L . Lauwers Mathematical Social Sciences 40 2000 1 –39
7
satisfy all but one of Chichilnisky’s axioms. Of course, as the circle is non-contractible these maps cannot satisfy all of the axioms.
1 1
1 1
• Let q [ S . The constant map S 3 ? ? ? 3 S
→
S : p∞q is continuous and
anonymous.
1
• Represent the circle S
by the interval [0, 2 p[, i.e. a point q on the circle is
1
represented by the positive angle a measured from the point 1, 0 [ S . Then the
following maps 1
] a , . . . , a ∞ a 1 ? ? ? 1 a ,
1 n
1 n
n
5
a , . . . , a ∞median ha , . . . , a j
1 n
1 n
are anonymous and unanimous. Continuity is violated: points close to but at different sides of 1, 0 generate divergent outcomes. Similarly, the Pareto condition is
violated. In case the circle is restricted to its positive part i.e. a [ [0, p 2], the
second map satisfies anonymity, the Pareto condition, continuity, and Arrow’s independence axiom Nitzan, 1976.
• The map
1 1
1
S 3 S →
S : p , p ∞ n
i p 2 p i p 1 2 2 i p 2 p i p ,
s d
1 2
1 2
1 1
2 2
where n normalizes a non-zero vector, i.e. nq 5 q
iqi, satisfies continuity, weak Pareto hence unanimity, and non-dictatorship, and violates anonymity.
3. The resolution theorem
3.1. Statement and definitions We repeat the resolution theorem and explain the topological concepts that appear in
its statement.
Theorem 3.1.1 Chichilnisky and Heal, 1983. Let the preference space 3 be a path-connected parafinite CW-complex. Then a necessary and sufficient condition for the
existence of a Chichilnisky rule for each number of individuals is that 3 is contractible.
8 L
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8
First, we explain the notion of CW-complex. Such a complex refers to a topological space that is built in stages, each stage being obtained from the preceding by adjoining
cells of a given dimension. Hereby, a topological space Y is said to be obtained from X
,
by adjoining , -cells with , [ N if Y 2 X is the disjoint union of open subsets e
l
with l running over an index set L each of which is homeomorphic to the
, ,
,
~ , -dimensional open ball B , R , i.e. there exists a continuous one-to-one corre-
spondence
, ,
, 2
2 2
,
~ w : B 5
h x [ R u ixi 5 x 1 ? ? ? 1 x , 1j →
e
l 1
, l
, 21
, ,
~ for which the inverse map
w : e →
B is also continuous. There are no restrictions
l l
on the cardinalities of the index sets L , L , . . . ; they might be empty. It is also
1 2
,
assumed that for each , and each l [ L the map w
extends to a continuous map
, l
, ,
, 2
2 2
,
] f : B 5
hx [ R u ixi 5 x 1 ? ? ? 1 x 1j →
e
l 1
, l
, ,
, 21 ,
,
] ~
with e the closure of e , that maps S 5 B 2 B
into X.
l l
Definition 3.1.2. A Hausdorff space X is said to be a CW-complex if it can be expressed
`
X 5 X satisfying
k 50 k
• X , X , ? ? ? , X , ? ? ? ,
1 k
• X is a discrete topological space,
• X
with k [ N is obtained from X by attaching a collection of k 1 1-cells,
k 11 k
• X is supposed to have the weak topology: a subset A of X is closed if and only if
, ,
] A e is closed and compact for each cell e .
l l
A complex X is said to be finite if only a finite number of cells are involved. It is said to be of dimension k if in the above construction X 5 X . A finite CW-complex of
k 2k 11
dimension k can be embedded in R
. Finally, X is said to be parafinite if, for every k, there are only a finite number of k-cells. A parafinite complex can be embedded in the
`
countably infinite dimensional Euclidean space R .
Properties 3.1.3. Maunder 1996, Ch 7 observes that
,
• a CW-complex is closure finite: for each cell e the intersection of all subcomplexes
l 9
containing this cell is a finite subcomplex,
, m
, ,
m m
~ ~
• for two cells e and e
we have w B w B is empty unless , 5 m and l 5 m,
l m
l m
• in general the product of two CW-complexes is not a CW-complex.
,
Examples 3.1.4. i Let us verify that the , -sphere X 5 S is a CW-complex. Let
X 5 hs 5 21, 0, . . . , 0j be a singleton and add one ,-cell. It is sufficient to consider
8
With respect to this limitation, Heal 1983 and Baryshnikov 1997 argue that the class of CW-complexes is large enough to cover most if not all spaces of preferences that arise naturally in economics. On the other
hand, Horvath 1995 points at some shortcomings of this class and lists sufficient conditions for a larger class of topological spaces in order to admit Chichilnisky rules.
9
The initials CW stand for Closure finite and Weak topology e.g. Gray, 1975.
L . Lauwers Mathematical Social Sciences 40 2000 1 –39
9
, ,
, 21
a continuous map f : B →
X that maps the boundary S into the point s .
ii All smooth , -manifolds, all polyhedra, and spaces homeomorphic to a polyhedron are CW-complexes. iii The comb space CS, defined by
2
CS 5 x, y [ R u0 x, y 1 and
y ± 0 implies x [ h0, 1, 12, 13, . . . j ,
h j
and embedded in the Euclidean plane is not a CW-complex. Indeed, the set X of
zero-cells contains the collection h0, 1, 1, 1, 12, 1, . . . , 1n, 1, . . . j which is
not discrete. Next, we clarify the contractibility condition.
Definition 3.1.5. A topological space X is said to be contractible if there exists a point x [ X and a continuous map
q : X 3 I →
X with I 5 [0, 1] such that ;x [ X:
qx, 0 5 x and
qx, 1 5 x . In words, contractibility means that the identity map can be continuously deformed to a
constant map. This concept of continuous deformation can be generalized to arbitrary maps.
Let X, Y be two topological spaces. Two continuous maps f, g: X
→ Y are said to be
homotopic f
. g if there exists a continuous map q : X 3 I
→ Y
such that q., 0 5 f. and q., 1 5 g.. Hence, X is contractible if there exists a point
x [ X such that the maps Id: X
→ X: x ∞xand
C : X →
X: x∞x
x
are homotopic.
Two spaces X, Y are said to be homotopic if there exist continuous maps f : X
→ Y
and g: Y →
X such that g + f . 1 and f + g . 1 . In this case the map f and g is said to
X Y
be a homotopy equivalence. The relation ‘is homotopic to’ induces an equivalence relation on the class of all topological spaces, the equivalence classes are called
homotopy types. Observe that homeomorphic spaces belong to the same homotopy type,
10
the converse is not true. Also, a space is contractible if and only if it has the same
homotopy type as a one-point space e.g. Spanier, 1966.
,
Examples 3.1.6. i A convex set X , R is contractible: define qx, t 5 1 2 tx 1 tx
with x [ X arbitrary and t [ I.
10
The spaces and
are homotopic but not homeomorphic.
10 L
. Lauwers Mathematical Social Sciences 40 2000 1 –39
1 , 21
ii The circle S is not contractible. More general, the sphere S is not contractible
cf. Section 3.2.
1
iii When one single point p is dropped, the remaining space S 2 p is contractible. The second space is homeomorphic to the first one and is therefore contractible.
iv The comb space is not a CW-complex but is contractible. The map q : CS 3
I →
CS: x, y, t∞x, 1 2 ty is a homotopy from the identity to the projection to the x-axis. This projection is homotopic to a constant map.
v The double comb space is not a CW-complex and is not contractible Maunder, 1996, Ex 7 5 5. To provide some intuition: the central point 0, 0 is the limit of the
sequences h0, 1kj and h0, 2 1kj and a continuous deformation of this space into
one single point pushes this central point to the left extreme 21, 0 as well as to the right extreme 1, 0.
L . Lauwers Mathematical Social Sciences 40 2000 1 –39
11
3.2. How to prove the resolution theorem? This subsection develops a strategy to prove the ‘only if part’ of Theorem 3.1.1: if 3
admits topological aggregators then 3 is contractible. The proof is not constructive in the sense that a contraction map
q is defined but follows an algebraic topological route. The intuition is as follows: contractibility of a space means that there are no holes in it,
an n-dimensional hole can be detected through the non-triviality of the nth homotopy group, hence in order to prove contractibility one has to show that all homotopy groups
are trivial.
We start with the definition of closed paths and of the first homotopy group. Let X be a topological space and let x [ X. A closed path at x in X is a continuous
map a: I
→ X for which
a0 5 a1 5 x . Two paths are considered equivalent if they are homotopic. The equivalence class containing the closed path
a is denoted by [a]. Next, we define an operator upon the set of paths: given a couple of paths, how to create
a new one? The construction goes as follows. Let a and b be two closed paths, the new
closed path a ? b travels with a doubled speed first along a, then along b:
a2t 0 t 1 2,
a ? b: I →
X: t∞
H
b2t21 1 2 t 1.
This product respects the above defined equivalence: when [ a] 5 [a9] and [b] 5 [b9]
then [ a ? b] 5 [a9 ? b9].
Definition 3.2.1. The first homotopy group or fundamental group p X, x , ? of the
1
space X is the set of homotopy classes of closed paths in X at x equipped with the operation ‘?’. The neutral element is the class e 5 [
g ] that contains the constant path g , i.e.
g t 5 x for all t [ I; and the inverse of a class [a] is the class of the closed path
21
a defined by
21
a : I
→ X: t∞
a1 2 t, i.e. travelling backwards along
a.
Properties 3.2.2.
• The group
p X, x depends upon the point x , but in case the space X is
1
path-connected the reference to the base point x becomes redundant and all the first homotopy groups are isomorphic. Indeed, a closed path
a [ p X, x corresponds to
1 21
b ? a ? b [
p X, y .
1
12 L
. Lauwers Mathematical Social Sciences 40 2000 1 –39
• Let X, Y be two topological spaces. Then, a continuous map f : X
→ Y induces a
11
homomorphism f :
p X, x →
p Y, fx : [a]∞[ f + a].
1 1
• Also, the homotopy group
p X 3 Y of the product is isomorphic to the product
1
p X 3 p Y of the homotopy groups.
1 1
• Finally, homotopic spaces have isomorphic homotopy groups.
Examples 3.2.3. i A contractible space is homotopic to a point and has a trivial
1
fundamental group. ii The fundamental group p S of a circle is isomorphic to Z.
1
Indeed, consider the closed path a: t∞cos 2pt, sin 2pt.
1 21
Any closed path in S is homotopic to a multiple of a or a
. The element [ a] is said to
1 1
12 2
3
be generator of p S and p S Z. For the higher dimensional spheres S , S , . . .
1 1
the fundamental groups are trivial. Intuitively: a space has a non-trivial fundamental group as soon as the space exhibits
one-dimensional holes, the number of which is reflected in the number of generators of the fundamental group.
In order to study higher dimensional holes, higher homotopy groups have been designed. Let , [
N , let X be a topological space, and let x [ X. Consider the set
,
p X, x of homotopy classes of continuous maps from I →
X that map the border
, ,
, ,
~ ≠I 5 I 2 I into x . Again, two classes [
a] and [b] in p X, x can be multiplied: the
,
class [ a] ? [b] contains the map
a2t , t , . . . , t 0 t 1 2,
1 2
, 1
,
a ? b: I →
X: t , t , . . . , t ∞
H
1 2
,
b2t 21, t , . . . , t 1 2 t 1.
1 2
, 1
This multiplication is well defined and turns p X, x , ? into a group: the , th homotopy
,
group. In addition to the properties in the previous list, the higher homotopy groups are abelian.
Examples 3.2.4. i All homotopy groups of a contractible space are trivial. ii For a
, ,
sphere, we have p S Z: the ‘one’ ,-dimensional hole in S
corresponds to a
, ,
non-trivial , th homotopy group with ‘one’ generator. Furthermore, p S 5
h0j for
k 13
k , ,. Observe that homotopic spaces generate isomorphic homotopy groups. The converse
statement, alas, is not true: all homotopy groups of the double comb space are trivial Maunder, 1996, Ex 7 5 5 and yet this space is not homotopic to a point. In case of
CW-complexes however, life becomes much easier:
11
Let G , ? and G , ? be two groups. A homomorphism is a map h: G →
G that satisfies ha ? b 5 ha ? hb
1 2
1 2
for all a, b [ G . If, in addition, h is a one-to-one correspondence, h is said to be an isomorphism.
1 12
1 2
1 2
From the fact that the fundamental groups of S and B are different, it follows that S and B are not homotopic. This implies the two-dimensional version of Brouwer’s fixed point theorem e.g. Massey, 1977.
13 ,
2
The statement ‘ p S 5
h0j for k . , ’ is false: e.g. p S Z Spanier, 1966.
k 5
2
L . Lauwers Mathematical Social Sciences 40 2000 1 –39
13
Theorem 3.2.5 Whitehead, 1949. Let f : X
→ Y be a continuous map between two
path-connected CW-complexes X and Y. If f induces isomorphisms f : p X
→ p Y for
, ,
all ,, then f is a homotopy equivalence.
Proof. See Maunder 1996, Thm 7 5 4.
As a matter of fact this theorem makes the family of CW-complexes the most tractable family for homotopy theory. In addition, for every path-connected topological space X
there exists a CW-complex K and a continuous map f : K →
X that induces isomorphisms at each homotopy level. The following lemma, a consequence of the previous theorem,
summarizes our strategy in proving the ‘only if’ part of the resolution theorem.
Lemma 3.2.6 Maunder, 1996, Corr 8 3 11. Let X be a path-connected CW-complex. Then X is contractible if and only if all homotopy groups
p X, p X, . . . , p X, . . .
1 2
k
are trivial.
Remark 3.2.7. In view of certain pathological spaces such as the double comb space, it is only in terms of CW-complexes that the statements ‘being contractible’ and ‘having
no holes’ are synonymous.
3.3. Proof of the resolution theorem The ‘only if part’.
2 3
n
Let 3 be a path-connected parafinite CW-complex, and let F , F , . . . , F , . . . be a sequence of Chichilnisky rules one for every n 2. Hence, for all k [
N there exists a sequence of homomorphisms
n n
F : p 3 5p 3 3 p 3 3 . . . 3 p 3
→ p 3 ,
k k
k k
k n times
with n 5 2, 3, . . . . Note that we used property 3.2.2: the homotopy group of a product is the product of the homotopy groups.
In view of Lemma 3.2.6, it is sufficient to show that all the homotopy groups are trivial, i.e.
p 3 5 h0j for all k [ N . This is proven by contradiction.
k
Suppose that p 3 5 ? ? ? 5 p
3 5 h0j and that p 3 is the lowest non-trivial
1 , 21
,
homotopy group. It will turn out that the group p 3 is abelian Step 1 and finitely
,
generated Step 2. Group theory provides us with a full classification of these objects, and the desired contradiction will be obtained Step 3.
Step 1. The homotopy groups of level two or higher are abelian by definition. That
also the first homotopy group p 3 is abelian follows from the existence of one single
1 n
Chichilnisky rule say, F . Indeed, let a and b be two classes in p 3 , e is the neutral
1 14
element. By unanimity and anonymity we have
k times 14
We write k 3 a for a ? a ? . . . ? a.
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n n
n n
a 5 F a, a, . . . , a 5 F a, e, . . . , e ? F e, a, e, . . . , e ? ? ? ? ? F e, e, . . . , e, a
n
5 n 3 F a, e, . . . , e
n
5 F n 3 a, e, . . . , e. Hence,
n n
a ? b 5 F n 3 a, e, . . . , e ? F n 3 b, e, . . . , e
n n
5 F n 3 a, e, . . . , e ? F e, n 3 b, e, . . . , e
n
5 F n 3 a, n 3 b, e, . . . , e
n
5 F n 3 b, n 3 a, e, . . . , e 5 b ? a.
As a consequence, we are allowed to write the homotopy groups as additive groups: p 3 , 1 with 0 the neutral element, and 2 a the inverse of a. Observe that the above
k
calculations can be done at any homotopy level: the identity
n times n
n n
g 5 n 3 F g, 0, . . . , 0 5F g, 0, . . . , 0 1 ? ? ? 1 F g, 0, . . . , 0 1
holds for all g [ p 3 and for all k [ N . The existence of an n-rule on 3 implies that
k
multiplication by n is a group isomorphism on all homotopy levels. Step
2. Since 3 is parafinite, Hurewicz theorem implies that p 3 is finitely
, 15
generated, i.e. there exists a finite subset
hg , g , . . . , g j , p 3 of generators such
1 2
m ,
that any g in p 3 can be expressed not necessary in a unique way in the form
,
g 5 z g 1 z g 1 ? ? ? 1 z g ,
1 1
2 2
m m
with z , z , . . . , z [ Z. The order of such a generator g is either infinite, i.e. for
1 2
m k
different integers z and z9 in Z the elements zg and z9g in p 3 are different; either
k k
,
finite, i.e. there exists an m [ N for which mg 5 0.
k
Step 3. Let the space 3 admit an n-rule for any natural number. Let g be a generator
of the lowest non-trivial homotopy group p 3 .
,
Suppose that g is of infinite order. The group G hgj generated by g is isomorphic to Z.
Eq. 1 states that g is divisible by n for all n 5 2, 3, . . . . But then g has to be 0 because 0 is the only element in
Z that can be divided by any integer. Hence, the group p 3 ,
,
being abelian, only contains elements of finite order and is therefore a finite group. Next, suppose that g ± 0 is of finite order and let m 2 be the smallest natural number such
that mg 5 0. Let a satisfy Eq. 1 with n 5 m, i.e. g 5 ma. In case the group G haj
generated by a and the group G hgj generated by g coincide, it follows that a is a
multiple of g. But then we have ma 5 0, this is in contradiction with g ± 0. In case the groups G
haj and Ghgj do not coincide, then Ghgj , Ghaj. Hence, in the set of generators the element g can be dropped and a might be added. Then, apply the previous
15
Hurewicz theorem states that in case all homotopy levels up to , 2 1 are trivial, then the , th homotopy group is isomorphic to the , th homology group Maunder, 1996, Th 8 3 7. Since the homology groups of a
path-connected parafinite CW-complex are finitely generated, the assertion follows.
L . Lauwers Mathematical Social Sciences 40 2000 1 –39
15
reasoning upon the generator a. As p 3 is a finite group, this process ends in the
,
desired contradiction. Conclude that p 3 5
h0j. Hence, all homotopy groups are trivial
,
and 3 is contractible. The ‘if part’ there exist Chichilnisky rules ‘if ’ the space of preferences is
contractible. Conversely, let 3 be a contractible parafinite CW-complex. If 3 is convex
`
in R , then the convex mean
1
c
] F : P , . . . , P ∞
P 1 ? ? ? 1 P
1 n
1 n
n
`
defines an n-rule for all n 2. Otherwise, let K 3 be the convex hull of 3 in R . Then
K 3 is constructed by adding a finite number of new cells of each dimension and is a CW-complex with 3 as a subcomplex. Since both spaces 3 and K 3 are contractible,
the inclusion map i: 3 →
K 3 is a homotopy equivalence use Theorem 3.2.5. This implies the existence of a continuous map r: K 3
→ 3 for which the restriction r
u is
3
equal to the identity map on 3 e.g. Lundell and Weingram, 1969, Thm IV 3 1; or
16
Rohlin and Fuchs, 1981, p. 119. Now, the following maps are well defined and satisfy
all the Chichilnisky axioms:
c
i F
r n
n c
3 →
K 3 →
K 3 →
3 : P∞r + F P,
for n 2. 3.4. Remarks
Remark 3.4.1. Some of the ideas in the above proof also appear in Eckman 1954 and
´ in Candeal and Indurain 1995. Eckman studied topological n-means and proved the
resolution theorem for polyhedra a special class of CW-complexes. In his article, however, there is no sign of a social choice interpretation. The proof in Candeal and
´ Indurain avoids Hurewicz theorem footnote 15, but needs a more general classifica-
tion result of abelian groups.
Remark 3.4.2. Baigent 1984, 1985 suggests a slightly different approach. He stresses
n
the anonymity condition and considers profiles in 3 that are identical up to a
permutation as equivalent. Let [P] denote the unique equivalence class that contains
]
the profile P and its permutations. Obviously, it is sufficient to define a rule F on the
]
n
quotient space 3 of these equivalence classes. In order to give content to the notion of continuity, this quotient space needs to be equipped with a topology. A natural
]
n
candidate is the quotient topology, i.e. the largest topology on 3 that makes the map
16
To provide some intuition: as 3 is contractible the convex hull is constructed from 3 through attaching cells ‘on the outside’ of 3. The map r has to push these new cells towards the border of 3. See remark 3.4.6 for
an illustration.
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]
n n
p: 3 →
3 : P∞[P] continuous. The whole set-up generates a commutative diagram:
F n
3 →
3 ↓
p ↓
Id
]
]
F n
3 →
3, ]
i.e. each continuous and anonymous map F defines a continuous map F and vice versa. As a consequence, the original Chichilnisky approach and this alternative approach
both produce the same results Le Breton and Uriarte, 1990b; Zhou, 1997.
Remark 3.4.3. The proof of the resolution theorem does not use any additional properties of the space 3 and its topology except that it is a path-connected parafinite
CW-complex Chichilnisky, 1996a. The model presented in Section 2.1 only serves as an example of how to construct such a space.
Remark 3.4.4. The contractibility condition in the resolution theorem provides a domain restriction for which the topological aggregation problem is solvable. It tests
whether there is a way of ‘deforming continuously’ the space of preferences into a single point or preference. Such a continuous deformation, however, might be rather
wild and interpreting such transformations might be difficult.
Heal 1983 interprets contractibility in terms of some sort of limited agreement. For example, in the contractible space obtained from the sphere after deleting one single
1
point cf. 3.1.6 iii: S 2 p there is a limited agreement: individuals agree on not moving in that particular direction. It seems hard to extend this interpretation to an arbitrary
contractible space cf. 3.1.6 iii: the spiral .
Remark 3.4.5. Observe that step 3 in the proof of the ‘only if part’ is valid as soon there exists a topological aggregation rule for any prime number.
Chichilnisky 1996b isolates the implications of the existence of one single n-rule with n a prime number: it follows that all the homotopy groups consist out of elements of
finite order relatively prime to n. See also Candeal et al. 1992 and Candeal and ´
Indurain 1995.
Remark 3.4.6. In order to illustrate the ‘only if part’ in the proof, we construct a topological aggregation rule on the space ML of monotone linear preferences. Linearity
implies constant gradient fields and monotonicity implies that the gradient vector is positive. Consequently, ML can be represented as the positive part of a sphere and is
contractible.
The following picture shows the situation in the two-dimensional two-person case. The convex hull KML is obtained from ML after attaching one 1-cell the line segment
] p , p [ and one 2-cell the interior shaded area. There are many maps from the
1 2
convex hull KML to ML that extend the identity map on ML. There are as many Chichilnisky rules on ML. The picture considers projections r: KML
→ ML with base
point R. The aggregated preferences of the profile p , p are drawn. Observe that the
1 2
2
second rule G is ‘biased towards’ the preference p .
1
L . Lauwers Mathematical Social Sciences 40 2000 1 –39
17
Remark 3.4.7. Some spaces 3 of preferences allow for an alternative construction of a topological aggregator. The idea is to make use of a continuous identification map
b: 3
→ U where U is a certain class of utility functions. In case the selection
b has a continuous projection map
p: U →
3 : u∞P defined by a, b [ P if and only if ua ub, the following diagram appears:
F n
3 →
3 ↓
b ↑
p
U
F n
U →
U,
n
where b is extended to 3 : bP , P , . . . , P 5 bP , bP , . . . , bP . Obviously,
1 2
n 1
2 n
U
an aggregation rule F defined on the space U induces an aggregation rule F on the
space 3. As the construction of F is based upon the selection map b only ordinal
information is used.
U
The existence of such a diagram and an aggregation rule F was proved for the
,
following spaces of preferences. The set A , R of alternatives is assumed to be convex
and compact. •
The space of continuous and strictly Schur-convex preferences, i.e. continuous
k
preferences P such that for all x [ R with ox fixed and for all bistochastic matrices
i
B we have: x, Bx [ P implies that B is a permutation matrix, equipped with a suitable topology Le Breton et al., 1985.
• The space 3
of strictly quasi-concave preferences a subspace of the space of all
sco
continuous preferences on A endowed with the topology closed convergence Le Breton and Uriarte, 1990a; Chichilnisky, 1991.
• The space of continuous monotone preferences Allen, 1996; see also Chichilnisky
and Heal, 1983, Ex 1; Chichilnisky, 1996a. Note that 3
allows for thick indifference curves and satiation points. Remark 4.2.3
sco
returns to this issue.
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Remark 3.4.8. Candeal et al. 1992, Efimov and Koshevoy 1994, and Horvath 1995 study Chichilnisky rules on other structures such as topological vector-spaces,
semi-lattices, metric spaces, . . . We list three simple examples of spaces admitting a Chichilnisky rule although they are not contractible or not a CW-complex:
• 3 5 the set of rational numbers and F 5 convex mean,
• 3 5 the set of irrationals and F 5 minimum,
• 3 5 the comb space CS and F x , y , . . . , x , y 5 min
hx j, min hy j.
s d
1 1
n n
k k
, ,
4. Linear preferences and spheres
