U and that

(3.7) X n U0 n ( X — U0) = 0 .

It follows that there is a neighborhood V0 of Y 0 such that gk( V0) a U

q

for almost all 7c.

Since lim Yn <=. Y 0, we infer that there exists an index n0 such that

71 = 0 0

Yn cz V0 for every n0.

It follows that for n > n0

gk{ Yn) cz U0 for almost all 7c.

Since every neighborhood of the set Л' ( Y 0) contains gk( Yn) for almost all 7c, we infer that A ' ( Y n) cz U0 cz U for every n > n0. Hence

l i m d ' ( Y J c A ' { Y 0),

n =oo

that is condition 3° is satisfied and the proof of Theorem (3.1) is finished.

(3.8)

Co r o l l a r y.

I f X, Y are two compacta and if Sh(X) = Sh(Y), then there exists a one-to-one correspondence A between the sets of com­

ponents □ (X) and D ( Y ) such that the corresponding components have the same shape and if X 0, X x, ... e □ {X), then lim J № c J

0

if and only if ÏÏm H (XJ с Л( Х0).

7 1 = 0 0

§ 4. Two quasi-homeomorphic compacta. Now let us construct two quasi-homeomorphic compacta for which we shall prove later (in § 5) that their shapes are different.

Let A a, where 0 < a < \, denote the circle given in the set E 3 r\Q by the equations:

(^x— è)2+ (^

2

— i

) 2

= b жз = ! + « ,

(7)

and let B a denote (for 0 <

a

< |) the simple closed curve given in Еъ n Q by the parametric equations:

®«,

3

(Z)), for 0 < Z < 2

tt

, where

r ajl(Z) == | + | c o s 2 Z + -^ c fC o s 2 Z -c o s Z , жа>2(/) = ~ + | s i n 2 ^ + — a -sin 2 Z -c o sZ , x a, 3(Z) = i + a + i a - s i n * .

It is easy to see that the map ha assigning to every point (cos/, sinZ, 0 , 0 , . . . ) (where 0 < Z < 2n) of the unit-circle S1 c B2 with center 0, the point xa(t) is a homeomorphism

ha : S1 Ba for 0 < a < \ and

h0: 8 1 -> M

0

maps onto the circle A 0 with the degree

2

.

Let us denote by p a the projection of Ba onto A a given by the formula PaK(Z)) = (| + |cos2Z, l + lsin2Z, |+a) for 0 < Z < 2

tc

.

Consider the sets:

OO O O

A = A 0 и IJ 4»-»> B = A o и U #>-».

>

1 = 2

n

= 2

It is clear, that Л and J5 are 1-dimensional compacta lying in the Hil­

bert cube Q and that A has as its components the sets A 0 and A 3~n for n = 2 , 3 , . . . Moreover, for every 0 < Z < 2n the coordinate xa>3(t) of the point xa{t)eBa satisfies the inequality

Ê +

a

Jsa

^ #a>3(Z) ^ g + a + ^ a- It follows that

I + £ 3 " » < * , - * , ( / ) and | + Л з - (»+

1

) >®з- (Я+

1

)>з(

0

, and since j^3-w > 1|3_(и+1), we infer that

B3- n n B3-(n+k) = 0 for n =

2

,3 , ... and h = 1 , 2 , . . .

It is also clear, that B3- n n A {) = 0 , and consequently M

0

and B3-n, n =

1

,

2

, . . . , are components of the compactum Б.

Consider now a sequence of disjoint 3-dimensional cubs Q

i

, Q 2,.*.

lying in the set I2

3

n Q - ( l u B) and converging to a point a belonging O O

to the set Q— (А и 5 ) - ( и Q-j. Let Yn denote a subset of Qn homeomor-

phic with B. Setting i=1

(8)

(4.1) X = А и [J Yn и (а),

71=1

О О

(4.2) Г = Б и и Г . и ( в ) ,

I 71=1

we get two compacta X and Y lying in Еъ n Q . Let us prove that they are quasi-homeomorphic.

It is clear, that for every positive e there exists an index m so great

m

that the set A z-n is a retract of the set A by a retraction rs satisfying 71 = 1

the condition

q(x,

re(x))

<

for every point x e A .

Ш

Moreover, there exists a homeomorphism hm of the set (J A z- n

m , n— 1

onto the set U B z- n . Setting <p — hmrE, we get a map of the set A onto

да n

= 1

the set (J B z-n satisfying the condition

71=1 m

< 5 1 < £ for every point ye (J B z~n.

71=1

Moreover, there exists a homeomorphism ip of the set X —A onto

m

the set Y — (J B 3-n. Setting 71=1

f e(x) = (p(x) for every point x e A , f e(x) = ip(x) for every point x e X —A ,

we get a map f £ of X onto Y satisfying the condition

ô [ f 7 1(y)] < « for every point y e Y .

In order to define a map gs of Y onto X satisfying the condition (4.3) <5[Sfe - 1( ^ ) ] < e for every point x e X ,

consider a natural m so great that for every n > m the projection p z~n

of the set Bz-n onto the set A z~n satisfies the condition

6[p~ln{X)] < £ for every point x e A %_n.

m

N ow let us consider a homeomorphism hm of the set U Bz-n onto

да и

= 1

the set U A z—n . I t suffices to set тг=1

ge{y) — Р

з

- n(y) for every point y e B 3- n and ''n = m + l , w + 2 , . . . ,

_ 771

9e(y) = K( y) for every point ye (J Б 3_ » , 7г=1 О О

for every point y e Y — {J B3~n,

g.(y) =

у

71=1

(9)

in order to obtain a map ge of Y onto X satisfying the condition (4.3).

Thus X and Y are quasi-homeomorphic.

§ 5. X and Y have different shapes. We shall show more, actually that the relation S h ( Y ) > S h ( X ) does not hold.

Otherwise, there exist two fundamental sequences f = { f k, X , Y } , g = { g k, Y , X }

such that g I - ix . It follows, by Theorem (3.1), that there exist two functions

Л: П С Х ) - * П ( Г ) and A' : □ ( Y) -> □ (X)

such that Л'A{ Xf ) = X 0 for every X 0e □ (X) and that the maps f k and gk constitute two fundamental sequences

{fkj -^o? Л ( X „)}, {gk, A { X f ) , X 0}, satisfying the relation

{ffkfk ? ? ^ o} — ix0 •

Let us denote by p the projection of Q onto E3 n Q given by the formula

P (^

1

, ^

2

? ^4 ? • • • ) =

(*®1

>

^ 2

? ^3 ? ^ J • • • ) *

It is easy to see that setting f k = pfk and gk = pgk for к = 1 , 2 , . . . , one gets two fundamental sequences {fk, X 0, Л ( Х 0)}, {gk, A ( X 0), X 0}

homotopic to { f k, X 0, A ( X 0)} and {gk, A ( X 0) , X Q} respectively, hence satisfying the relation {g'kf k, X 0, X 0} - iXo. It follows that we can assume that the maps f k: Q -+Q and gk: Q ->Q satisfy the condition

(6.1) f k(Q) cz E3 n Q , gk(Q) a E 3 ел Q for every fc = 1 , 2 , . . . Consider the component A 0 of the set X. Then A 0 = И т 1 3- Й. Since Л'(щ(У)) = D ( X ) , there exists, for every n =

1

,

2

, . . . , a component

Yn of Y such that A ' ( Yn) = A z-n. Since Y is compact, we infer that there exist a component Y

0

of Y and a sequence of indices n1< nz< ... such that

fhfiYn. cz Yo = A ( A 0).

j = oo

It is clear that all components Yn are different, and consequently the component Y

0

of Y is not isolated (that is every neighborhood of it contains points belonging to components of Y different from Y0). More­

over, Y

0

= A' (M0) implies that Y 0 > A 0 and consequently Y

0

Ф («)• Now

F

let us observe that for every two non-isolated components Y', Y " of Y, different from (a), there exists a homeomorphism mapping Y onto itself and mapping Y' onto Y " . It follows, that in the sequel we can restrict ourselves to the case when Y

0

= A 0, i.e. to the case when A 0 = A ’ { A0).

3 — P ra ce m a tem a ty c zn e X I V

(10)

Since ПшУи. c Ÿ 0 = А 0, we infer that for almost all к the set Y nk

7 = 0 0

belongs to the sequence B3- i , B3-

2

, ... Replacing the sequence {rij} by one suitably selected of its subsequences, we can assume that Yn. = B3-mj, where limwf — oo.

7

=

00

Consider now a toroidal neighborhood V of the circle A 0 in the set JEb n Q. Then B3- mj- <= V for almost all j, and we infer by (5.1) that for almost all к the map f kfA3-n. is homotopic in F to a map of the set A 3-nj yith values in the set B3- mj.

Now let us observe that the construction of the curve B3- n implies that every map / of a circle S into a subset of B3-n а V is homotopic in V with a map of S into A 0 of an even degree. In particular, f k/A3- nj for almost all к is homotopic in V with a map of A 3- nj into A 0 of an even degree. It follows that gkf kjA 3-n. can not be homotopic in V (for almost all k) to the identity map of the set A 3-nf , contrary to the relation

{ffkfk f -A3—nj ?

Thus the supposition that

8

h ( Y ) > S h ( X ) leads to a contradiction and our proof is finished.

References

[1] K . B o r s u k , Concerning the notion of the shape of compacta, Proceedings of the Symposium on Topology and its Applications, held in Herceg Novi in 1968.

[2] — Concerning homotopy properties of compacta, Fund. Math. 62 (1968), pp. 223*254.

[3] C. K u r a t o w s k i , and S. U la m , Sur un coefficient lié aux transformations con­

tinues d’ensembles, ibidem 20 (1933), pp. 244-253.

[4] S. M a r d e s ic and J. S e g a l, e-mappings onto polyheclra, Trans. Amer. Math.

Soc. 109 (1963), pp. 146-164.

Cytaty

Powiązane dokumenty

The theorem im- plies that if there exist counterexamples to the conjecture in C 2 then those of the lowest degree among them fail to satisfy our assumption on the set {f m = 0} (it

I would like to thank Professors Peter Pflug and W lodzimierz Zwonek for their valuable

4.5.. Denote this difference by R.. In a typical problem of combinatorial num- ber theory, the extremal sets are either very regular, or random sets. Our case is different. If A is

The new tool here is an improved version of a result about enumerating certain lattice points due to E.. A result about enumerating certain

Besides these the proof uses Borel–Carath´ eodory theorem and Hadamard’s three circles theorem (the application of these last two theorems is similar to that explained in [4], pp..

Although it can be deduced from the general statements on Hirzebruch surfaces that these scrolls are not isomorphic we give here a simple direct argument..

This generalisation of gauge theory uses the notion of coalgebra ψ-bundles, which was proposed by the author at his lecture delivered at the Banach Center Minisemester on Quantum

In fact, we know the correspondence, at least generically, in the case of second order linear ordinary differential equations of Fuchsian type (with a large parameter) and we