Title: Theory / Unimportant / Math / Analysis / Real Pugh/{ / 2 - Topology
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Name: The \((\epsilon,\delta)\)-condition
Definition: For each \(\epsilon>0\) and each \(p\in M\) there exists \(\delta>0\) such that if \(x\in M\) and \(d_M(x,p)<\delta\) then \(d_N(fx,fp)<\epsilon\).
Page: 065
Chapter: 2

Name: A limit point of \(S\)
Definition: \(p\in M\) such that there exists a sequence \((p_i)_{i\in\mathbb N}\) wich converges to \(p\).
Page: 065
Chapter: 2
Section: 3
Variables: \(M\) a metric space
\(S\subseteq M\).

Name: An open subset of \(M\)
Definition: A subset \(S\subseteq M\) such that, for all \(p\in S\), there exists \(r>0\) such that for all \(q\in M\) with \(d(p,q)<r\), \(q\in S\).
Page: 066
Chapter: 2
Section: 3
Variables: \(M\) a metric space

Name: A closed subset of \(M\)
Definition: A subset of \(M\) which contains all of its limits.
Page: 066
Chapter: 2
Section: 3
Variables: \(M\) a metric space

Name: Clopen subset
Definition: Open and closed subset.
Page: 067
Chapter: 2
Section: 3
Variables: \(M\) a metric space

Name: A topology of \(M\)
Definition: contains \(\emptyset\)
Page: 067
Chapter: 2
Section: 3
Variables: \(M\) a set.
Definition2: contains \(M\)
Conjdef: subset of \(M\)
Definition3: closed under finite intersection
Definition4: closed under all unions

Name: Property of class of closed sets of \(M\)
Page: 068
Section: 2
Index: 7
Variables: \(M\) a metric space
Kind: Cor
Closure: contains
Under: \(\emptyset\) and \(M\)

Name: closed sets of \(M\)
Page: 068
Section: 2
Index: 7
Variables: \(M\) a metric space
Kind: Cor
Under: intersection
Condition: arbitrary
Under2: union
Condition2: finite

Name: A Cauchy sequence
Definition: \((p_k)_{k\in\mathbb N}\in M\) such that for all \(\epsilon>0\), there exists \(N\) such that \(k,n\ge N\) imply \(d(p_k,p_n)<\epsilon\).
Page: 077
Chapter: 2
Section: 3
Variables: \(M\) a metric space

Name: A complete space
Definition: A metric space where all Cauchy sequences converge.
Page: 078
Chapter: 2
Section: 3

Name: Compact set
Page: 80, 081
Section: 4
Chapter: 2
Index: 28, 32, 36
Kind: , theo
Under: subset
Under2: union
Condition2: finite
Under3: cartesian product
Under4: intersection
Under5: taking boundary
Under6: Function
Condition6: continuous

Name: The diameter of \(S\)
Definition: \(\max\left\{d(x,y):x,y\in S\right\}\).
Page: 082
Chapter: 2
Section: 4
Variables: \(M\) a metric space and \(S\subseteq M\).

Smaller0: \(M\) is compact
Greater0: \(f\) is bounded
Page: 084
Chapter: 2
Section: 4
Variables: \(M\subseteq\mathbb R\)
\(f:M\to\mathbb R\)
Index: 37
Kind: Cor
Smaller1: \(f\) continuous
Greater1: \(f\) assumes max and min values
Connect Greater0: and

Name: A function which embeds \(M\)
Definition: A function \(h:M\to N\) which is a homeomorphism from \(M\) to \(hM\).
Page: 085
Chapter: 2
Section: 4
Variables: \(M,N\) metric spaces.

Name: A proper subset of \(M\)
Definition: A subset of \(M\) which is distinct from \(\emptyset\) and \(M\).
Page: 086
Chapter: 2
Section: 5

Name: The disjoint union of \(A\) and \(B\)
Page: 086
Chapter: 2
Section: 5
Variables: \(A,B\) sets.
Context: Sets
Notation: \(A\sqcup B\)

Name: Connected space
Definition: Not disconnected space.
Page: 086
Chapter: 0
Section: 2

Name: Connected set
Page: 087
Section: 5
Chapter: 2
Index: 43, 49, 50
Kind: Theorem
Extra: , 89, 90
Under: function
Condition: continuous
Under2: Closure
Under3: union of sets
Condition3: arbitrary (assuming they share a common point)
Under4: product, for the product topology
Condition4: arbitrary
Under5: product, for the box topology
Condition5: finite
Under6: intersection
Not7: not

Name: A path-connected topological space
Definition: A topological space \(M\) such that, for each \(p,q\in M\), there is a path from \(p\) to \(q\) in \(M\).
Page: 090
Chapter: 2
Section: 5
Kind: Definition

Name: Topologist's sine curve
Definition: \(\{(x,\sin (1/x)): 0<x\le1/\pi\}\cup\{(0,y):-1\le y\le 1\}\).
Page: 091
Chapter: 2
Section: 5
Kind: Example

Name: The boundary of a set \(S\).
Definition: Difference between closure and interior of \(S\)
Page: 092
Chapter: 2
Section: 6
Variables: \(M\) a topological space and
\(S\subseteq M\).
Notation: \(\partial S\)

Definition: The set of cluster point of \(S\).
Page: 093
Chapter: 2
Section: 6
Variables: \(M\) a topological space
\(S\subseteq M\)
Notation: \(S'\)

Name: A condensation point of \(S\)
Definition: A point \(p\in M\) such that \(M_r p\) contains uncountably many points of \(S\) for all \(r>0\).
Page: 092
Chapter: 2
Section: 6
Variables: \(M\) a topological space,
\(S\subseteq M\)

Name: A perfect subset of \(M\)
Definition: A set \(S\subseteq M\) such that \(S'=S\).
Page: 094
Chapter: 2
Section: 6
Variables: \(M\) a topological space
French: Sous-ensemble parfait de \(M\)

Name: An everywhere uncountable subset of \(M\)
Definition: A set \(S\subseteq M\) such that each \(r\)-neighborhood is uncountable
Page: 095
Chapter: 2
Section: 6
Variables: \(M\) a metric space.
Index: 58
Kind: Cor

Name: Continuous functions
Page: 097
Section: 6
Chapter: 2
Index: 61
Kind: Cor
Under: maxima, minima
Under2: absolute values,
Under3: sums
Under4: differences
Under5: quotients
Under6: Polynomials

Name: Bounded set
Definition: A metric set \(S\) such that \(S\subseteq M_rp\) for some point \(p\) and \(r>0\).
Page: 097
Chapter: 2
Section: 6

Name: Bounded function
Definition: A function to a metric space whose range is bounded.
Page: 098
Chapter: 2
Section: 6

Name: Continuous bijective map from bounded to unbounded set
Chapter: 2
Section: 6
Example: Tangeant from \((-\pi/2,\pi/2)\) to \(\mathbb R\).

Name: A covering of \(S\)
Definition: A collection \(\mathcal U\) of subset of \(M\) such that \(S\subseteq\bigcup_{U\in\mathcal U}U\).
Page: 098
Chapter: 2
Section: 7
Variables: \(M\) be a metric space
\(S\subseteq M\)
Kind: Definition

Name: \(\mathcal U\) reduces to \(\mathcal V\)
Definition: \(\mathcal V\subseteq \mathcal U\).
Page: 098
Chapter: 2
Section: 6
Variables: \(M\) be a metric space and
\(S\subseteq M\).
\(\mathcal U\) and \(\mathcal V\) cover \(S\).
Name2: \(\mathcal V\) is a subcovering of \(S\)

Name: An open covering
Definition: A covering composed of open sets.
Page: 098
Chapter: 2
Section: 6

Name: A covering compact
Definition: A set whose every open covering reduces to a finite subcovering.
Page: 098
Chapter: 2
Section: 6
Variables: \(T\) a topological space
\(A\subseteq T\).

Name: Scrap
Definition: An element of a covering.
Page: 099
Chapter: 2
Section: 6
Context: Topology
Name2: Patch

Question: Properties of the Cantor space
Answer: uncountable
Page: 106, 107
Chapter: 2
Section: 8
Answer2: contains no interval
Index: 67, 68, 69
Kind: Theo, Cor, Theo
Answer3: nowhere dense
Answer4: outer mesure 0

Name: Fat Cantor set
Definition: The intersection of \(C^n\) where \(C^{n+1}\) is \(C^n\) without the \(\frac{1}{n!}\)-the portion of each segment.
Page: 108
Chapter: 2
Section: 8

Name: Peano Curve
Definition: A continuous path in the plane which is space-filling.
Page: 112
Chapter: 2
Section: 9
Index: 72
Kind: Theorem

Name: A space-filling set
Definition: A set with a non-empty interior.
Page: 112
Chapter: 2
Section: 9

Name: A Cantor Piece
Definition: A nonempty clopen subset of a Cantor space.
Page: 112
Chapter: 2
Section: 9
Variables: \(C\) a Cantor space.

Name: Ambiently homeomorphic subsets of \(\mathbb R^m\)
Definition: Two subsets \(A,B\subseteq \mathbb R^m\) such that there is an homeomorphism in \(\mathbb R^m\) sending \(A\) to \(B\).
Page: 115
Chapter: 2
Section: 9

Name: A tame Cantor space in \(\mathbb R^m\)
Definition: A Cantor subset of \(\mathbb R^m\) such that there is an ambient homeomorphism \(h:\mathbb R^m\to\mathbb R^m\) that carries the standard Cantor set \(C\) onto \(M\).
Page: 116
Chapter: 2
Section: 9

Name: A wild cantor space
Definition: A Cantor set which is not tame.
Page: 117
Chapter: 2
Section: 9
Example: Antoine's Necklace

Name: Antoine's Necklace
Definition: The intersection of the \(A^i\), where \(A^0\) is a solid torus and \(A^{n+1}\) is \(A^n\) where each tori are replaced by a necklace of 20 tori.
Page: 117
Chapter: 2
Section: 9

Name: Completion Theorem
Definition: Every metric space can be completed
Page: 119
Chapter: 2
Section: 10
Index: 80
Kind: Theorem

Question: The least upper bound of \(S\) and its greatest lower bound have the property
Answer: to belong to the closure of \(S\).
Page: 093
Chapter: 2
Section: 6
Variables: \(S\subseteq \mathbb R\) non-empty bounded.
Index: 55
Kind: Cor

Name: Compact sets
Definition: A set \(A\subseteq M\) such that every sequence in \(A\) has a subsequence that converges to a limit in \(A\).
Page: 081
Chapter: 2
Section: 4
Context: Metric topology
Kind: Example, Definition
Example: Finite set
Example2: closed unit ball in \(\mathbb R^n\)
Example3: \(\{\frac 1n:n\in \mathbb N\}\cup\{0\}\)
Example4: the Hawaiian earring

Name: A cluster point of \(S\)
Definition: There is a sequence of distinct points in \(S\) that converges to \(p\),
Page: 093
Chapter: 2
Section: 6
Variables: \(M\) a topological space,
\(S\subseteq M\).
Index: 52
Kind: Theo
Definition2: Each neighborhood of \(p\) contains infinitely many points of \(S\)
Conjdef: A point \(p\in M\) such that, equivalently
Definition3: Each neighborhood of \(p\) contains at least 2 points of \(S\)
Extra: Condensation is given in the book, but false according to WP and Munkres's Topolgy
Definition4: Each neighborhood of \(p\) contains at least one points of \(S\) other than \(p\).

Name: Continuity conditions
Definition: \((\epsilon,\delta)\)-condition
Page: 072
Chapter: 2
Section: 3
Variables: \((X,d)\) a metric space
Index: 11
Kind: Theorem
Definition2: conservation of convergence of sequence
Definition3: preimage of closed(open) set is closed(open).
Extra: for all \(x\in X\), \(U\) a neighborhood of \(f(x)\), \(f^{-1}(U)\) contains an open neighborhood of \(x\) ?

Name: A Cantor-space
Definition: compact
Page: 112
Chapter: 2
Section: 9
Definition2: nonempty
Conjdef: A set which is
Definition3: perfect
Definition4: totally disconnected

Smaller0: \((a_i)\) is bounded
Greater0: \((a_i)\) has a convergent subsequence in \(S\)
Page: 080, 6
Chapter: 2, 0 Stillwell
Section: 4, 1
Variables: \(S\subseteq\mathbb R^m\)
\((a_i)\in S\).
Index: 31
Kind: Theo
Assuming: \(S\) is closed
Name: Bolzano-Weierstrass theorem
Counterexample: \(a_i=i/2\) if \(i\) even, \(0\) otherwise

Name: The limit set of \(S\)
Definition: the set of limits of \(S\).
Page: 068, 70
Chapter: 2
Section: 3
Variables: \(M\) a metric space,
\(S\subseteq M\).
Notation: \(\overline S\)
Index: 10
Kind: Theo
Definition2: the smallest closed set containing \(S\).
Abbreviation: \(\text{lim }S\)
Extra: Closure?

Page: 058
Section: 1
Variables: \(E\) a set
Chapter: 2
Name: The discrete metric on \(E\)
Definition: \(\left\{\begin{array}{l}1 \text{ if } x\ne y\\ 0\text{ if }x=y\end{array}\right.\)

Smaller0: \((u_i)\) converges to \(\lambda\)
Greater0: \((v_i)\) converges to \(\lambda\)
Page: 060
Chapter: 2
Section: 1
Variables: \(M\) a metric space.
\((u_i)\) be a convergent sequence and
\((v_i)\) a subsequence of \((u_i)\).
\(\lambda\in M\).
Kind: Theorem

Smaller0: \(S\) is open
Greater0: the complement of \(S\) is closed
Page: 066
Chapter: 2
Section: 3
Variables: \(M\) a metric space
\(S\subseteq M\).
Index: 5
Kind: Theorem
Smaller to greater: iff
Hide middle: yes

Definition: \(\{q\in M: d(p,q)<r\}\).
Page: 068
Chapter: 2
Section: 3
Variables: \(M\) be a metric space
\(p\in M\)
\(r>0\).
Notation: \(M_rp\)
Index: 8
Kind: Theo
Typ: open set

Definition: \(\{p\in M:f(p)\in V\}\).
Page: 071
Chapter: 2
Section: 3
Variables: \(M,N\) be metric space
\(f:M\to N\)
\(V\subseteq N\)
Notation: \(f^{\text{pre}}(V)\)

Page: 075
Section: 3
Notation: \(d_E((x,y),(x',y'))\)
Variables: \((X,d),(X',d')\) metric space.
Chapter: 2
Name: The Euclidean metric on the product of those spaces
Definition: \(\sqrt{d_X(x,x')^2+d_Y(y,y')^2}\)
Metric: yes

Page: 075
Section: 3
Notation: \(d_{\text{max}}((x,y),(x',y'))\)
Variables: \((X,d),(X',d')\) metric space.
Chapter: 2
Definition: \(\max\{d_X(x,x'),d_Y(y,y')\}\)
Metric: yes

Page: 075
Section: 3
Notation: \(d_{\text{sum} }((x,y),(x',y'))\)
Variables: \((X,d),(X',d')\) metric space.
Chapter: 2
Name: The sum metric on the product of those space
Definition: \(d_X(x,x')+d_Y(y,y') \)
Metric: yes

Smaller0: \((u_i)\) and \((u_i')\) converge
Greater0: \((u_i,u'_i)\) converges for \(d_E\)
Page: 075
Chapter: 2
Section: 3
Variables: \((X,d), (X',d')\) two metric spaces
\((u_i)\in X\), \((u'_i)\in X'\) two sequences
Index: 17
Kind: Prop
Smaller to greater: iff
Smallest: \((u_i,u'_i)\) converges for \(d_{\text{sum}}\)
Greatest: \((u_i,u'_i)\) converges for \(d_{max}\)

Smaller0: \(f,g\) are continuous
Greater0: \(f\times g\) is continuous for \(d_E\)
Page: 075
Chapter: 2
Section: 3
Variables: \(M,N,X,Y\) metric spaces,
\(f:M\to N\).
\(g:X \to Y\)
Index: 18
Kind: Cor
Smaller to greater: iff
Smallest: \(f\times g\) is continuous for \(d_{\text{sum}}\)
Intermediate: \(f\times g\) is continuous for \(d_{\text{max}}\)

Question: The metrics \(d_{\text{max} }\), \(d_E\) and \(d_{\text{sum} }\) have the property to be
Answer: continuous.
Page: 075
Chapter: 2
Section: 3
Index: 21
Kind: Cor

Question: \(n\) is
Answer: continuous.
Page: 076
Chapter: 2
Section: 3
Variables: \(n\) a norm.
Index: 21
Kind: Cor

Name: A disc in the max metric
Definition: a square.
Page: 077
Chapter: 2
Section: 3

Name: A disc in the sum metrics
Definition: a rhombus.
Page: 077
Chapter: 2
Section: 3

Smaller0: \((p_k)\) converges
Greater0: \((p_k)\) is a Cauchy sequence
Page: 077
Chapter: 2
Section: 3
Variables: \(M\) a metric space
\((p_k)_{k\in\mathbb N}\in M\)
Smaller to greater: iff

Smaller0: \(S\) is closed
Greater0: \(S\) is complete
Page: 078
Chapter: 2
Section: 3
Variables: \(M\) a metric space
\(S\subseteq M\)
Index: 24
Kind: Theo
Smaller1: \(M\) complete
Context: Subset of metric space
Construction: Let \((s_i)_{i\in\mathbb N}\in S\) Cauchy.
\(M\) is complete, so there is a limit \(l\) to \((s_i)\) in \(M\)
\(S\) is closed, so \(l\in S\). I.e. \((s_i)\) has a limit in \(S\)

Smaller0: \(T\) is compact
Greater0: \(T\) is closed
Page: 079
Chapter: 2
Section: 4
Variables: \(T\) a metric space
Index: 26
Kind: Theo
Greater1: \(T\) is bounded

Question: Topologically set \([a,b]\) is
Answer: compact
Page: 079
Chapter: 2
Section: 4
Variables: \(a,b\in\mathbb R\).
Index: 27
Kind: Theo

Smaller0: the \(S_i\)'s are compact
Greater0: \(\bigcap S_i\) is compact
Page: 082
Chapter: 2
Section: 4
Variables: \((S_i)\) sets.
Index: 34
Kind: Theorem
Hide middle: yes
Smaller1: the \(S_i\)'s are decreasing
Greater1: \(\bigcap S_i\) is nonempty

Smaller0: the \(S_i\)'s are compact
Greater0: \(\bigcap S_i\) is a singleton
Page: 083
Chapter: 2
Section: 4
Variables: \((S_i)\) sets.
Index: 35
Kind: Cor
Hide middle: yes
Smaller1: the \(S_i\) are decreasing and the diameter tends to 0

Smaller0: \(M\) is compact
Greater0: \(f\) is uniformily continuous
Page: 085
Chapter: 2
Section: 4
Variables: \(M,N\) metric spaces
\(f:M\to N\).
Index: 42
Kind: Theo
Smaller1: \(f\) is continous

Name: Disconnected topological space
Definition: A topological space which is the disjoint union of two clopen sets.
Page: 086
Chapter: 2
Section: 5
Variables: \(M\) a topological space.

Smaller0: \(f\) is continuous
Greater0: \(f\) has the intermediate value property
Page: 087
Chapter: 2
Section: 5
Variables: \(M\) a topological space,
\(f:M\to\mathbb R\).
Index: 45
Kind: Cor
Smaller1: \(M\) connected

Name: A path from \(p\) to \(q\) in \(M\)
Definition: A continuous function \(f:[a,b]\to M\) such that \(f(a)=p\) and \(f(b)=q\).
Page: 090
Chapter: 2
Section: 5
Variables: \(M\) a topological space and
\(p,q\in M\).
Kind: Definition

Smaller0: \(S'\subseteq S\)
Greater0: \(S\) is closed
Page: 093
Chapter: 2
Section: 6
Variables: \(M\) a topological space
\(S\subseteq M\)
Index: 54
Kind: Cor
Smaller to greater: iff
Hide middle: yes

Smaller0: \(S\cup S'\)
Greater0: \(\overline S\)
Page: 093
Chapter: 2
Section: 6
Variables: \(M\) a topological space and \(S\subseteq M\).
Index: 53
Kind: Prop
Smaller to greater: =

Smaller0: \(S\) is closed
Greater0: \(S\) is totaly bounded
Page: 081, 100
Chapter: 2
Section: 4
Variables: \(S\subseteq\mathbb R^m\)
Index: 33, 69
Kind: Theo, Theo
Smaller to greater: iff
Smaller1: \(S\) is bounded
Name: Heine-Borel
Greater1: \(S\) is complete
Intermediate: \(S\) is compact
French: Théorème de Borel Lebesgue

Name: Compact sets
Definition: A set \(A\subseteq M\) such that every sequence in \(A\) has a subsequence that converges to a limit in \(A\).
Page: 081
Chapter: 2
Section: 4
Kind: Example, Definition
Example: Finite set
Example2: closed unit ball in \(\mathbb R^n\)
Example3: \(\{\frac 1n:n\in \mathbb N\}\cup\{0\}\)
Example4: the Hawaiian earring

Smaller0: nonempty
Greater0: \(S\) is uncountable
Page: 094
Chapter: 2
Section: 6
Variables: \(M\) be a metric space and
\(S\subseteq M\).
Index: 56
Kind: Theo
Smaller1: perfect
Extra: hypothèse sur M?
Prefix Smaller: \(S\) is
Smaller2: complete

Question: Cardinality of \(\mathbb R\), of \([a,b]\) is
Answer: Uncountable.
Page: 094
Chapter: 2
Section: 6
Index: 57
Kind: Cor

Question: The arithmetic operations of \(\mathbb R\) have the property to be
Answer: continuous.
Page: 095
Chapter: 2
Section: 6
Index: 59
Kind: theo

Question: Topological property of polynomials:
Answer: continuous
Page: 097
Chapter: 2
Section: 6
Index: 62
Kind: Cor

Smaller0: \(A\) is contained in a metric space
Greater0: covering compact
Page: 100
Chapter: 2
Section: 6
Index: 69
Kind: Theo
Hide middle: yes
Prefix Greater: \(A\) is
Greater1: sequentially compact
Connect Greater0: iff \(A\) is

Name: A total bounded set
Definition: A set \(A\) such that, for each \(\epsilon>0\), \(A\) is covered by a finite number of \(\epsilon\)-neighboorhood.
Page: 103
Chapter: 2
Section: 7
Variables: \(A\) a set

Name: The Cantor set
Definition: The intersection of the \(C^n\)'s with \(C^n=\bigcup_{i=0}^{3^{n-1} }\left[\frac{3i}{3^n},\frac{3i+1}{3^n}\right]\).
Page: 105
Chapter: 2
Section: 8
Definition2: The set of numbers of the form \((0,2)^\omega\) in base 3
Conjdef: , formally, equivalently
Name2: The middle-third Cantor set

Smaller0: the domain is \(\mathbb R\)
Greater0: \(C\) is ambiantly homeomorphic
Page: 115
Chapter: 2
Section: 9
Variables: \(C\) a Cantor space
Index: 78
Kind: Theo
Hide middle: yes
Smaller1: the domain is \(\mathbb R^2\)
Connect Smaller0: or

Question: \(C\) and \(C'\) are
Answer: homeomorphic.
Page: 114
Chapter: 2
Section: 9
Variables: \(C,C'\) Cantor spaces.
Index: 75
Kind: Cor

Question: The Convex Hull of \(S\) is
Answer: a segment.
Page: 115
Chapter: 2
Section: 9
Variables: \(S\subseteq\mathbb R\)

Name: Totally disconnected space
Definition: Each \(p\in M\) has arbitrarily small clopen neighborhoods.
Page: 105, 106
Chapter: 2
Section: 8
Variables: \(M\) a metric space
Example: \(\mathbb Q\)
Example2: discrete spaces.

Question: Any Cantor space is homeomorphic to
Answer: the standard middle-thirds Cantor set \(C\).
Page: 112
Chapter: 2
Section: 9
Index: 73
Kind: Theo
Name: Moore-Kline Theorem

Name: A distance on \(S\)
Definition: Positive definite
Page: 058
Chapter: 2
Section: 1
Variables: \(S\) a set
Definition2: symmetric
Conjdef: A function \(S^2\to S\)
Definition3: satisfies the triangle inequality
Denoted by: \(d\)

Smaller0: \(V\) is open(closed) in \(N\)
Greater0: \(V=N\cap U\) for some \(U\) open(closed) in \(M\)
Page: 074
Chapter: 2
Section: 3
Variables: \(M\) a metric space
\(N\subseteq M\) open(closed)
\(V\subseteq N\).
Index: 13, Cor 14
Smaller to greater: iff
Name: Inheritance principle

Smaller0: non empty
Greater0: \(M\) is everywhere uncountable
Page: 095
Chapter: 2
Section: 6
Variables: \(M\) a set
Index: 58
Kind: Cor
Smaller1: perfect
Prefix Smaller: \(M\) is
Smaller2: complete
Smaller3: metric space

Question: Example of nonempty clopen set \(\mathbb Q\) different from \(\mathbb Q\)
Answer: \((-\pi,\pi)\)
Page: 067
Chapter: 2
Section: 3

Name: Sequences without accumulation point in a closed bounded complete metric space. It shows that they are not compact
Page: 081
Chapter: 2
Section: 4
Example: Any infinite discrete metric \(M\) space with \((x_i)\in M\) a sequence of distinct element
Example2: \(C([0,1],\mathbb R)\) with distance \(d(f,g)=\max\{|(f(x)-g(x))|\}\) and the sequence \(x^n\).

Smaller0: \(S\) is closed
Greater0: to belong to \(S\)
Page: 093
Chapter: 2
Section: 6
Variables: \(S\subseteq \mathbb R\) non-empty bounded.
Index: 54
Kind: Cor
Hide middle: yes
Prefix Greater: the least upper bound and the greatest lower bound of \(S\) have the property

Smaller0: \(d_E(x,y)\)
Greater0: \(d_{\text{sum} }(x,y)\)
Page: 41
Chapter: 1
Section: 6
Variables: \(X,Y\) metric spaces
\(x\in X, y\in Y\)
Index: 16
Kind: Prop
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)
Smallest: \(d_{\text{max} }(x,y)\)
Greatest: \(2d_{\text{max} }(x,y)\)

Name: The set of complete set
Page: 78
Section: 2
Chapter: 3
Index: theo
Kind: 24
Extra: Let \((s_i)_{i\in\mathbb N}\in S\) Cauchy.
\(M\) is complete, so there is a limit \(l\) to \((s_i)\) in \(M\)
\(S\) is closed, so \(l\in S\). I.e. \((s_i)\) has a limit in \(S\)
Under: subset
Condition: closed