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Marco Manetti

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Topology

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UNITEXT - La Matematica per il 3+2

Volume 91

Editor-in-chief

A. Quarteroni

Series editors

L. Ambrosio
P. Biscari
C. Ciliberto
M. Ledoux
W.J. Runggaldier

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148 8 More Topics in General Topology �

Proof Define x ≤ y if x = λ(x, y), and let’s show that (X,≤) is well ordered and
λ = min. As λ(x, x) = λ(x) = x for every x ∈ X , ≤ is reflexive. If x ≤ y and
y ≤ x then x = λ(x, y) and y = λ(x, y) by definition, so x = y. If x ≤ y and y ≤ z
then

x = λ(x, y) = λ(λ(x),λ(y, z)) = λ(x, y, z) = λ(λ(x, y),λ(z)) = λ(x, z)

whence x ≤ z. At last, if A ⊂ X is a non-empty subset and a = λ(A) ∈ A, then
A = {x} ∪ A for every x ∈ A, and so

a = λ(A) = λ(A ∪ {x}) = λ(λ(A),λ(x)) = λ(a, x),

implying a ≤ x . �
Theorem 8.11 Any set X admits a function λ : P(X)′ → X such that:
1. λ(A) ∈ A for every A ∈ P(X)′;
2. λ(A ∪ B) = λ(λ(A),λ(B)) for every A, B ∈ P(X)′.
Proof If X = ∅ there’s nothing to prove. If X �= ∅ we consider the family A of
pairs (E,λE ) where E ⊂ X is non-empty and λE : P(E)′ → X satisfies properties
1 and 2. For x ∈ X we always have ({x}, {x} �→ x) ∈ A, showing that A is not
empty. Let’s introduce onA the order relation (E,λE ) ≤ (F,λF ) ⇐⇒ E ⊂ F and
λE (E ∩ A) = λF (A) for every A ⊂ F such that A ∩ E �= ∅.

With the aid of Zorn’s lemma we shall prove that A has maximal elements. Take
a chain C in A and define the pair (C,λC ) as follows:

C =

{E | (E,λE ) ∈ C}

λC (A) = λE (A ∩ E) for some (E,λE ) ∈ C such that A ∩ E �= ∅.

The reader can easily check that (C,λC ) ∈ A is an upper bound for C.
So take a maximal element (M,λM ) and suppose by contradiction that there is an

m ∈ X − M . Consider (N ,λN ), where N = M ∪ {m}, λN ({m}) = m and λN (A) =
λM (A ∩ M) for every A ⊂ N different from ∅ and {m}. Since (M,λM ) < (N ,λN ),
we’ve contradicted the maximality of (M,λM ). �

We point out that condition 1 in Theorem 8.11 is equivalent to the axiom of choice
(see Exercise 2.13).

Corollary 8.12 (Zermelo’s theorem)Every non-empty set X admits a well-ordering
≤ such that the cardinality of {y ∈ X | y < x}, for any x ∈ X, is strictly less than
|X |.
Proof ByTheorem8.11 andLemma8.10 X admits awell-ordering�.Define L(x) =
{z ∈ X | z ≺ x}, for x ∈ X . If the cardinality of L(x) is less than that of X , we can
take � as our ≤. Otherwise let a ∈ X be the minimum of the non-empty set

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8.3 Zermelo’s Theorem 149

{x ∈ X | |L(x)| = |X |}.

By construction L(a) has the cardinality of X , and � induces on L(a) a well-
ordering with the additional property that |L(x)| < |L(a)| for every x ∈ L(a). Now
it suffices to take an arbitrary bijection f : X → L(a) and set

x ≤ y ⇐⇒ f (x) � f (y).

Exercises

8.4 Let K ⊂ R2 be a connected set. Prove that if K has at least two points then it
has the cardinality of R.

8.5 Let A ⊂ R2 be path disconnected. Show that |R2 − A| = |R|.
8.6 (K) Let C be the family of closed sets in R2 with the cardinality of R.
1. If C, D ∈ C and C ∪ D = R2, prove that C ∩ D ∈ C. (Hint: if neither is contained

in the other, use Exercise 8.5.)
2. Prove that C has the cardinality of R. (Hint: R2 is second countable.)
3. Fix an order ≤ on C that satisfies the hypotheses of Zermelo’s theorem. One says

that U ⊂ C is a lower set if, for any given x ∈ U , y ≤ x =⇒ y ∈ U . Denote by A
the set of triples (U , f, g) where U is a lower set in C and f, g : U → R2 satisfy:
(a) f (U) ∩ g(U) = ∅;
(b) f (C), g(C) ∈ C for every closed set C ∈ U .
Show that A, ordered by extension, contains maximal elements. Use statement
2. to conclude that there exist disjoint subsets A, B ⊂ R2 such that C ∩ A �= ∅,
C ∩ B �= ∅ for every C ∈ C.

4. Using the previous points and Exercise 8.4 prove that there exists a dense con-
nected subset A ⊂ R2 such that any path α : [0, 1] → A is constant.

Remark 8.13 The reader familiar with measure theory may also prove that the sub-
sets A, B in Exercise 8.6 are not Lebesgue measurable.

8.7 (K, ♥) Let X be an infinite set. Prove that there is a family A of subsets of X
such that:

1. |A| = |X | for every A ∈ A;
2. |A ∩ B| < |X | for every A, B ∈ A, A �= B;
3. |X | < |A|.
(Hint: consider a maximal family among those satisfying conditions 1 and 2 and with
cardinality larger than or equal to |X |.)

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308 Index

free - of groups, 248
- in a category, 178
- of topological spaces, 58

Product of paths, 181
Projection

- formula, 22
Projective space

complex -, 96
real -, 95

Properly discontinuous, action -, 204
Property

finite-intersection -, 75
universal -

- - of identifications, 88
- - of quotient spaces, 90
- - of free groups, 244
– of free products, 248

Q
Quotient

- map, 30
- set, 30
- space, 90
of two sets, 23

R
Refinement, 134

- function, 134
star -, 138

Relation, 30
equivalence - , 30
order -, 33

Retraction, 173
deformation -, 173

S
Sard’s theorem, 192
Saturated,- subset, 22
Second countability axiom, 105
Section, 200
Semi-local simply connectedness, 233
Separ able, - topological space, 105
Separated subsets, 57
Separation axioms, 139
Sequence, 109

Cauchy -, 112
convergent -, 109
generalised -, 125
limit of a -, 109
limit point of a -, 109

Set, 21
convex -, 66
directed -, 125
ordered -, 33
- countable, 26
- countably infinite, 26
- finite, 21
- infinite, 21
star-shaped -, 68
totally ordered -, 33
well-ordered -, 147

Simple path, 161
Singleton, 21
Space of paths, 181
Sphere

combing a -, 272
unit -, 12

Standard simplex, 178
Stereographic projection, 13
Stone’s theorem, 136
Sub-basis

- of a topological space, 129
canonical - of a product, 131

Subcover, 71
Subsequence, 109
Subset

dense -, 44
closed -, 9, 40
locally closed -, 57
open -, 39

Subspace
relatively compact -, 116
topological -, 55

Supremum, 25

T
T0,T1,T2,T3,T4, 140
T1, topological space -, 45
Tietze

- extension theorem, 157
- separation axioms, 140

Topological
manifold, 136
monoid, 196

Topological space, 40
compactly generated -, 78
Baire -, 117
compact -, 72
connected -, 64
disconnected -, 64
fully normal -, 138
Hausdorff -, 140

Page 315

Index 309

irreducible -, 155
locally compact -, 98
metrisable -, 53
Noetherian -, 155
normal -, 138
paracompact -, 134
path connected -, 65
regular -, 140
second countable -, 105
separable -, 105
sequentially compact, 110

Topology, 39
cofinite -, 40
discrete -, 40
Euclidean -, 40
finite complement -, 40
indiscrete -, 40
metric -, 51
pointwise convergence -, 129
product -, 58
quotient -, 90
Standard -, 40
subspace -, 55
trivial -, 40
upper -, 40
Zariski -, 42

Total space of a covering space, 201
Tree, 161, 256
Tukey’s lemma, 35

Tychonov’s theorem, 131, 151

U
Ultrafilter, 150
Urysohn

- lemma, 157
- metrisability theorem, 141

V
Van Kampen’s theorem, 192, 193, 239, 251,

275

W
Wallace’s theorem, 76

Y
Yoneda’s lemma, 264

Z
Zariski topology, 42
Zermelo

- postulate, 32
- theorem, 148

Zorn’s lemma, 33, 146