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 .
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.
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 >
0such 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(ж
) - & 0+ Д
1(ж)-&1+ . . . + Ато(а?)-&то for every point x e X , where
U x ) = _____________ i t
lEA_____________ _ 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.
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:
1° 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
0of X . Then lim g ( f k(a), Y) = 0,
k=oo
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
0denote the component of У containing b. Then for every neighborhood V0 of Y
0there is a neighborhood F of У such that the component of V containing Y
0lies 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
1ponent У
0of 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
0of 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
0such that
gk( VQ) c U'0 for almost all Jc.
Moreover, (3.2) implies that f k( X 0) <= F
0for 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.
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
qfor 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
0if 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 = ! + « ,
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
0maps 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 = 2n
= 2It 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
0and 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
3n 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
(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
= 1the 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
= 1the 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
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
0of 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
0of 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
0Ф («)• 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
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
=
00Consider 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
8h ( 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.