Title: Theory / Unimportant / Math / Topology / Munkres(+2 if from paper) / 1 Introduction
Tags: fromMath mathToDef mathToSee closed hasSeenConjDef examplesSwitchExample1 counterExample topology automaticImplies fromImplication fromRelation list mid1516974248815 toUnary récupéré EquivalentButNoIntermediate addIntermediateCard closedContinuous continuous equivalentHand metric toBinaryFunction typWTF

Definition: The minimal uncoutable well-ordered set.
Page: xii
Chapter: Intro
Notation: \(S_\Omega\)

Name: Axiom of choice
Definition: Given a collection \(\mathcal A\) of disjoint nonempty set, there exists a set \(C\) consisting of exactly one element from each element of \(\mathcal A\).
Page: 059
Chapter: 1
Section: 9

Name: The section of \(X\) by \(a\)
Definition: \(\{x\in X\mid x<a\}\).
Page: 066
Chapter: 1
Section: 10
Variables: \(X\) an ordered set
\(a\in X\).
Notation: \(S_a\)
Kind: Def

Name: Well-ordering theorem
Definition: If \(A\) is a set, there exists an order relation on \(A\) that is a well-ordering.
Page: 065
Chapter: 1
Section: 10

Name: Zorn's lemma
Definition: Every simply ordered subset of \(A\) has an upper bound in \(A\) implies \(A\) has a maximal element.
Page: 70
Chapter: 1
Section: 11
Variables: \(A\) be a set that is strictly partially ordered.

Name: A topology on \(X\)
Definition: \(\emptyset,X\in\mathcal T\)
Page: 78
Chapter: 2
Section: 12
Variables: \(X\) a set
Definition2: \(\mathcal T\) is closed under union
Conjdef: A collection \(\mathcal T\) of subsets of \(X\) having the following properties
Definition3: \(\mathcal T\) is closed under finite intersection.

Page: 77, 79
Name: The discrete topology
Set: \(X\)
Chapter: 2
Section: 12, 13
Variables: \(X\) a set
Kind: Ex
Base: singletons
Opens: all subset
Metric: \(d(x,x)=0\) and \(d(x,y)=1\) for \(x\ne y\)
Index: 2, 3
Extra: 18 p Ex 1, Ex 3

Page: 077
Name: The finite complement topology of \(X\)
Chapter: 2
Section: 13
Variables: \(X\) a set
Kind: Ex
Opens: The empty set and the set whose complement is finite.
Index: 3

Name: \(\mathcal T\) is coarser than \(\mathcal T'\)
Definition: \(\mathcal T\subseteq \mathcal T'\).
Page: 77, 81, 122
Chapter: 2, 2
Section: 13, 20
Variables: \(X\) a set
\(\mathcal T, \mathcal T'\) two topologies on \(X\).
Index: , 3, 2
Kind: Def, Lem, Lem
Definition2: For each \(x\in X\) and each basis element \(B\in\mathcal B\) containing \(x\), there is a basis element \(B'\in \mathcal B'\) such that \(x\in B'\subseteq B\)
Definition3: For each \(\epsilon>0\), for each \(x\in X\), there exists \(\delta>0\) such that \(B_{d'}(x,\delta)\subset B_d(x,\epsilon)\) (assuming metrical topologies)
Name2: \(\mathcal T'\) is finer than \(\mathcal T\)

Name: A basis over \(X\)
Definition: For each \(x\in X\), there is at least one element \(B\in\mathcal B\) such that \(x\in B\)
Page: 78
Chapter: 2
Section: 13
Variables: \(X\) a set
Definition2: If \(x\in B_1\cap B_2\) then there exists \(B_3\in\mathcal B\) such that \(x\in B_3\subseteq B_1\cap B_2\).
Conjdef: such that
DefType: A set \(\mathcal B\subseteq\mathcal P(X)\)

Name: The topology generated by \(\mathcal B\)
Definition: The collection of all unions of elements of \(\mathcal B\)
Page: 80
Chapter: 2
Section: 13
Variables: \(X\) be a set
\(\mathcal B\) a basis over \(X\)
Index: 1
Kind: Lem

Name: A basis for the topology of \(X\)
Definition: \(\mathcal T\) is generated by \(\mathcal B\)
Page: 80
Chapter: 2
Section: 13
Variables: \(X\) be a set
\(\mathcal T\) a topology on \(X\)
Kind: Lem
Definition2: for each open set \(U\in\mathcal T\) and \(x\in U\), there is an element \(C\in\mathcal C\) such that \(x\in C\subseteq U\)
DefType: A subset \(\mathcal C\subseteq\mathcal T\)

Page: 81, https://en.wikipedia.org/wiki/Real_line#As_a_topological_space
Name: The standard topology of \(\mathbb R\)
Chapter: 2
Section: 13
Kind: Def
Base: \((a,b)\)
Opens: Countable union of open intervals
Locally compact: yes
Fundamental group: zero

Page: 82
Name: The K-topology
Set: Reals
Chapter: 2
Section: 13
Base: sets of the form \((a,b)\) and \((a,b)\setminus\{\frac 1n\mid n\in\mathbb Z_+\}\).
Notation: \(\mathbb R_K\)
Name2: Smirnov's topology

Name: The topology generated by the subbasis \(\mathcal S\)
Definition: The collection of all unions of finite intersections of elements of \(S\).
Page: 82
Chapter: 2
Section: 13
Variables: \(X\) a set
\(\mathcal S\) a set of subsets of \(X\)
Kind: Def

Definition: \(\{x\in X\mid a\le x\le b\}\)
Page: 84
Chapter: 2
Section: 14
Variables: \(X\) be an ordered set
\(a,b\in X\).
Notation: \([a,b]\)

Definition: \(\{x\in X\mid a< x < b\}\)
Page: 84
Chapter: 2
Section: 14
Variables: \(X\) be an ordered set
\(a,b\in X\)
Notation: \((a,b)\)

Name: \([a,b)\)
Definition: \(\{x\in X\mid a\le x < b\}\)
Page: 84
Chapter: 2
Section: 14
Variables: \(X\) be an ordered set
\(a,b\in X\)

Name: \((a,b]\)
Definition: \(\{x\in X\mid a< x \le b\}\)
Page: 84
Chapter: 2
Section: 14
Variables: \(X\) be an ordered set
\(a,b\in X\).

Question: The order topology of \(\mathbb Z_+\) is
Answer: discrete
Page: 85
Chapter: 2
Section: 14
Index: 3
Kind: Ex

Question: The order topology of \(\{1,2\}\times \mathbb N\) with the dictionnary order has basis:
Answer: the singletons \(\{(a,b)\}\) for \((a,b)\ne(2,0)\) and the sets
\(\{(2,0)\}\cup(\{1\}\times (b,+\infty))\) for any \(b\in\mathbb N\). Thus it is not discrete because of the point \(\{2,0\}\).
Page: 85
Chapter: 2
Section: 14
Index: 4
Kind: Ex

Definition: \(\{x\in X\mid x<a\}\).
Page: 85
Chapter: 2
Section: 14
Variables: \(X\) be an ordered set
\(a\in X\)
Notation: \((-\infty,a)\)
Kind: Def

Definition: \(\{x\in X\mid a\le x\}\).
Page: 85
Chapter: 2
Section: 14
Variables: \(X\) be an ordered set
\(a\in X\)
Context: The set
Notation: \([a,\infty)\)
Kind: Definition

Definition: \(\{x\in X\mid a< x\}\).
Page: 85
Chapter: 2
Section: 14
Variables: \(X\) be an ordered set
\(a\in X\)
Context: The set
Notation: \((a,\infty)\)
Kind: Definition

Name: The rays of \(X\)
Definition: The sets of the form \([a,\infty)\), \((a,\infty)\), \((-\infty,a)\) and \((-\infty,a]\)
Page: 85
Chapter: 2
Section: 14
Variables: \(X\) be an ordered set
Kind: Definition

Page: 86, 88, not in book
Name: The product topology
Set: \(X\times Y\)
Chapter: 2
Section: 15
Variables: \(X,Y\) be topological space.
Kind: Theo
Base: \(U\times V\) for \(U\subseteq X\) and \(V\subseteq Y\) open sets.
Index: 1, 2, , not in book
Definition: The coarsest topology such that the functions \(\pi_i\) are continuous.
Base2: \(U\times V\) for \(U\in B\) and \(V\in C\) for \(B\) and \(C\) two basis.
Prebase: \(\pi_X^{-1}(U)\) and \(\pi_Y^{-1}(V)\) for \(U\) and \(V\) open sets.

Definition: The function sending \((x_0,x_1)\in X_0\times X_1\) to \(x_i\).
Page: 87
Section: 15
Chapter: 2
Variables: \(X_0,X_1\) be topological space
Notation: \(\pi_i\)
Name: The projection functions
Index: Def
Jective: onto/surjective

Smaller0: \(\pi_1^{-1}(Y)\)
Greater0: \(X_0\times Y\)
Page: 87
Chapter: 2
Section: 15
Variables: \(X_0,X_1\) be topological space
\(Y\subseteq X_1\).
Smaller to greater: =

Name: A simple order on \(B\)
Definition: A strict order on \(B\) such that for all distinct \(x\) and \(y\), either \(x\prec y\) or \(y\prec x\).
Page: 68
Chapter: 1
Section: 11
Variables: \(B\) a set.
Extra: total

Smaller0: \(U\) is open in \(Y\)
Greater0: \(U\) is open in \(X\)
Page: 89, 92
Chapter: 2
Section: 17
Variables: \((X,\mathcal T)\) be a topology
\(U\subseteq Y\subseteq X\)
Index: 2, 2
Kind: Lem, Theo
Hide middle: yes
Smaller1: \(Y\) is open in \(X\)
Extra: Implication in Lem. 16.2.

Smaller0: \(U\) is closed in \(Y\)
Greater0: \(U\) is closed in \(X\)
Page: 89, 92
Chapter: 2
Section: 17
Variables: \((X,\mathcal T)\) be a topology
\(U\subseteq Y\subseteq X\).
Index: 2, 2
Kind: Lem, Theorem
Hide middle: yes
Smaller1: \(Y\) is closed in \(X\)
Extra: Implication in Lem. 16.2.

Smaller0: the product topology on \(A\times B\)
Greater0: the topology \(A\times B\) inherits as a subspace of \(X\times Y\)
Page: 89
Chapter: 2
Section: 16
Variables: \((X,\mathcal T)\)
\((Y,\mathcal U)\) be topologies
\(A\subseteq X\)
\(B\subseteq Y\).
Index: 3
Kind: theo
Smaller to greater: is equal to

Name: Subset whose order topology is not equal to the subspace topology
Page: 90
Chapter: 2
Section: 16
Kind: Ex
Example: The subset \([0,1)\cup \{2\}\) of the order topology of \(\mathbb R\).

Smaller0: the topology \(Y\) inherits as a subspace of \(X\)
Greater0: the order topology on \(Y\)
Page: 91
Chapter: 2
Section: 16
Variables: \((X,\prec)\) be an ordered set
\(Y\subseteq X\)
Index: 4
Kind: Theorem
Smaller to greater: equals
Hide middle: yes
Assuming: \(Y\) is convex

Name: A convex subspace
Definition: A set \(Y\subseteq X\) such that, for all \(a,b\in Y\) with \(a<b\), \([a,b]_X\subseteq Y\).
Page: 90
Chapter: 2
Section: 16
Variables: \((X,\prec)\) be an ordered set

Name: The set of closed sets
Page: 94
Section: 17
Chapter: 2
Index: 1(3)
Variables: \(X\) be a topological space
Kind: Theo
Under: union
Condition: finite
Under2: intersection

Smaller0: closed in \(\mathcal T_Y\)
Greater0: the intesection of \(Y\) and of a closed set of \(X\)
Page: 94
Chapter: 2
Section: 17
Variables: \((X,\mathcal T)\) be a topological space
\(Z\subseteq Y\subseteq X\)
Kind: Theorem
Smaller to greater: iff
Prefix Greater: \(Z\) is

Name: Interior of \(A\)
Definition: The union of all open sets contained in \(A\).
Page: 95
Chapter: 2
Section: 17
Variables: \((X,\mathcal T)\) be a topological space and \(A\subseteq X\).
Abbreviation: Int \(A\)

Name: Closure of \(A\)
Definition: The intersection of all closed sets contaning \(A\).
Page: 95, 96, 130
Chapter: 2, 2, 2
Section: 17, 17, 21
Variables: \((X,\mathcal T)\) be a topological space
\(A\subseteq X\).
Notation: \(\overline A\)
Index: ,5 , 2
Kind: ,Theo, Lem
Definition2: Union of \(A\) and its boundary
Definition3: Union of \(A\) and its limit points
Abbreviation: Cl\(A\)
Extra: Distinct from limit, because it also contains \(a\)
Extra from wp
Definition4: the set of \(x\) such that every neighborhoods of \(x\) intersects \(A\)
Definition5: there is a sequence of point of \(A\) converging to \(x\) (assuming first-countable space)
Definition6: the set of basis element containing \(x\) intersects \(A\) (assuming the topology has a basis)

Smaller0: \(\text{Int }A\)
Greater0: \(\overline A\)
Page: 95
Chapter: 2
Section: 17
Variables: \((X,\mathcal T)\) be a topological space
\(A\subseteq X\).
Smaller to greater: \(\subseteq\)
Greater to smaller: \(\supseteq\)
Intermediate: \(A\)

Smaller0: The closure of \(A\) in \(Y\)
Greater0: the intersection of \(Y\) and of the closure of \(A\) in \(X\)
Page: 95
Chapter: 2
Section: 17
Variables: \((X,\mathcal T)\) be a topological space and \(A,Y\subseteq X\).
Index: 4
Kind: Theo
Smaller to greater: equals
Hide middle: yes

Name: Two sets which intersect.
Definition: Two sets \(A,B\) such that \(A\cap B\ne\emptyset\).
Page: 96
Chapter: 2
Section: 17
Variables: \(A,B\) sets.

Name: A neighborhood of \(x\)
Definition: An open set containing \(x\).
Page: 96
Chapter: 2
Section: 17
Variables: \(X\) a topological space
\(U\subseteq X\)
\(x\in X\)

Name: A limit point of \(A\)
Definition: which belongs to the closure of \(A-\{x\}\).
Page: 97
Chapter: 2
Section: 17
Variables: \((X,\mathcal T)\) be a topological space,
\(A\subseteq X\).
Definition2: every neighborhood of \(x\) contains infinitely many points of \(A\) (assuming \(X\) satisfies the T1 axiom)
Conjdef: A point \(x\), equivalently
Name2: cluster point \(A\)
Name3: accumulation point of \(A\)

Definition: The set of limit point of \(A\).
Page: 97
Chapter: 2
Section: 17
Variables: \((X,\mathcal T)\) be a topological space
\(A\subseteq X\).
Context: Topology
Notation: \(A'\)
Index: 6
Kind: Theo
Extra: overline a

Name: Topology with not-closed singletons and where a sequence may converge to many points
Page: 98
Chapter: 2
Section: 17
Example: \(\{\{a,b,c\},\{\emptyset,\{a,b\},\{b,c\},\{b\},\{a,b,c\}\}\).

Name: A Hausdorff space
Definition: A topological space \((X,\mathcal T)\), such that for each distinct pair \(x_1,x_2\in X\), there exists distinct neighborhood of \(x_1\) and of \(x_2\).
Page: 98
Chapter: 2
Section: 17
Kind: Def
Name2: Separated space
French2: Espace séparé

Smaller0: \(A\) is finite
Greater0: \(A\) is closed
Page: 99
Chapter: 2
Section: 17
Variables: \((X,\mathcal T)\) be a topological space
\(A\subseteq X\).
Index: 8
Kind: Theorem
Smaller1: \(X\) is Hausdorff
Construction: Given \(x,y\in X\), let \(C_{x,y}\) a closed set containing \(x\) and not \(y\).
\(A=\bigcup_{x\in A}\bigcap_{y\not\in A}C_{x,y}\)
Counterexample: An infinite set with finite complement topology.

Name: A set which satisfies the \(T_1\) axiom
Definition: A set \(X\) such that every finite point set is closed.
Page: 99
Chapter: 2
Section: 17
Variables: \(X\) a topology
Index: 9
Kind: Theo

Question: The number of point to which \((x_i)\) converges is
Answer: at most one.
Page: 99
Chapter: 2
Section: 17
Variables: \(X\) be a topological space
\((x_i\in X)_{i\in\mathbb N}\)
Index: 10
Kind: Theorem
Assuming: \(X\) is a Hausdorff space

Smaller0: \(<\) is a simple order
Greater0: Hausdorff
Page: 100
Chapter: 2
Section: 17
Variables: \((X,<)\) be an ordered set.
Index: 11
Kind: Theo
Prefix Greater: the separability class of the order topology of \(X\) is

Name: The class of Hausdorff set
Page: 100, 116, 142
Section: 17, 19, 22
Chapter: 2, 3, 3
Index: 11, 4, 3
Kind: Theo, Cor
Extra: Chapitre 2, Section 19, Theorem 3, Page 114,
Under: subspace
Under2: product topology
Condition2: arbitrary
Under3: box product
Under4: taking the quotient topology when the image is Hausdorff
Other fields (not shown on cards):
Prefix5: Adding open set?

Name: A countinuous map
Definition: A map, such that for each \(V\subseteq Y\) open, \(f^{-1}(V)\) is open
Page: 102
Chapter: 2
Section: 18
Variables: \(X,Y\) be topological spaces

Name: An homeomorphism from \(X\) to \(Y\)
Definition: continuous whose inverse is continuous.
Page: 105, 72
Chapter: 2, 2 (real pugh)
Section: 18, 3,
Variables: \(X,Y\) be topological spaces
Index: , 12
Kind: , Cor
Definition2: such that, for each \(V\subseteq X\), \(f(V)\) is open iff \(V\) is open.
Conjdef: A bijection \(f:X\to Y\)
Definition3: bijection of set of open/closed sets
Counterexample: The function \(x\mapsto (\cos 2\pi t,\sin 2\pi t)\), \([0,1)\to S^1\).

Name: A topological property
Definition: A property such that, when \(X,Y\) are both homeomorph topologies and \(X\) satisfies \(\mathcal P\) then \(Y\) satisfies \(\mathcal P\).
Page: 105
Chapter: 2
Section: 18
Variables: \(\mathcal P\) be a property on topological spaces.

Name: Imbedding
Definition: \(f:X\to Y\), which is an homeomorphism from \(X\) to \(f(X)\).
Page: 105
Chapter: 2
Section: 18
Variables: \(X,Y\) be topological spaces
\(f:X\to Y\)
Name2: Topological imbedding

Name: Continuous injective function which is not an imbedding
Page: 104
Chapter: 2
Section: 18
Example: The function \(x\mapsto (\cos 2\pi t,\sin 2\pi t)\), \([0,1)\to \mathbb R^2\).

Definition: A function such that, for each \(x,y\in X\), \(f(x)=f(y)\).
Page: 105
Section: 18
Chapter: 2
Variables: \(X,Y\) be topological spaces.
Name: Constant function
Index: Theorem
Kind: 2a
Continuous: Continuous

Question: The inclusion function has the property to be
Answer: continuous.
Page: 107
Chapter: 2
Section: 18
Variables: \(X\) be topological spaces
\(A\) a subspace of \(X\)
Index: 2(b)
Kind: Theorem

Name: Continuous functions
Page: 108, 110, 116
Section: 18, 19
Chapter: 2, 2
Index: 2(d), 4, 5
Variables: \(X,Y\) be topological spaces.
\(f:X\to Y\) continuous
Kind: Theorem, Theo
Under: restriction of it to a subspace of the domain of definition.
Under2: product, sum
Condition2: finite
Under3: product topology
Not4: not
Under4: box topology
Under5: composition
Closure6: making X
Under6: finer
Closure7: making Y
Under7: coarser

Question: \(f\) as a function \(X\to Z\) is
Answer: continuous.
Page: 108
Chapter: 2
Section: 18
Variables: \(X,Y\) be topological spaces
\(f:X\to Y\) continuous
\(f(X)\subseteq Z\subseteq Y\)
Index: 2(e)
Kind: Theorem

Smaller0: \(f=g\) where they are both defined
Greater0: the function which maps \(x\in A\) to \(f(x)\) and \(x\in B\) to \(g(x)\) is continuous
Page: 108
Chapter: 2
Section: 19
Variables: \(X,Y\) be topological spaces.
Let \(A,B\subseteq X\)
Let \(f:A\to Y,g:B\to Y\) continuous.
Index: 3
Kind: Theo
Hide middle: yes
Smaller1: \(A,B\) both open or both closed
Name: The Pasting lemma

Name: A \(\mathbf J\)-tuple of elements of \(X\)
Definition: A function \(\mathbf x:\mathbf J\to X\).
Page: 113
Chapter: 2
Section: 19
Variables: \(\mathbf J\) be an index set
\(X\) a set.
Notation: \((x_\alpha)_{\alpha\in \mathbf J}\)
Kind: Definition

Name: \(\alpha\)-th coordinate of \(\mathbf x\)
Definition: \(\mathbf x(\alpha)\).
Page: 113
Chapter: 2
Section: 19
Variables: \(\mathbf J\) be an index set and \(X\) a set
\(\mathbf x\) be a \(\mathbf J\)-tuple of elements of \(X\).
Notation: \(x_\alpha\)
Kind: Definition

Definition: The set of \(\mathbf J\)-tuple of elements of \(X\).
Page: 113
Chapter: 2
Section: 19
Variables: \(\mathbf J\) be an index set
\(X\) a set.
Notation: \(X^{\mathbf J}\)
Kind: Definition

Name: The Cartesian product of this indexed family
Definition: The set of all \(\mathbf J\)-tuples of elements \((x_\alpha)_{\alpha\in \mathbf J}\) of \(X\) such that \(x_\alpha\in A_\alpha\).
Page: 113
Chapter: 2
Section: 19
Variables: \(\mathbf J\) be an index set
\(\{A_\alpha\}_{\alpha\in\mathbf J}\) be an indexed family of sets
\(X=\bigcup_{\alpha\in \mathbf J} A_\alpha\).
Notation: \(\Pi_{\alpha\in\mathbf J}A_\alpha\)
Kind: Definition

Page: 114
Name: The box topology
Set: \(\Pi_{\alpha\in\mathbf J}X_\alpha\)
Chapter: 2
Section: 19
Variables: \(\mathbf J\) be an index set
\(\{X_\alpha\}_{\alpha\in\mathbf J}\) be an indexed family of sets.
Kind: Definition
Base: \(\Pi_{\alpha\in\mathbf J}U_\alpha\) for \(U_\alpha\) open in \(X_\alpha\).

Definition: The function \(X^{\mathbf J} \to X \) sending \((x_\alpha)_{\alpha\in\mathbf J}\) to \(x_\beta\).
Page: 114
Chapter: 2
Section: 19
Variables: \(\mathbf J\) be an index set
\(\beta\in\mathbf J\) and \(X\) a set
Notation: \(\pi_\beta\)
Kind: Definition

Page: 116
Chapter: 2
Section: 19
Variables: \(\mathbf J\) be an index set
\(\{X_\alpha\}_{\alpha\in\mathbf J}\) be an indexed family of sets with subspace \(A_\alpha\).
Notation: \(\Pi_{\alpha\in \mathbf J}A_\alpha\)
Index: 3
Kind: Theorem
Typ: A subspace of \(\Pi_{\alpha\in \mathbf J}X_\alpha\)
Assuming: Box and product topology

Smaller0: \(\Pi_{\alpha\in \mathbf J}\overline{A_\alpha}\)
Greater0: \(\overline{\Pi_{\alpha\in \mathbf J}A_\alpha}\)
Page: 116
Chapter: 2
Section: 19
Variables: \(\mathbf J\) be an index set
\(\{X_\alpha\}_{\alpha\in\mathbf J}\) an indexed family of sets
\(A_\alpha\) subspaces of \(X_\alpha\)
Index: 5
Kind: Theorem
Smaller to greater: is equal to
Assuming: In both products are box or both products are product topology

Name: Function such that all elements are continuous but which is not continuous for the box topology
Page: 116
Chapter: 2
Section: 19
Index: 5
Kind: Theorem
Example: \(\Pi_{i\in\mathbb N}\mathbb R\) and \(f:t\mapsto (t,t,\dots)\).

Name: The distance between \(x,y\)
Page: 119
Chapter: 2
Section: 20
Variables: \((X,d)\) be a metric space and \(x,y\in X\).
Notation: \(d(x,y)\)
Kind: Definition

Name: The \(\epsilon\)-ball, centred at \(x\)
Definition: The set of \(y\in X\) such that \(d(x,y)<\epsilon\).
Page: 119
Chapter: 2
Section: 20
Variables: \((X,d)\) be a metric space,
\(x\in X\) and
\(\epsilon>0\).
Notation: \(B_d(x,\epsilon)\)
Kind: Definition

Page: 119, https://en.wikipedia.org/wiki/Metric_space#Open_and_closed_sets,_topology_and_convergence
Name: The metric topology
Set: \(X\)
Chapter: 2
Section: 20
Variables: \((X,d)\) be a metric space.
Kind: Definition
Base: \(B_d(x,\epsilon)\) for \(x\in X\) and \(\epsilon{\gt}0\).
Definition: Coarsest topology on \(M\) relative to which the metric \(d\) is a continuous map
T4/normal Hausdorff: yes
Sequentially compact: the compact sets

Name: A metrizable space
Definition: A topological space whose topology is generated by a metric on \(X\).
Page: 120
Chapter: 2
Section: 20
Kind: Definition

Name: A bounded metric space
Definition: A metric set \((X,d)\) such that there exists \(M\in\mathbb R\) such that for all \(a,b\in X\), \(d(a,b)<M\).
Page: 121
Chapter: 2
Section: 20
Kind: Definition

Name: The diameter of \(X\)
Definition: \(\text{sup}\{d(a,b)\mid a,b\in X\}\).
Page: 121
Chapter: 2
Section: 20
Variables: \((X,d)\) be a bounded metric space.
Kind: Definition

Name: The standard bounded metric corresponding to \(d\)
Definition: \((x,y)\mapsto\text{min}(1,d(x,y))\).
Page: 121
Chapter: 2
Section: 20
Variables: \((X,d)\) be a metric space.
Notation: \(\overline d\)
Kind: Definition

Name: The norm of \(\mathbf x\)
Definition: \(\sqrt{\sum_{i=1}^nx_i^2}\).
Page: 122
Chapter: 2
Section: 20
Variables: \(\mathbf x\in\mathbb R^n\).
Notation: [$]\norm{\mathbf x}[/$]
Kind: Definition

Name: The Euclidean metric on \(\mathbb R^n\)
Definition: [$]\defFun{\mathbb R^n}{\mathbb R^{\ge0}}{(x,y)}{\norm{x-y}}[/$].
Page: 122
Chapter: 2
Section: 20
Variables: \(\mathbf x,\mathbf y\in\mathbb R^n\).
Notation: \(d\)
Kind: Definition

Name: The square metric
Definition: [$]\defFun{\mathbb R^n}{\mathbb R^{\ge0}}{(x,y)}{\max\{|x_i-y_i|\mid 0<i\le n\}}[/$].
Page: 122
Chapter: 2
Section: 20
Variables: \(\mathbf x,\mathbf y\in\mathbb R^n\).
Context: \(\mathbb R^n\)
Notation: \(\rho\)
Kind: Definition
Intuition: Square is the object, not the function.

Page: 123
Name: The topology on \(\mathbb R^n\)
Chapter: 2
Section: 20
Kind: Theorem
Metric: Topology generated by the Euclidean/square metric
Index: 3

Page: 124
Section: 20
Notation: \(\overline\rho\)
Variables: \(\mathbf J\) be an index set.
\((X,d)\) a metrized set
Chapter: 2
Name: The uniform metric on \(X^{\mathbf J}\)
Definition: [$]\defFun{X^{\mathbf J}}{\mathbb R^{\ge0}}{(\mathbf x,\mathbf y)}{\text{sup}\{\overline d(x_\alpha,y_\alpha)\mid\alpha\in\mathbf J\}}[/$].
Kind: Def
Metric: A bounded metric

Smaller0: the box topology
Greater0: the uniform topology
Page: 115, 124
Chapter: 2
Section: 20
Variables: \(\mathbf J\) be an index set
\(\mathbb R^{\mathbf J}\)
Index: 1, 4
Kind: Theo, Theorem
Smaller to greater: is finer than
Greater to smaller: is coarser than
Greatest: the product topology

Smaller0: \(\mathbf J\) is finite
Greater0: the box topology
Page: 113, 122
Chapter: 2
Section: 19, 20
Variables: \(\mathbf J\) be an index set.
\(\mathbb R^{\mathbf J}\).
Index: 4
Kind: Theorem
Smaller to greater: iff
Greater1: the uniform topology
Connect Greater0: equals
Greater2: product topology

Smaller0: \(\mathbf J\) is at most countable
Greater0: metrizable
Page: 127
Chapter: 2
Section: 20
Variables: \(\mathbf J\) be an index set
\((X,d)\) a metrizable set
\(\mathcal T\) its topology.
Index: 5, 21
Kind: Theorem
Smaller to greater: iff
Prefix Greater: \(\mathcal T^{\mathbf J}\) is
Extra: Ch 2 21 Ex 2
introuvable

Smaller0: the topology \((A,d_{\mid A\times A})\)
Greater0: the topology \((X,d)_{\mid A}\)
Page: 129
Chapter: 2
Section: 21
Variables: \((X,d)\) be a metric space
\(A\subseteq X\).
Smaller to greater: is equal to

Smaller0: there is a sequence of point of \(A\) converging to \(x\)
Greater0: \(x\in\overline A\)
Page: 130, https://math.stackexchange.com/questions/409317/when-is-an-accumulation-point-not-the-limit-of-some-sequence-in-a-topological-sp/409321
Chapter: 2
Section: 21
Variables: \(X\) be a topological space
\(A\subset X\).
Kind: Lemma
Name: Sequence lemma
Extra: There are actually two sequence lemmas, for first-countable it's an equivalence; otherwise it's an implication
Counterexample: Any point in a countable complement topology in an uncountable set
Counterexample2: The order topology of \([0,\omega_1]\) with \(\omega_1\) the first uncountable ordinal, and \(x=\omega_1\)

Smaller0: \((x_i)\) converges to \(x\)
Greater0: \(\left(f(x_i)\right)\) converges to \(f(x)\)
Page: 130
Chapter: 2
Section: 21
Variables: \(X,Y\) be topological spaces,
\(f:X\to Y\)
\((x_i\in X)_{i\in\mathbb N}\).
Index: 3
Kind: Theorem
Smaller1: \(f\) is continuous
Counterexample: \(f\) constant, \(x_i\) not converging to \(x\)

Smaller0: \(x\in\overline A\)
Greater0: there is a sequence of point of \(A\) converging to \(x\)
Page: 130, 190
Chapter: 2, 4
Section: 21, 30
Variables: \(X\) be a topological space
\(A\subset X\)
Index: 2, 1
Kind: Lemma, Theo
Smaller to greater: iff
Hide middle: yes
Assuming: \(X\) is first-countable (e.g. metrizable)
Name: Sequence lemma
Extra: There are actually two sequence lemmas, for first-countable it's an equivalence; otherwise it's an implication

Name: \(X\) have a countable basis at the point \(x\)
Definition: There is a countable collection \(\{U_n\}_{n\in\mathbb Z_+}\) of neighborhoods of \(x\) such that any neighborhoood \(U\) of \(x\) contains at least one of the \(U_n\).
Page: 130
Chapter: 2
Section: 21
Variables: \(X\) a topological space
\(x\in X\).

Name: \(X\) satisfies the first countability axiom
Definition: \(X\) have a countable basis at the point \(x\) for all \(x\in X\).
Page: 130, 190
Chapter: 2, 4
Section: 21, 30
Variables: \(X\) a topological space
Name2: \(X\) is first-countable

Name: Continuous real functions
Page: 131
Section: 21
Chapter: 2
Index: 4
Kind: Lemma
Under: addition, substraction, multiplication and quotient.

Smaller0: \(f_n\) converges uniformly
Greater0: the limits of \((f_n)_n\) is continuous
Page: 132
Chapter: 2
Section: 21
Variables: \(X,Y\) be metric topological space
\(f_n:X\to Y\).
Index: 6
Kind: Theorem
Hide middle: yes
Smaller1: each \(f_n\) is continuous

Name: A subset of \(X\) saturated with respect to \(f\)
Definition: A set \(C\subseteq X\) such that \(f^{-1}(f(C))\subseteq C\).
Page: 137
Chapter: 2
Section: 22
Variables: \(X,Y\) topological spaces
\(f:X\to Y\) surjective
Kind: Definition

Name: An open map
Definition: \(f:X\to Y\) such that for each \(U\subseteq X\) open, \(f(U)\) is open.
Page: 137
Chapter: 2
Section: 22
Variables: \(X,Y\) topological spaces.
Kind: Definition

Name: Quotient map
Definition: surjective
Page: 137
Chapter: 2
Section: 22
Variables: \(X,Y\) topological spaces
\(f:X\to Y\)
Index: , 1
Kind: Definition, Ex
Definition2: open or closed
Example: The function \(x:[0,1]\cup [2,3]\to [0,2]\) sending \(x\in[0,1]\) to \(x\) and \(x\in[2,3]\) to \(x-1\)
Conjdef: A function which is
Counterexample: The function [$]\defFun{[0,1)\cup [2,3]}{[0,2]}{x}{x\qif x<1\qelse x-1}[/$].
\([2,3]\) is open in \(A\), saturated with respect to \(q\), and image not open in \(Y\)

Page: 138
Name: The quotient topology of \(A\) induced by \(p\)
Set: \(A\)
Chapter: 2
Section: 22
Variables: \(X\) a topological space
\(A\) a set,
\(p:X\to A\) a map
Kind: Definition
Opens: \(U\subseteq A\) such that \(p^{-1}(U)\) is open
Locally compact: implied by \(X\) compact
Definition: the only topology such that \(p\) is a quotient
Path connected: implied by \(X\) is path-connected
Assuming: \(p\) surjective
Connected: implied by \(X\) is connected
T1/accessible/Tichonov/Fréchet topology: If and only if every equivalence class of ~ is closed in X

Question: Its quotient topology is
Answer: \(\left(\{a,b,c\},\{\emptyset,\{a\},\{b\},\{a,b\},\{a,b,c\}\}\right)\)
Page: 138
Chapter: 2
Section: 22
Variables: \(p:\mathbb R\to \{a,b,c\}\) with
\(p(x)=a\) if \(a>0\),
\(p(x)=b\) if \(x<0\)
\(p(0)=c\).
Index: 3
Kind: Ex

Page: 139
Name: The quotient space of \(X\)
Set: \(X^*\)
Chapter: 2
Section: 22
Variables: \((X,\mathcal T)\) be a topological space and
\(X^*\) a partition of \(X\).
Kind: Definition
Opens: The identification space of \(X\)
Definition: the quotient topology induced by the function which sends each element to its class.
Order: The decomposition space of \(X\)

Smaller0: \(\{(x,y)\mid x^2+y^2\le1\} \)
Greater0: \(S^2=\{(x,y,z)\mid x^2+y^2+z^2=1\}\)
Page: 139
Chapter: 2
Section: 22
Index: 4
Kind: Ex
Hide middle: yes
Prefix Greater: \(X^*\) is homeomorphic to
Smaller1: \(\left\{\{(x,y)\}\mid x^2+y^2<1\right\}\cup\left\{\{(x,y)\mid x^2+y^2=1\} \right\} \)
Prefix Smaller: \(X\) is
Connect Smaller0: and \(X^*\) is

Smaller0: \([0,1]^2\)
Greater0: the torus
Page: 139
Chapter: 2
Section: 22
Index: 5
Kind: Ex
Hide middle: yes
Prefix Greater: \(X^*\) is homeomorphic to
Smaller1: \(\left\{\{(x,y)\}\mid x,y\in(0,1)\right\}\cup\left\{\{(x,0),(x,1)\},\{(0,x),(1,x)\}\mid x\in(0,1) \right\}\cup \left\{\{(x,y)\mid x,y\in\{0,1\} \}\right\}\)
Prefix Smaller: \(X\) equals
Connect Smaller0: and \(X^*\) equals

Smaller0: \(A\) is open or closed
Greater0: the restriction of \(p\) to \(A\) is a quotient map
Page: 140
Chapter: 2
Section: 22
Variables: \(p:X\to Y\) a quotient map and \(A\subseteq X\).
Index: 1
Kind: Theo
Hide middle: yes
Smaller1: \(p\) is an open map or a closed map
Connect Smaller0: or

Name: Open maps
Page: 141
Section: 22
Chapter: 2
Extra: composition not mentionned, but straightforward
domain restriction ?
Under: product
Under2: composition

Page: 137, 143 and ?
Chapter: 2
Section: 22
Index: 1, 7,
Kind: Ex, Ex
Q: Counter example for closure by product:
-[...].
A: Let \(X=\mathbb R\),
\(X^*=\{\{\mathbb Z_+\}\}\cup\left\{\{x\}\mid x\in\mathbb R\setminus \mathbb Z_+\right\}\) with quotient map \(p\),
let \(i:\mathbb Q\to\mathbb Q\) the identity function.
Then \((p,i):X\times\mathbb Q\to X^*\times\mathbb Q\)

Page: 137, 143 and ?
Chapter: 2
Section: 22
Index: 1, 7,
Kind: Ex, Ex
Name: Quotient map
Q: [...]
A: restricting the domain of definition of the map
Q: [...]
A: finite product
Q: [...]
A: composition
Q: [...]
A: not
Q: [...]
A: not
Q: [...]
A: closed

Smaller0: each element of the partition \(X^*\) is closed
Greater0: \(X^*\) satisfies the \(T_1\) axiom
Page: 141
Chapter: 2
Section: 22
Variables: \(X\) be a topological space
\(X^*\) a quotient space
Index: second-to-last paragraph

Name: Continuous functions
Page: 140
Section: 22
Chapter: 2
Index: 3
Kind: Cor
Under: composition

Question: Surjective continuous map which is not a homeomorphism
Answer: Let [$]g\colon\defFun{\left\{[0,1]\times n\mid n\in\mathbb Z_+\right\} }{ \left\{\{ (x,x/n)\mid x\in[0,1]\}\mid n\in\mathbb Z_+\right\} }{(x,n)}{(x,x/n)}[/$].
The closed sequence \(((\frac 1n),n)_{n\in\mathbb N}\) is sent to \((\frac 1n,1)_{n\in\mathbb N}\) which is not closed.
Page: 141
Chapter: 2
Section: 22
Index: 6
Kind: Ex

Name: A connected set
Definition: A set which admits no separation.
Page: 146
Chapter: 3
Section: 23
Variables: \(X\) a topological space
Index: 1
Kind: Lemma

Smaller0: \(X\) is connected
Greater0: \(X\) contains at most one element
Page: 147
Chapter: 3
Section: 23
Variables: \(X\subseteq\mathbb Q\)
Index: 4
Kind: Ex
Smaller to greater: iff

Smaller0: \(Y\) is connected
Greater0: \(Y\) is a subset of some \(A_i\)
Page: 147
Chapter: 3
Section: 23
Variables: \(X\) a topological space
\(Y\subseteq X\)
\(A_1,A_2\subseteq Y\) a separation of \(C\)
Index: 2
Kind: Lemma
Counterexample: \(A_1=\{x<0\}\), \(A_2=\{x>0\}\), \(Y=\{1,2\}\)

Smaller0: the \(A_i\)'s have a point in common
Greater0: \(\bigcup A_i\) is connected
Page: 148
Chapter: 3
Section: 23
Variables: \(X\) a topological space
\((A_i)\subseteq X\) connected subspaces.
Kind: Theo
Hide middle: yes
Counterexample: \(A_i=\{e ^{\frac{ix\pi}{3}}\mid x\in[n,n+1]\}\)

Name: The set of connected subset of \(X\)
Page: 149, 150
Section: 23
Chapter: 3
Index: 5, 4,, 6
Variables: \(X\) a topology
Kind: Theo, Theo, Theo
Extra: Ex 6, 7,
Under: image by a continuous maps
Under2: Cartesian product with the product topology
Condition2: arbitrary
Under3: Cartesian product with the box topology
Condition3: finite
CounterExample3: \(\mathbb R^\omega\) takes the open subsets of all bounded sequences of real numbers and of all unbounded sequences.
Under4: intersection
Condition4: with a common point
Under5: preimage
Condition5: ?
Under6: union
Condition6: with a common point ?
Under7: adding a subset of the closure of the set

Name: Function with \(A\) open, \(B\) closed, continuous on \(A\) and \(B\), equal on \(A\cap B\) and not continuous on \(A\cup B\)
Page: 107
Chapter: 2
Section: 19
Index: 8
Kind: Ex
Example: Let \(A=\mathbb R^{<0}\) and \(B=\mathbb R^{\ge0}\), \(k:\mathbb R\to\mathbb R\), sending \(x\in A\) to \(x-2\) and \(x\in B\) to \(x+2\)

Smaller0: \((x_i)\) converges to \(x\)
Greater0: \((f(x_i))\) converges to \(f(x)\)
Page: 128
Chapter: 2
Section: 21
Variables: \(X,Y\) be topological spaces,
\(x,(x_i)\in X\) and
\(f:X\to Y\).
Index: 3
Kind: Theo
Assuming: \(f\) continuous

Smaller0: \(x\) satisfies the first countability axiom
Greater0: \((x_i)\) converges to \(x\)
Page: 128
Chapter: 2
Section: 21
Variables: \(X,Y\) be topological spaces,
\(f:X\to Y\)
Index: 3
Kind: Theo
Hide middle: yes
Smaller1: (for each sequence \((x_i)\in X\), \(\lim f(x_i)=f(x)\) implies \(\lim x_i=x\))
Extra: contre exemple x hausdorff non métrisable
Countable basis is stated below
continuous ?

Name: A linear continuum
Definition: \((x,y)\) is not empty for any \(x<y\)
Page: 151
Chapter: 3
Section: 24
Definition2: \(L\) a set with at least 2 elements,
Conjdef: \((L,<)\)
Definition3: \(L\) has the least upper bound property;

Smaller0: \(L\) is a linear continuum
Greater0: connected
Page: 151
Chapter: 3
Section: 24
Variables: \((L,<)\) an ordered set
Index: 1
Kind: Theo
Hide middle: yes
Prefix Greater: its rays, intervals, are

Name: Linear continuum different from the subsets of \(\mathbb R\)
Page: 153
Chapter: 3
Section: 24
Index: 1,2
Kind: Ex
Example: Ordered square
Example2: \(X\times[0,1)\) for dictionarry order and \(X\) well ordered
Example3: Example: \(\mathbb R\) and \(\overline{\mathbb R}\).
counter example \(\mathbb {N,Q}\)
Example4: Long line

Name: A path from \(x\) to \(y\)
Definition: A continuous function \(f:[a,b]\to X\) with \(f(a)=x\), \(f(b)=y\) for some \(a<b\in\mathbb R\).
Page: 153
Chapter: 3
Section: 24
Variables: \(X\) a topological space
\(x,y\in X\)
Kind: Def

Name: A path connected topological space
Definition: A space such that, each pair of points is connected by a path.
Page: 153
Chapter: 3
Section: 24
Kind: Def

Smaller0: \(X\) is path-connected
Greater0: \(X\) is connected
Page: 154, 159, 154
90, 91
Chapter: 3, 3
2
Section: 24, 25, 24
5
Variables: \(X\) a topological space
Index: ,3, 6, 7
51
Kind: Def, Ex, Ex, Ex
Ex, theo Real analysis, Pugh
Counterexample: Topologist's sine curve
Counterexample2: \(I_o^2\)

Q: [...] is path-connected implies [\(f(X)\) and \(X\)] is path-connected
A: \(X\); \(f(X)\)
Page: 154
Chapter: 3
Section: 24
Variables: \(X,Y\) topological spaces and \(f:X\to Y\) continuous.
Index: 5
Kind: Ex

Page: 154
Name: the (closed) topologist's sine curve
Set: The closure of \(S=\{(x,\sin(1/x))\mid 0<x\le 1\}\), that is, \(S\cup\{0\}\times[-1,1]\)
Chapter: 3
Section: 24
Kind: Ex
Metric: Standard metric
Index: 7
Compact: yes
Name2: Warsaw sine curve
Path connected: no

Page: 154
Name: the extended topologist's sine curve
Set: The closed topologist sine curve and \([0,1]\times\{1\}\)
Chapter: 3
Section: 24
Kind: Ex
Metric: Standard metric
Index: 7
Context: Récupéré
Locally compact: yes
Dense: no

Name: A path-component of \(X\)
Definition: An equivalence class for the relation \(\{(x,y)\mid \text{there exists a path-connected subspace of $X$ containing $x$ and $y$} \}\)
Page: 157
Chapter: 3
Section: 25
Variables: \(X\) a topological space
Kind: Def

Name: A component of \(X\)
Definition: An equivalence class for the relation \(\{(x,y)\mid \text{there exists a connected subspace of $X$ containing $x$ and $y$} \}\)
Page: 157
Chapter: 3
Section: 25
Variables: \(X\) a topological space
Kind: Def

Name: The set of (path)-components
Definition: A partition of \(X\) and each (path)-connected subspace of \(X\) intersects a single component.
Page: 157
Chapter: 3
Section: 25
Variables: \(X\) a topological space
\(\mathcal X\subseteq\mathcal P(X)\)
Index: 1, 2
Kind: Theo

Question: The components of \(X\) are
Answer: closed.
Page: 158
Chapter: 3
Section: 25
Variables: \(X\) a topological space

Smaller0: \(X\) has a finite number of components
Greater0: open
Page: 158
Chapter: 3
Section: 25
Variables: \(X\) a topological space
Hide middle: yes
Prefix Greater: the components of \(X\) are

Question: The (path)-components of \(\mathbb Q\) are
Answer: the singletons.
Page: 158
Chapter: 3
Section: 25
Index: 1
Kind: Ex

Question: The components (respectively, path components) of the closure of the topologist's sin curve are
Answer: itself (respectively, the curve, and \(\{0\}\times[-1,1]\)).
Page: 158
Chapter: 3
Section: 25
Index: 2
Kind: Ex

Name: A topological space \(X\) locally (path)-connected at a point \(x\)
Definition: A topological space \(X\) such that, for each open \(U\) containing \(x\), there exists a (path)-connected open \(V\subseteq U\) containing \(x\).
Page: 159
Chapter: 3
Section: 25
Kind: Def

Name: A locally (path)-connected topological space
Definition: which is (path)-connected at each of its points.
Page: 159
Chapter: 3
Section: 25
Index: 4
Kind: Def, Theo
Definition2: for every open \(U\subseteq X\), each path-component of \(U\) is open in \(X\)
DefType: A topological space \(X\)

Name: Not connected but locally connected
Page: 159
Chapter: 3
Section: 25
Kind: Ex
Example: \([-1,0)\cup (0,1]\)

Name: Not connected, not locally connected
Page: 159
Chapter: 3
Section: 25
Kind: Ex
Example: \(\mathbb Q\)

Name: Connected and locally connected
Page: 159
Chapter: 3
Section: 25
Kind: Ex
Example: \(\mathbb R\)

Name: A (open) cover of \(Y\)
Definition: \(Y\subseteq\bigcup_{A\in\mathcal A}A\)
Page: 162
Chapter: 3
Section: 26
Variables: \(X\) a topological space
\(Y\subseteq X\)
Kind: Def
Definition2: the elements of \(\mathcal A\) are open
Name2: A (open) covering of \(Y\)
DefType: A set \(\mathcal A\subseteq\mathcal P(X)\)

Name: A compact space
Definition: each open cover of \(X\) contains a finite subcollection that also covers \(X\)
Page: 162, 167
Chapter: 3
Section: 26
Index: , 9
Kind: Def, Theo
Definition2: for each collection \(\mathcal C\) of closed sets in \(X\), with the finite intersection property, \(\bigcap\mathcal C\) is nonempty.
Conjdef: A topological space \(X\) such that, equivalently

Smaller0: \(Y\) is compact
Greater0: every covering of \(Y\) by sets open in \(X\) contains a finite subcollection covering \(Y\)
Page: 162
Chapter: 3
Section: 26
Variables: \(X\) a topological space
\(Y\subseteq X\)
Index: 1
Kind: Lem
Smaller to greater: iff
Extra: Note the \(Y\) and \(X\)

Smaller0: \(X\) is Hausdorff
Greater0: \(Y\) is closed
Page: 162
Chapter: 3
Section: 26
Variables: \(X\) a topological space
\(Y\subseteq X\).
Kind: Theo
Smaller1: \(Y\) is compact
Connect Smaller0: and

Page: 164, not in the book
Chapter: 3
Section: 26
Index: 6
Kind: Ex
Name: Topology such that all subsets are compact
Q: [...]
[...]
A: The finite complement topology; trivial topology

Name: The compact sets
Page: 164, 165 https://en.wikipedia.org/wiki/Tychonoff's_theorem
https://en.wikipedia.org/wiki/Box_topology#Example_-_Failure_at_compactness
Section: 26
Chapter: 3
Index: 5, 7
Kind: Theo
Under: continuous map
Under2: product topology
Under3: box product
Condition3: finite
CounterExample3: \(\{0,1\}^\omega\) with discrete topology. It is not even limit compact
Under4: intersection?

Smaller0: \(f\) is bijective continuous
Greater0: \(f\) is a homeomorphism
Page: 165
Chapter: 3
Section: 26
Variables: \(X,Y\) topological space and
\(f:X\to Y\).
Index: 6
Kind: Theo
Smaller1: \(X\) compact
Smaller2: \(Y\) Hausdorff

Name: Tychonoff Theorem
Definition: Compact sets are closed under arbitrary products
Page: 167
Chapter: 3
Section: 26

Name: A collection of subsets which have the finite intersection property
Definition: A set of subsets of \(X\) such that the intersection of finite subcollection is nonempty.
Page: 167
Chapter: 3
Section: 26
Variables: \(X\) a topological space.

Smaller0: \(X\) is compact
Greater0: \(\bigcap C_i\) is nonempty
Page: 168
Chapter: 3
Section: 26
Variables: \(X\) a topological space
\((C_i)i_{i\in\mathbb N}\subseteq X\)
Smaller1: the \(C_i\) are closed
Smaller2: \((C_i)i_{i\in\mathbb N}\) decreasing

Smaller0: \(X\) is compact
Greater0: there exists \(c,d\in X\) such that \(f(c)\le f(x)\le f(d)\) for every \(x\in X\)
Page: 172
Chapter: 3
Section: 27
Variables: \(X\)
\((Y,<)\)
\(f:X\to Y\)
Index: 4
Kind: Theo
Hide middle: yes
Smaller1: \(f\) continuous

Definition: \(\text{inf}\{d(x,a)\mid a\in A\}\).
Page: 173
Section: 27
Chapter: 3
Variables: \((X,d)\) a metric space,
\(A\subseteq X\) nonempty and
\(x\in X\).
Notation: \(d(x,A)\)
Index: Def
Continuous: Continuous in \(x\)

Smaller0: \(X\) is compact
Greater0: \(\mathcal A\) admits a Lebesgue number
Page: 175
Chapter: 3
Section: 27
Variables: \((X,d)\),
\(\mathcal A\) an open covering.
Index: 5
Kind: Lem
Name: Lebesgue number lemma

Smaller0: \(X\) is compact
Greater0: \(f\) is uniformly continuous
Page: 174
Chapter: 3
Section: 27
Variables: \((X,d_X)\), \((Y,d_Y)\)
\(f:X\to Y\).
Index: 6
Kind: Theo
Hide middle: yes
Smaller1: \(f\) continuous

Name: An isolated point
Definition: A point \(x\in X\) such that \(\{x\}\) is open.
Page: 174
Chapter: 3
Section: 27
Variables: \(X\) a topological set
Kind: Def

Name: A limit point compact set
Definition: A set \(X\) such that every infinite subset of \(X\) has a limit point.
Page: 176
Chapter: 3
Section: 28
Kind: Def
Name2: A set which has the Bolzano-Weierstass property
Name3: A set which is Fréchet compact

Smaller0: compactness
Greater0: limit point compactness
Page: 176, 177
Chapter: 3, 3
Section: 28, 28
Index: 1, 1.2
Kind: Theo, Ex,
Extra: checked
Counterexample: Let \(Y=(\{a,b\},\{\emptyset,\{a,b\}\})\) (trivial topology).
\(X=\mathbb N\times Y\). Every nonempty subset of \(X\) has a limit point. The covering by \(\{n\}\times Y\) has no finite subcovering.
Counterexample2: \(S_\Omega\), the least uncountable simple ordered set. Not countable since no largest element. Each infinite subset contains a countable subset, which admits a least upper bound, hence limit point compact.

Name: A subsequence of \((x_i)_{i\in\mathbb N}\)
Definition: \((x_{y_i})_{i\in\mathbb N}\) for some infinite increasing sequence \((y_i)_{i\in\mathbb N}\).
Page: 177
Chapter: 3
Section: 28
Variables: \((x_i)_{i\in\mathbb N}\) an infinite sequence.
Kind: Def

Name: Sequentially compact topology
Definition: A topological space such that every sequence of points has a convergent subsequences.
Page: 177
Chapter: 3
Section: 28
Kind: Def

Smaller0: compact
Greater0: sequentially compact
Page: 177
Chapter: 3
Section: 28
Variables: \(X\) a topological space
Index: 2
Kind: Theo
Smaller to greater: iff
Hide middle: yes
Prefix Greater: \(X\) is
Assuming: \(X\) is metrizable
Intermediate: \(X\) is limit point compact

Smaller0: \(X\) is first-countable (e.g. metrizable)
Greater0: \(f\) is continous
Page: 130, 190
Chapter: 2, 4
Section: 21, 30
Variables: \(X,Y\) be topological spaces and
\(f:X\to Y\)
Index: 3, 1(b)
Kind: Theo, Theo
Smaller1: \(f(\lim x_i)=f(x)\) for each sequence \(x_i\to x\)
Extra: contre exemple non métrisable ?
Connect Smaller0: and

Name: A set with the least upper bound property
Definition: An ordered set \((L,<)\) such that every nonempty \(A_0\subseteq A\) that is bounded above has a least upper bound.
Page: 25
Chapter: 1
Section: 3

Name: A basis element
Definition: An element of \(\mathcal B\)
Page: 076
Chapter: 2
Section: 13
Variables: \((X,\mathcal T)\) be a topology.
\(\mathcal B\) a basis of \(\mathcal T\)

Name: A separation of \(X\)
Definition: A two-partition of \(X\) such that no sets contains a limit point of the other.
Page: 146
Chapter: 3
Section: 23
Variables: \((X,\mathcal T)\) a topology
Index: 1
Kind: Lem

Name: The Lebesgue number for \(\mathcal A\)
Definition: A number \(\delta>0\), such that for each subset of \(X\) having diameter less than \(\delta\), there exists an element of \(\mathcal A\) containing it.
Page: 173
Chapter: 3
Section: 27
Variables: \((X,d)\),
\(\mathcal A\) an open covering.
Index: 5
Kind: Lem

Name: The space is locally compact space at \(x\)
Definition: There is some compact subspace \(C\) of \(X\) that contains a neighborhood of \(x\).
Page: 180
Chapter: 3
Section: 29
Variables: \((X,\mathcal T)\) a topological space,
\(x\in X\)
Kind: Def

Question: The number of one point compactification is
Answer: 1 up to homeomorphism.
Page: 181
Chapter: 3
Section: 29
Variables: \((X,\mathcal T)\) a topology
Index: 1
Kind: Theo
Assuming: \(X\) is Hausdorff

Name: The one-point compactification of \(X\)
Definition: \(Y-X\) is a singleton,
Page: 183
Chapter: 3
Section: 29
Variables: \((X,\mathcal T)\) a topology,
Kind: Def
Example: homeomorphic to a circle
Example2: the Riemann Sphere
Assuming: \(X\) is Hausdorff
DefType: A compactification \(Y\) of \(X\)
Applied to: \(\mathbb R\)
Applied to2: \(\mathbb C\)

Name: A compactification of \(X\)
Definition: \(X\) is a proper subspace of \(Y\)
Page: 183
Chapter: 3
Section: 29
Variables: \((X,\mathcal T)\) a topology,
Index: Def
Definition2: \(Y\) is compact
Conjdef: A space \(Y\) satisfying the following conditions
Definition3: the closure of \(X\) in \(Y\) is \(Y\)
Definition4: \(Y\) is Hausdorff
Assuming: \(X\) is Hausdorff

Question: The one-point compactification of \(\mathbb R\) is
Answer: homeomorphic to the circle.
Page: 183
Chapter: 3
Section: 29
Index: 4
Kind: Ex

Question: The one-point compactification of \(\mathbb R^2\) is
Answer: homeomorphic to the sphere.
Page: 183
Chapter: 3
Section: 29
Index: 4
Kind: Ex

Question: The one-point compactification of \(\mathbb C\) is
Answer: the Riemann sphere.
Page: 183
Chapter: 3
Section: 29
Index: 4
Kind: Ex

Name: The set of locally compact Hausdorff space
Page: 183
Section: 29
Chapter: 3
Kind: Cor
Under: taking subsets
Condition: open or closed
CounterExample: \(\mathbb Q\) is a non locally compact space of \(\mathbb R\)
Under2: product topology
Condition2: finite
CounterExample2: \(\mathbb R^\omega\), any neighborhood of \((0)\) contains an infinite number of \(\mathbb R\), which is not compact
Under3: union
CounterExample3: \(\mathbb Q\) is the countable union of \(\{\frac pq\}\)
Under4: box topology
CounterExample4: \((X_i,\mathcal T_i)_i\) topologies
\(\vec x=(x_i\in X_i)_i\)
Take \(K\) a compact containing a neighborhood of \(\vec x\).
\(\{\prod_{j<i}X_j\times(X_i\setminus\{x_i\})\prod_{j>i}\{X_j\}\}\cup\{\{x\}\}\) is an infinite open covering without finite covering

Name: A directed set
Definition: A partially ordered set \((J,\preceq)\) such that, for each pair \(a,b\in J\), there exists an element \(c\) such that \(a\preceq c\), \(b\preceq c\).
Page: 185
Chapter: 3
Section: 29

Name: A cofinal set in \(J\)
Definition: A set \(K\subseteq J\) such that, for all \(a\in J\), there exists \(b\in K\) such that \(a\preceq b\).
Page: 185
Chapter: 3
Section: 29
Variables: \((J,\preceq)\) a partially ordered set
Index: 2

Name: A net
Definition: A function from directed set into a topological space.
Page: 185
Chapter: 3
Section: 29
Intuition: generalization of sequence from domain \(\mathbb N\) to arbitrary ordered set

Name: A net which converges to \(x\)
Definition: A net \((J,\preceq)\) to \(X\) such that, for each neighborhood \(U\) of \(x\), there exists \(\alpha\in J\) such that \(\alpha\preceq \beta\implies x_\beta\in U\).
Page: 185
Chapter: 3
Section: 29
Variables: \((X,\mathcal T)\) a topology,
\(x\in X\)

Name: \((x_\alpha)\) converges to \(x\)
Page: 185
Chapter: 3
Section: 29
Variables: \((X,\mathcal T)\) a topology,
\(x\in X\)
\(J\) a directed set
\((x_\alpha)_{\alpha\in J}\in X\) a net
Notation: \(x_\alpha\to x\)

Smaller0: the standard topology
Greater0: K-topology
Page: 80
Chapter: 2
Section: 13
Smaller to greater: is coarser than
Greater to smaller: is finer than
Context: \(\mathbb R\)
Hide sides: yes

Page: 122
Name: The uniform topology
Set: \(X^{\mathbf J}\)
Chapter: 2
Section: 20
Variables: \(\mathbf J\) be an index set.
\((X_j,d_j)_{j\in\mathbf J}\) a metrized set
Metric: \(\overline\rho(\vec x,\vec y)=\sup\{d_{j}(x_j,y_j)\mid j\in\mathbf J\}\)

Question: Number of topologies on \(A\) such that \(p\) is a quotient
Answer: exactly one
Page: 138
Chapter: 2
Section: 22
Variables: \(X\) a topological space
\(A\) a set
\(p:X\to A\) a surjective map

Name: The ordered square
Definition: \([0,1]^2\) with dictionnary order
Page: 88
Chapter: 2
Section: 16

Smaller0: \(U\) is a path component
Greater0: \(U\) lies in a component of \(X\)
Page: 161
Chapter: 2
Section: 25
Variables: \(X\) a topological space
\(U\subseteq X\)
Index: 5
Kind: Theo

Name: A condensation point of \(A\)
Definition: A point of \(X\) such that each of its neighboors contains uncontably many point of \(A\).
Page: 97
Chapter: 2
Section: 17
Variables: \((X,\mathcal T)\) be a topological space,
\(A\subseteq X\).
Extra: Not actually in the book. Usefull for the card's author to remember the difference between both notions

Smaller0: \(x\) is a condensation point
Greater0: \(x\) is a limit point
Page: 97
Chapter: 2
Section: 17
Variables: \((X,\mathcal T)\) be a topological space,
\(A\subseteq X\),
\(x\in X\)
Hide sides: yes

Smaller0: there is a sequence of points converging to \(x\)
Greater0: \(x\in\overline A\)
Page: 190
Chapter: 4
Section: 30
Variables: \(X\) a topological space
\(A\subseteq X\)
\(x\in X\)
Index: 1 (a)
Kind: Theo

Name: A topology satisfying the second countability axiom
Definition: A topology with a countable basis
Page: 190
Chapter: 4
Section: 30
Index: , 1
Kind: Def, Ex
Example: \(\mathbb R^\omega\) has basis \(\Pi_{n\in \mathbb Z_+}U_n\), with finitely many \(U_n\)'s being an interval with rational bounds, and every other \(U_n\)'s being \(\mathbb R\)
Name2: A second-countable topology

Smaller0: second countability axiom
Greater0: first coutability axiom
Page: 190
Chapter: 4
Section: 30
Counterexample: \(\mathbb R^\omega\) for the uniform topology
Hide sides: yes

Smaller0: \(A\) discrete
Greater0: \(A\) is countable
Page: 190
Chapter: 4
Section: 30
Variables: \(X\) a topological space
\(A\subseteq X\)
Index: 2
Kind: Example
Smaller1: \(X\) has a countable basis

Name: The class of first-countable topologies
Page: 191
Section: 30
Chapter: 4
Index: 2
Kind: Theo
Under: subspace
Under2: product topology
Condition2: countable

Name: The class of second-countable topologies
Page: 191
Section: 30
Chapter: 4
Index: 2
Kind: Theo
Under: subspace
Under2: product topology
Condition2: countable

Name: A dense subset of \(X\)
Definition: A set \(A\subseteq X\) such that \(\overline A=X\).
Page: 191
Chapter: 4
Section: 30
Variables: \(X\) a topological space
Kind: Def
Extra: every open intersect this set ?

Smaller0: \(X\) has a countable basis
Greater0: a countable subcovering
Page: 191
Chapter: 4
Section: 30
Variables: \(X\) a topological space
\(C\) an open covering of \(X\)
Index: 3 (a) (b)
Kind: Theo
Prefix Greater: \(C\) admits
Greater1: countable
Connect Greater0: and there is a dense subset of \(X\) which is

Name: A Lindelöf space
Definition: A topological space in which every open covering contains a countable subcovering
Page: 192
Chapter: 4
Section: 30
Intuition: Compactness, where finite is replaced by countable

Name: A separable space
Definition: A space with a countable dense subset
Page: 192
Chapter: 4
Section: 30

Smaller0: \(X\) is a Lindelöf space
Greater0: \(X\) is separable
Page: 192
Chapter: 4
Section: 30
Variables: \(X\) a topology
Index: , 3
Kind: , Ex
Smaller to greater: iff
Assuming: \(X\) metrizable
Counterexample: \(\mathbb R_\ell\). It is a separable topology with dense subset \(\mathbb Q\), with countable basis \([x,x+1/n)\) at \(x\), is Lindelöf but is not second-countable
Smallest: \(X\) is second-countable

-Separable: Having a countable dense subset
-A separation: A two-partition of \(X\) such that no sets contains a limit point of the other,
-The separation axiom: stating that some kind of sets can be separated.
Page: 192
Chapter: 4
Section: 30
Variables: \(X\) a topology

Q: Product of Lindelöf spaces, which is not Lindelöf: [...]
A: The product of two lower limit topology.
An open covering is \(\mathbb R^2\setminus\{(x,-x)\mid x\in\mathbb R\}\), \([a,b)\times[-a,d]\). It has no countable subcovering
Page: 193
Chapter: 4
Section: 30
Index: 4, 5
Kind: Ex
Name: Lindelöf spaces
Q: Product of Lindelöf spaces, which is not Lindelöf: [...]
Subspace of Lindelöf spaces, which is not Lindelöf: [...]
A: The product of two lower limit topology.
An open covering is \(\mathbb R^2\setminus\{(x,-x)\mid x\in\mathbb R\}\), \([a,b)\times[-a,d]\). It has no countable subcovering; \([0,1]\times(0,1)\) as a subspace of \(I_o^2\). \(\{x\}\times(0,1)\) is an open covering without countable subcovering.

Page: 193
Chapter: 4
Section: 30
Index: 4, 5
Kind: Ex
Name: Lindelöf spaces
Q: [...],
A: binary product
Q: [...]
A: subspace
Q: [...]
A: not
Q: [...]
A: not

Name: The Sorgenfrey plane
Definition: The product of two lower limit topology
Page: 193
Chapter: 4
Section: 30
Kind: Ex

Name: A \(G_\delta\) set
Definition: A countable intersection of open sets of \(X\)
Page: 194
Chapter: 4
Section: 30
Variables: \(X\) a topological space
Index: 1a
Kind: Exercice

Name: A regular topological space
Definition: singletons are closed, and either
Page: 195, 196, https://en.wikipedia.org/wiki/Regular_space
Chapter: 4
Section: 31
Index: , 1(a)
Kind: Def, Lem
Definition2: (for each closed set \(B\) and point \(x\) not in \(B\), there are disjoint open sets containing \(x\) and \(B\) respectively or
Conjdef: A topological space such that
Definition3: for each \(x\in X\) with neighborhood \(U\), there exists a neighborhood \(V\) such that \(\overline V\subseteq U\))
Name2: A T3-space
Counterexample: \(\mathbb R_K\) takes the point \(0\) and the set \(\{\frac 1k\mid k\in\mathbb N\}\)
Construction: 1=>2
\(x\in U\) a neighborhood. \(X-U\) is closed, without \(x\), thus there exists \(V_x\ni x\) and \(V_{X-U}\) disjoint containing \(x\) and \(X-U\) respectively. Thus (not clear)
2=>1
\(x\in X-B\), thus
there is a neighborhood \(x\in V\subseteq\overline V\subseteq X-B\). Then \(V\) and \(X-\overline V\) are disjoint open set containing \(x\) and \(B\)

Name: A normal topological space
Definition: singletons are closed and for each disjoint closed sets \(B_1\) \(B_2\), there are disjoint open sets containing each \(B_i\).
Page: 195, 196, 198, https://en.wikipedia.org/wiki/Normal_space
Chapter: 4
Section: 31
Index: , 1, 2
Kind: Def, Lem, Ex
Definition2: singletons are closed and for each \(A\subseteq X\) closed, with neighborhood \(U\), there exists a neighborhood \(V\) of \(A\) such that \(\overline V\subseteq U\)
Example: \(\mathbb R_\ell\)
Conjdef: A topological space such that
Name2: A space satisfying T4 axiom
Example2: metrizable space
Counterexample: \(\mathbb R_l^2\)
Construction: 1=>2
\(A\subseteq U\) a neighborhood of \(A\). \(X-U\) is closed, without \(A\), thus there exists \(V_A\supseteq x\) and \(V_{X-U}\supseteq X-U\) disjoint containing \(A\) and \(X-U\) respectively. Thus (not clear)
2=>1
\(A\in X-B\), thus
there is a neighborhood \(A\subseteq V\subseteq\overline V\subseteq X-B\). Then \(V\) and \(X-\overline V\) are disjoint open set containing \(A\) and \(B\)
Counterexample2: Trivial space

Smaller0: \(X\) is normal
Greater0: \(X\) is regular
Page: 195, 198
Chapter: 4
Section: 31,
Variables: \(X\) a topological space
Index: 3
Kind: Def, Ex
Counterexample: \(\mathbb R_\ell^2\) (square of Lower limite topology) is a product of regular space, thus regular.
\(\{(r,r)\mid r\not\in\mathbb Q\}\) and \(\{(r,r)\mid r\in\mathbb Q\}\) are disjoint closed subset not separable by any open set (non-trivial proof)
Hide sides: yes

Name: The separation axioms
Page: 195, https://en.wikipedia.org/wiki/Separation_axiom
Chapter: 4
Section: 31
Example: The axiom for Hausdorff, regular and normal spaces
Name2: Tychonoff separation axioms
Intuition: points or sets that are distinguishable or separated in some weak sense must also be distinguishable or separated in some stronger sense.
French: Axiome de séparation

Name: Regular spaces
Page: 195
Section: 31
Chapter: 4
Index: 2 (b)
Kind: Theo
Under: box product
Condition: arbitrary
Under2: subspace
Under3: product topology

Page: 197
Chapter: 4
Section: 31
Index: 1
Kind: Theo, Ex
Name: Normal spaces
Q: Product: [...]
A: \(\mathbb R_l^2\)
Q: [...]
A: product
Q: [...]
A: subspace
Q: [...]
A: not closed
Q: [...]
A: not closed

Name: A closed map
Definition: \(f:X\to Y\) such that for each \(U\subseteq X\) closed, \(f(U)\) is closed.
Page: 137
Chapter: 2
Section: 22
Variables: \(X,Y\) topological spaces.
Kind: Definition

Name: Closed set
Definition: A set which contains all of its limit points.
Page: 098
Chapter: 2
Section: 17
Variables: \(X\) a topological space
Index: 7
Kind: Cor
Definition2: A set whose complement is open.
Example: \(\emptyset\)
Example2: \(X\)

Name: The unit circle
Definition: The set \((\sin x,\cos x)\) for \(x\in\mathbb R\)
Page: 106
Chapter: 18
Section: 1
Notation: \(S^1\)
Definition2: The set of \((x,y)\) such that \(|x|^2+|y|^2=1\)

Name: A metric
Definition: \(d(x,y)\ge 0\), the equality holds iff \(x=y\).
Page: 119
Chapter: 2
Section: 20
Variables: \(X\) a set
Kind: Definition
Definition2: \(d(x,y)=d(y,x)\) and
Conjdef: A function \(d:X\times X\to\mathbb R\) such that
Definition3: \(d(x,y)+d(y,z)\ge d(x,z)\).

Name: A saturated subset of \(X\)
Definition: \(C\) contains all sets \(p^{-1}(y)\) that it intersects.
Page: 137
Chapter: 2
Section: 22
Variables: \((X,\mathcal T),(Y,\mathcal U)\) topologies
\(p:X\to Y\) a surjective map
Definition2: of the form \(p^{-1}(D)\) for some \(D\subseteq Y\)
Conjdef: A set \(C\subseteq X\) such that, equivalently
Definition3: \(C=p^{-1}(p(C))\)

Name: Continuous function
Definition: for each open subset \(V\) of \(Y\), \(f^{-1}(V)\) is open.
Page: 102, 104, 108, 129
Chapter: 2
Section: 18, 21
Variables: \(X,Y\) be topological spaces
Index: , 1, 2(f), 1
Kind: , Theo, Theo, Theo
Definition2: for each subset \(U\) of \(X\), \(f(\overline U)\subseteq\overline{f(U)}\)
Conjdef: A function \(f:X\to Y\) such that
Definition3: for each closed subset \(V\) of \(Y\), \(f^{-1}(V)\) is closed.
Definition4: for each \(x\in X\) and each neighborhood \(V\) of \(f(x)\), there is a neighborhood \(U\) of \(x\) such that \(f(U)\subseteq V\)
Definition5: \(f\) is continous on each element of an open covering
Definition6: for all \(x\in X\), for all \(\epsilon>0\) there exists \(\delta>0\) such that, if \(d_X(x,y)<\delta\) then \(d_Y(f(x),f(y))<\epsilon\) (assuming metric).

Name: A locally compact space
Definition: A topological space locally compact at each of its point
Page: 182, 182, 183, 185, 185
Chapter: 3
Section: 29
Variables: \((X,\mathcal T)\) a topology,
Index: , 2, 1, 2, 4, 3
Kind: Def, Ex, Theo, Theo, cor, Ex
Definition2: given \(x\) and a neighborhood \(U\) of \(x\), there is a neighborhood \(V\) of \(x\) such that \(\overline V\) is compact and \(\overline V\subseteq U\) (assuming Hausdorff)
Example: \(\mathbb R^n\) for \(n\in\mathbb N\)
Definition3: there exists a one-point compactification of \(X\)
Example2: Ordered sets with least-upper bound
Definition4: \(X\) is homeomorphic to an open subspace of a compact Hausdorff space.
Typ: An Hausdorf space
Counterexample: \(\mathbb R^\omega\)
Counterexample2: \(\mathbb Q\)

Smaller0: \(T=\mathbb R^n\) Euclidean or square metric
Greater0: \(X\) is compact
Page: 173
Chapter: 3
Section: 27
Variables: \(T\) a topological space,
\(X\subseteq T\)
Kind: Theo
Hide middle: yes
Greater1: \(X\) is closed and bounded
Connect Greater0: iff
Counterexample: \(\mathbb Q\)

Smaller0: \(X\) is compact
Greater0: closed
Page: 173
Chapter: 3
Section: 27
Variables: \(\mathcal T\) a topology
\(X\subseteq\mathcal T\).
Kind: Theo
Smaller to greater: iff
Hide middle: yes
Prefix Greater: \(X\) is
Assuming: \(\mathcal T=\mathbb R^n\) for some \(n\)
Greater1: bounded
Connect Greater0: and \(X\) is
Context: Euclidean or square metric
Counterexample: Take an infinite set and discrete measure. It is bounded, closed but not compact

Smaller0: nonempty
Greater0: \(X\) is uncountable
Page: 176
Chapter: 3
Section: 27
Variables: \(X\) a topological space
Index: 7
Kind: Theo
Smaller1: compact
Prefix Smaller: \(X\)
Smaller2: Hausdorff
Smaller3: without isolated point

Name: A strongly continuous function
Definition: A function \(f:X\to Y\) such that \(A\subseteq Y\) is open iff \(f^{-1}(A)\) is open.
Page: 137
Chapter: 2
Section: 22
Variables: \((X,\mathcal T)\), \((Y,\mathcal U)\) be topologies.
Definition2: A function \(f:X\to Y\) such that \(A\subseteq Y\) is closed iff \(f^{-1}(A)\) is closed
Definition3: A continuous function open or closed (i.e. sending open/closed to open/closed)

Definition: The set \(\{x\in X\mid x\le a\}\).
Page: 085
Chapter: 2
Section: 14
Variables: \(X\) be an ordered set
\(a\in X\)
Notation: \((-\infty,a]\)
Kind: Definition

Smaller0: \(Z\) is closed in \(Y\)
Greater0: \(Z\) is compact
Page: 165
Chapter: 3
Section: 26
Variables: \(X\) a topological space
\(Z\subseteq Y\subseteq X\)
Index: 2
Kind: Theo
Smaller1: \(Y\) is compact
Connect Smaller0: and

Smaller0: \(\prec\) is strict
Greater0: there exists a maximal simply ordered subset \(B\) of \(A\)
Page: 069
Chapter: 1
Section: 11
Variables: \(A\) be a set;
\(\prec\) be a partial order on \(A\)
Name: The maximum principle

Smaller0: \(X\) satisfies the \(T_1\) axiom
Greater0: every neighborhood of \(x\) contains infinitely many points of \(A\)
Page: 099
Chapter: 2
Section: 17
Variables: \(X\) be a topological space;
\(A\subseteq X\)
\(x\in X\).
Index: 9
Kind: Theo
Hide middle: yes
Greater1: \(x\) is a limit point of \(A\)
Connect Greater0: iff
Extra: Counter-example ?

Smaller0: \(f(a)<r<f(b)\)
Greater0: there exists \(c\in X\) such that \(f(c)=r\)
Page: 154
Chapter: 3
Section: 24
Variables: \(X\) connected
\((Y,<)\) an ordered set
\(f:X\to Y\)
\(a,b\in X\), \(r\in Y\)
Kind: Theo
Hide middle: yes
Assuming: \(f\) continuous
Name: Intermediate value theorem

Smaller0: a closed interval
Greater0: compact
Page: 172
Chapter: 3
Section: 27
Variables: \((X,<)\),
\(S\subseteq X\)
Index: 1
Kind: Theo
Hide middle: yes
Prefix Greater: \(S\) is
Assuming: \(X\) has the least upper bound property

Name: A quotient map
Definition: strongly continuous
Page: 137
Chapter: 2
Section: 22
Variables: \(X,Y\) topological spaces.
Definition2: maps open saturated sets to open sets
Conjdef: A surjective function, equivalently

Question: Topologically, \(f_{\mid Y}\) is
Answer: continuous.
Page: 108
Chapter: 2
Section: 18
Variables: \(X,Y,Z\) be topological spaces and
\(Y\subseteq Z\).
\(f:X\to Y\) continuous.
Index: 2 d
Kind: Theorem

Name: Box product of continuous function which is not continuous
Page: 117
Chapter: 2
Section: 19
Kind: Ex
Example: \(\mathbf J=\mathbb N\),
\(A=X_\alpha=\mathbb R\),
\(f_\alpha(x)=x\)
Let \(B=\prod_{n\in\mathbb N}\left(-\frac 1n,\frac 1n\right)\), it is open in box product.
 \(f^{-1}(B)\) is not open

Name: Normal space
Page: 197
Section: 31
Chapter: 4
Not: not
Under: product
CounterExample: Sorgenfrey plane \(\mathbb R_\ell^2\) (square of lower limit topology) is not normal; but \(\mathbb R_\ell\) is normal
Under2: subspace
CounterExample2: Deleted Tychonoff plank is non-normal subset of the Tychonoff plank (which is normal)