Title: Theory / Unimportant / Math / Topology / Munkres(+2 if from paper) / 2 Algebraic=
Tags: récupéré fromMath mid1516974248815 toUnary closed examplesSwitchExample1 fromImplication mathToSee typToBinary Wrong_taylor_series defToAlgebra defToAlgebra2 fromRelation hasSeenConjDef toBinaryFunction toBinaryRelation

Page: 323, 324
Definition: The set of pairs \(f,f':X\to Y\) continuous such that there exists an homotopy from \(f\) to \(f'\)
Chapter: 9
Section: 51
Variables: \(X,Y\) topological spaces,
Notation: \(\simeq\)
Name: Homotopic maps
Kind: Def, Lem
Index: , 1
Context: Maps
Equivalence: Equivalence relation
Example: \((\cos t,\sin t) \) and \((\cos t,2\sin t)\) in \(\mathbb R^2\setminus\{0\}\)
counterExample: \((\cos t,\sin t) \) and \((\cos t,-\sin t)\) in \(\mathbb R^2\setminus\{0\}\)

Name: An homotopy from \(f\) to \(f'\)
Definition: A continuous map \(F:X\times I\to Y\) with \(F(x,0)=f(x)\) and \(F(x,1)=f'(x)\)
Page: 323
Chapter: 9
Section: 51
Variables: \(X,Y\) topological spaces,
\(f,f':X\to Y\) continuous
Kind: Def
Intuition: Deforming \(Y\) such that \(f\) becomes \(f'\)

Name: A nulhomotopic map
Definition: A map homotopic to a constant map
Page: 323, 350
Chapter: 9
Section: 51, 55
Variables: \(X,Y\) topological spaces,
Index: 4
Kind: Def, Cor
Counterexample: identity map \(S^1\to S^1\)
Counterexample2: inclusion map \(j:S^1\to\mathbb R^2\setminus 0\)

Name: A path homotopy between \(f\) and \(f'\)
Definition: An homotopy \(F:I\times I\to X\) between \(f\) and \(f'\) such that \(F(0,t)=f(0)\) and \(F(1,t)=f(1)\).
Page: 323
Chapter: 9
Section: 51
Variables: \(X\) a topological space
\(f,f':I\to X\)
Kind: Def
Assuming: \(f\) and \(f'\) have the same initial and final points

Page: 323, 324
Definition: A pair of path into \(X\), with a path homotopy between them.
Chapter: 9
Section: 51
Variables: \(X\) a topological space
Notation: \(\simeq_p\)
Name: Path homotopic paths
Kind: Def, Lem
Index: ,1
Equivalence: yes

Name: The straight-line homotopy between \(f\) and \(f'\)
Definition: \(F(x,t)=(1-t)f(x)+tf'(x)\)
Page: 325
Chapter: 9
Section: 51
Variables: \(X\) a topological space
\(f,f':X\to\mathbb R^n\)
Index: 1
Kind: Ex
Assuming: \(f(X)\) and \(g(X)\) belongs to a convex of \(R\)

Name: The punctured plane
Definition: \(\mathbb R^2-\{0\}\)
Page: 325
Chapter: 9
Section: 51
Notation: \(\mathbb R^2-0\)
Index: 2
Kind: Ex

Page: 326
Section: 51
Notation: \(f\star g\)
Variables: \(X\) a topological space
\(f,g\) pathes into \(X\)
Chapter: 9
Name: The product the paths \(f\) and \(g\)
Definition: \(\left\{\begin{array}{ll}f(2s)&\text{for }s\in \left[0,\frac 12\right]\\g(2s-1)&\text{for }s\in\left[\frac 12,1\right]\end{array}\right.\)
Kind: Ex
Index: 2
Context: Path, topology
Assuming: The final point of \(f\) is the initial point of \(g\)

Name: Associativity for groupoids
Definition: If \(f\star(g\star h)\) is defined, so is \((f\star g)\star h\) and they are equal
Page: 326
Chapter: 9
Section: 51
Variables: \(G\) a set
\(\star\) a partial function \(G\times G\to G\)
Index: 2 (1)
Kind: Theo

Name: Inverse axiom for groupoids
Definition: \(a\cdot a^{-1}\) and \(a^{-1}\cdot a\) are defined
Page: 327
Chapter: 9
Section: 51
Variables: \(G\) a set,
\(\star\) a partial function \(G\times G\to G\),
\(\cdot^{-1}:G\to G\)
Index: 2. (3)
Kind: Theo

Name: Left identity for groupoids
Definition: \(a^{-1}\star a\star b=b\)
Page: 327
Chapter: 9
Section: 51
Variables: \(G\) a set,
\(\star\) a partial function \(G\times G\to G\),
\(\cdot^{-1}:G\to G\)
Index: 1
Kind: Theo

Name: Right identity for groupoids
Definition: \(b\star a\star a^{-1}=b\)
Page: 327
Chapter: 9
Section: 51
Variables: \(G\) a set,
\(\star\) a partial function \(G\times G\to G\),
\(\cdot^{-1}:G\to G\)
Index: 1
Kind: Theo

Name: A groupoid
Definition: a set \(G\)
Page: 326
Chapter: 9
Section: 51
Index: 2
Kind: Theo
Definition2: an operation \(\cdot ^{-1}:G\to G\)
Example: For \(X\) a topological space:
The set of paths to \(X\) with the product \(\star\)
Conjdef: A tuple
Definition3: partial operation \(G\times G\to G\) which satisfies the associativity, identity and inverse axioms

Definition: The constant path [$]\defFun{I}{X}{\cdot}{x}[/$]
Page: 327
Chapter: 9
Section: 51
Variables: \(X\) a topological set,
\(x\in X\)
Context: Topology
Notation: \(e_x\)
Index: 2 (2)
Kind: Theo

Name: The reverse of \(f\)
Definition: \(x\mapsto f(1-x)\)
Page: 327
Chapter: 9
Section: 51
Variables: \(X\) a topological set,
\(f\) a path into \(X\)
Notation: \(\overline f\)
Index: 2 (3)
Kind: Theo

Question: The set of paths to \(X\) with the product \(\star\) is
Answer: a groupoid
Page: 327
Chapter: 9
Section: 51
Variables: \(X\) a topological space
Index: 2
Kind: Theo

Question: \([f_1]\star\dots\star [f_n]\)
Answer: equals \([f]\)
Page: 329
Chapter: 9
Section: 51
Variables: \(X\) a topological space
\(0=a_0<\dots< a_n=1\)
\(f\) a path into \(X\)
\(f_i\) the path that equals the positive linear map of \(I\) onto \([a_{i-1},a_i]\) followed by \(f\)
Index: 3
Kind: Theo

Name: A contractible topological space
Definition: A space whose identity map is nulhomotopic
Page: 330
Chapter: 9
Section: 51
Index: 3
Kind: Ex
Example: \(\mathbb R^n\)
Counterexample: \(\mathbb R^n\setminus\{0\}\)

Name: The standard definition of group theory are
Definition: omitted
Page: 330
Chapter: 9
Section: 52

Name: A loop based at \(x_0\)
Definition: A path in \(X\) that begins and ends at \(x_0\)
Page: 331
Chapter: 9
Section: 52
Variables: \(X\) a topological space
\(x_0\in X\)

Page: 331
Chapter: 9
Section: 52
Name: The fundamental group of \(X\) relative to the base point \(x_0\)
Set: The set of path-homotopy loop based at \(x_0\)
Variables: \(X\) a topological space
\(x_0\in X\)
Notation: \(\pi_1(X,x_0)\)
*: the \(\star\) operations of composition of loop
Kind: , Ex
Index: , 1
÷: Inverse of loop
Typ: A group
Name2: First homotopy group of \(X\)
Example: For \(X\) a convex subset of \(\mathbb R^n\) and \(x_0\) arbitrary, the group is trivial
Example2: For \(S^1\), it is isomorphic to \(\mathbb Z\)

Definition: [$]\defFun{\pi_1(X,x_0)}{\pi_1(X,x_1)}{\lbrack f\rbrack}{\lbrack \overline \alpha]\star[f]\star [\alpha\rbrack}[/$]
Page: 331, 332
Section: 52
Chapter: 9
Variables: \(X\) a topological space
\(\alpha\) a path in \(X\) from \(x_0\) to \(x_1\)
Notation: \(\hat\alpha\)
Index: Def, Theo
Kind: ,1
Morphism: Group isomorphism
Intuition: Using the path \(\alpha\) to go to \(x_1\), following the path \(f\) and then using \(\overline\alpha\) to go back to \(x_0\), thus transforming a \(x_1\) loop into a \(x_0\) loop

Smaller0: \(x_0,x_1\) are path connected
Greater0: \(\pi(X,x_0)\) is isomorphic to \(\pi(X,x_1)\)
Page: 332
Chapter: 9
Section: 52
Variables: \(X\) a topological space,
\(x_0,x_1\in X\)
Index: 2
Kind: Cor
Context: Algebraic Topology
Counterexample: Let \(x_0,x_1\) be in two disjoint identical copy of a connected element

Name: A simply connected space
Definition: A path-connected space such that \(\pi_1(X,x_0)\) is trivial for some \(x_0\in X\)
Page: 333
Chapter: 9
Section: 52
Variables: \(X\) a topological space
Notation: \(\pi_1(X,x_0)=0\)
Kind: Def

Smaller0: simply connected
Greater0: homotopic
Page: 333
Chapter: 9
Section: 52
Variables: \(X\) a topological space
\(f,g:I\to X\) with the same initial and final points
Index: 3
Kind: Lem
Prefix Greater: \(f,g\) are
Prefix Smaller: \(X\) is

Definition: The set of map \(X_0\to X_1\) sending \(x_0\) to \(x_1\)
Page: 333
Chapter: 9
Section: 52
Variables: \(X_0,X_1\) topological spaces
\(x_i\in X_i\)
Notation: \((X_0,x_0)\to (X_1,x_1)\)
Kind: Def

Name: The homomorphism induced by \(h\) relative to the base point \(x_0\)
Definition: [$]\defFun{\pi_1(X_0,x_0)}{\pi_1(X_1,x_1)}{\lbrack f]}{\lbrack h\circ f]}[/$]
Page: 333, 334
Chapter: 9
Section: 52
Variables: \(X_0,X_1\) topological spaces
\(x_i\in X_i\)
\(h:(X_0,x_0)\to (X_1,x_1)\)
Notation: \(h_\star\)
Index: , 4
Kind: Def, Theo
Intuition: Transforming a loop at \(x_0\) into another loop by applying \(h\) to it
Typ: Groupoid morphism
Notation2: \((h_{x_0})_\star\)

Name: A star convex subset of \(\mathbb R^n\)
Definition: A subset \(A\subseteq\mathbb R^n\) such that there exists a point \(a\in A\) such that for each \(b\in A\), \([a,b]\subseteq A\)
Page: 334
Chapter: 9
Section: 52
Index: 1
Kind: Exercice

Name: A retraction of \(X\) onto \(A\)
Definition: A continuous map \(X\to A\) which is the identity on \(A\)
Page: 348
Chapter: 9,
Section: 55
Variables: \(X\) a topological space
\(A\subseteq X\)
Index: 4
Kind: Def
Extra: And not retracTAtion

Name: A set evenly covered by \(p\)
Definition: there exists an open partition \(\{V_\alpha\}\) of \(p^{-1}(U)\) such that \(p:V_\alpha\to U\) is a homeomorphism
Page: 336
Chapter: 9
Section: 53
Variables: \(p:E\to B\) continuous
Kind: Def
Definition2: \(U\) is open
Conjdef: A set \(U\subseteq B\) such that
Assuming: \(p\) surjective

Name: A partition of \(p^{-1}(U)\) into slices
Definition: \(p\) is an homeomorphism on \(U\).
Page: 336
Chapter: 9
Section: 53
Variables: \(p:E\to B\) continuous surjective,
\(U\subseteq B\) a set evenly covered \(p\)
Kind: Def
DefType: A partition \(\{V_\alpha\}\) of \(p^{-1}(U)\)

Name: The set of evenly covered subsets of \(B\)
Page: 336
Section: 53
Chapter: 9
Variables: \(p:E\to B\) continuous surjective,
Kind: Def
Under: taking subset
Condition: open
CounterExample: by definition an evenly covered set must be open

Name: A covering map
Definition: every point \(b\in B\) has an evenly covered neighborhood.
Page: 336, 337, 338
Chapter: 9
Section: 53
Index: , 1, 2
Kind: Def, theo, ex
Example: [$]\defFun{\mathbb R}{S^1}{x}{(\cos x,\sin x)}[/$]
Example2: [$]\defFun{S^1}{S^1}{z}{z^2}[/$]
DefType: A continuous surjective map \(p:E\to B\)

Name: A covering space of \(B\)
Definition: A topological space which is the domain of definition of a covering map to \(B\).
Page: 336
Chapter: 9
Section: 53
Variables: \(B\) a topological space
Kind: Def
Denoted by: \(E\)

Question: \(p^{-1}(b)\)'s topology is
Answer: discrete
Page: 336
Chapter: 9
Section: 53
Variables: \(B\) a topological space,
\(b\in B\),
\(p:E\to B\) a covering map
Kind: Def

Smaller0: \(p\) is a covering map
Greater0: \(p\) is a local homeomorphism
Page: 336
Chapter: 9
Section: 53
Variables: \(p\) a continuous map
Index: last paragraph
Greater1: \(p\) is open
Counterexample: [$]\defFun{\mathbb R_+}{S^1}{x}{(\cos x,\sin x)}[/$]
Hide sides: yes

Question: \(p_{\mid p^{-1}(B_0)}\) is
Answer: a covering map.
Page: 338
Chapter: 9
Section: 53
Variables: \(p:E\to B\) a covering map,
\(B_0\subseteq B\) a subspace
Index: 2
Kind: Theo

Name: Covering map
Page: 338, 339
Section: 53
Chapter: 9
Index: 2, 3
Kind: Theo
Under: product
Condition: finite
Under2: subset
Condition2: saturated

Name: The torus
Definition: \(S^1\times S^1\)
Page: 339
Chapter: 9
Section: 53
Index: 4
Kind: Ex
Denoted by: \(T\)

Name: The figure-eight space
Definition: \((S^1\times \{a\})\cup(\{a\}\times S^1)\) for some \(a\in S^1\)
Page: 340
Chapter: 9
Section: 53
Index: 5
Kind: Ex
Definition2: Two copies of \(S^1\) with a point in common.
Definition3: Bouquet of 2-roses

Name: The infinite grid
Definition: The set of \((x,y)\in\mathbb R^2\) with \(x\) or \(y\) integral
Page: 340
Chapter: 9
Section: 53
Index: 5
Kind: Ex
Definition2: \((\mathbb Z\times\mathbb R)\cup(\mathbb Z\times\mathbb R)\)

Name: Standard homeomorphism of \(S^1\times\mathbb R_+\) with \(\mathbb R^2-0\).
Definition: \((x,t)\mapsto tx\)
Page: 340
Chapter: 9
Section: 53
Index: 6
Kind: Ex

Name: A lifting of \(f\)
Definition: A map \(g:X\to E\) such that \(p\circ g=f\)
Page: 342
Chapter: 9
Section: 54
Variables: \(X,E,B\) topological spaces,
\(p:E\to B\) a map
\(f:X\to B\) continuous
Notation: \(\tilde f\)

Smaller0: \(f,g\) are path homotopics
Greater0: end at the same points
Page: 344
Chapter: 9
Section: 54
Variables: \(X,E,B\) topological spaces,
\(p:E\to B\) a covering map
\(e_0\in E\),
\(f,g:[0,1]\to B\) continuous with same endpoints
Index: 3
Kind: Theo
Prefix Greater: the lifting of \(f\) and \(g\)
Greater1: are path homotopic

Definition: [$]\defFun{\pi_1(B,p(e_0))/p_*(\pi_1(E,e_0))}{p^{-1}(p(e_0))}{\lbrack f\rbrack}{\text{the end point $\tilde f(1)$ of $\tilde f$}}[/$] with \(\tilde f\) the lifting of \(f\)
Page: 346
Section: 54
Chapter: 9
Variables: \(X,E,B\) topological spaces,
\(p:E\to B\) a covering map
\(e_0\in E\),
Notation: \(\Phi\)
Index: Theo
Kind: 6(b)
Intuition: Similar to \(\phi\): given an homotopic-equivalence class of loop of \(\pi_1(B,b_0)\), it follows its pre-image in \(E\). The difference being that the equivalence relation consider that the image of loops in \(E\) are equivalent
Jective: Injective

Name: The lifting correspondence
Definition: [$]\defFun{\pi_1(B,p(e_0))}{p^{-1}(p(e_0))}{ {[f]} }{\text{the end point $\tilde f(1)$ of $\tilde f$}}[/$]
Page: 345
Chapter: 9
Section: 54
Variables: \(X,E,B\) topological spaces,
\(p:E\to B\) a covering map
\(e_0\in E\),
Kind: Def
Intuition: The other end of the loop \(f\), when it is applied on the covering space \(E\).
Denoted by: \(\phi\)

Smaller0: \(E\) is path connected
Greater0: surjective
Page: 345, 346
Chapter: 9
Section: 54
Variables: \(X,E,B\) topological spaces,
\(p:E\to B\) a covering map
\(e_0\in E\),
Index: 4,6 (b)
Kind: Theo
Prefix Greater: \(\phi\) is
Greater1: bijective
Connect Greater0: and \(\Phi\) is

Smaller0: \(E\) is simply connected
Greater0: bijective
Page: 345
Chapter: 9
Section: 54
Variables: \(X,E,B\) topological spaces,
\(p:E\to B\) a covering map
\(e_0\in E\),
Index: 4
Kind: Theo
Prefix Greater: the lifting correspondence is

Page: 345
Name: Circle
Chapter: 9
Section: 54
Kind: Theo
Index: 5
Notation: \(S^1\)
Fundamental group: \((\mathbb Z,+)\)

Definition: [$]\defFun{\pi_1(E,e_0)}{\pi_1(B,p(e_0))}{f:I\to[E]}{\defFun{I}{B}{x}{p(x)}}[/$]
Page: 346
Section: 54
Chapter: 9
Variables: \(X,E,B\) topological spaces,
\(p:E\to B\) a covering map
\(e_0\in E\),
Notation: \(p_*\)
Index: Theo
Kind: 6
Morphism: Mono
Intuition: Extension of \(p\) from points to paths

Question: The number of lifting of \(f\) to a path \(\tilde f\) in \(E\) beginning at \(e_0\)
Answer: is exactly one
Page: 324
Chapter: 9
Section: 54
Variables: \(E,B\) topological spaces,
\(p:E\to B\) a covering map
\(e_0\in E\),
\(f:[0,1]\to B\) continuous
Index: 1
Kind: Lem
Assuming: \(f(0)=p(e_0)\)
Construction: Take \(U\) a open covering of \(B\) each having a even covering.
Using Lebesgue number lemma to find a finite subset of \(U\) containing \(Im(f)\) (use \([0,1]\)'s distance over \(Im(F)\))
Let \(f(0)=s_0,\dots,s_n=f(1)\) such that \(f([s_{i},s_{i+1}])\) belongs to an open cover \(U_i\).
\(\tilde f\) is defined by induction on \(i\) from \(0\) to \(n-1\), with \(\tilde f(x)=p_{U_i}^{-1}(f(x))\)

Question: The number of lifting of \(f\) to a path \(\tilde f\) in \(E\) with \(\tilde f(0,0)=e_0\)
Answer: is exactly one
Page: 343
Chapter: 9
Section: 54
Variables: \(X,E,B\) topological spaces,
\(p:E\to B\) a covering map
\(e_0\in E\),
\(f:[0,1]\times [0,1]\to B\) continuous
Index: 2
Kind: Lem
Assuming: \(f(0,0)=p(e_0)\)

Question: The set of retraction of \(B^2\) onto \(S^1\) is
Answer: empty.
Page: 348
Chapter: 9
Section: 55
Index: 2
Kind: Theo
Name: No retraction theorem

Smaller0: \(h\) is nulhomotopic
Greater0: \(h\) extends to a continuous map \(k:B^2\to X\)
Page: 349
Chapter: 9
Section: 55
Variables: \(X\) a topological space
\(h:S^1\to X\)
Index: 3
Kind: Lem
Smaller to greater: iff
Intermediate: \(h_*\) is the trivial homomorphism of fundamental groups (i.e. its the constant morphism to the identity element)

Smaller0: continuous
Greater0: there are points of \(S^1\) pointing inward and outward
Page: 350
Chapter: 9
Section: 55
Variables: \(j:B^2\to\mathbb R^2\)
Index: 5
Kind: Theo
Smaller1: non vanishing
Prefix Smaller: \(j\) is

Name: A vector field on \(X\)
Definition: A continuous map \(X\to\mathbb R^2\)
Page: 350
Chapter: 9
Section: 55
Variables: \(X\) a set
Index: 5
Kind: Theo

Name: A non-vanishing map
Definition: A map whose image does not contains 0.
Page: 350
Chapter: 9
Section: 55
Index: 5
Kind: Theo

Smaller0: \(f\) continuous
Greater0: \(f\) admits a fixed-point
Page: 351
Chapter: 9
Section: 55
Variables: \(X\) a topological space
\(f:X\to X\)
Index: 6
Kind: Theo
Smaller1: \(X=B^2\)
Name: Brouwer fixed-point theorem

Name: A fixed point of \(f\)
Definition: An element \(x\in X\) such that \(f(x)=x\)
Page: 351
Chapter: 9
Section: 55
Variables: \(X\) a topological space
\(f:X\to X\)
Index: 6
Kind: Theo

Smaller0: \(A\) is \(3\times 3\)
Greater0: \(A\) has a positive real eigenvalue
Page: 351
Chapter: 9
Section: 55
Variables: \(A\) a real square matrix
Index: 7
Kind: Cor
Smaller1: \(A\) contains only positive elements
Construction: Take \(T\) the map induced by \(A\) on the octant of non-negative real.
[$]x\mapsto \frac{T(x)}{\norm{T(x)}}[/$] is a continuous map of an element homeomorphic to \(B_2\) to itself, thus has a fixed-point \(x\).
Take the eigenvalue [$]\norm{T(x)}[/$], since [$]x=\frac{T(x)}{\norm{T(x)}}[/$]

Smaller0: the elements of \(A\) have diameter at most \(\epsilon\), for \(\epsilon\) small enough
Greater0: three elements of \(\mathcal A\) intersects
Page: 352
Chapter: 9
Section: 55
Variables: \(T=\{(x,y)\mid x,y\ge 0, x+y\le 1\}\),
\(\mathcal A\) an open covering of \(T\)
Index: 8
Kind: Theo
Extra: read after paragraph 36 and 50

Smaller0: \(n>0\)
Greater0: \(p\) has at least one root
Page: 354
Chapter: 9
Section: 56
Variables: \(p(x)=x^n+\sum_{i=0}^{n-1} x^ia_i\)
Index: 1
Kind: Theo
Name: Fundamental theorem of algebra
Counterexample: \(x \mapsto0\)
French: théorème de d'Alembert(-Gauss)

Name: A root of \(p\)
Definition: An element \(x\) such that \(p(x)=0\)
Page: 354
Chapter: 9
Section: 56
Variables: \(p(x)=x^n+\sum_{i=0}^{n-1} x^ia_i\)
Index: 1
Kind: Theo

Name: The antipode of \(x\)
Definition: \(-x\)
Page: 356
Chapter: 9
Section: 57
Variables: \(n\in\mathbb N\)
\(x\in S^n\)
Context: Hypersphere

Name: Antipode-preserving function
Definition: A function \(S^n\to S^n\) such that \(h(-x)=-h(x)\).
Page: 356
Chapter: 9
Section: 57
Variables: \(n\in\mathbb N\)

Smaller0: \(h\) is not continuous
Greater0: \(h\) is not nulhomotopic
Page: 356
Chapter: 9
Section: 57
Variables: \(h:S^1\to S^1\)
Index: 1
Kind: Theo
Smaller to greater: or
Intermediate: \(h\) is not antipode-preserving

Smaller0: \(h\) is not continuous
Greater0: \(h\) is not antipode-preserving
Page: 357
Chapter: 9
Section: 57
Variables: \(h:S^2\to S^1\)
Index: 2
Kind: Theo
Smaller to greater: or

Question: there is a point \(x\in S^2\) such that
Answer: \(f(x)=f(-x)\)
Page: 358
Chapter: 9
Section: 57
Variables: \(f:S^2\to\mathbb R^2\)
Index: 3
Kind: Theo
Name: Borsuk-Ulam theorem for \(S^2\)
Construction: [$]g=\defFun{S^2}{S^1}{x}{\frac{[f(x)-f(-x)]}{\norm{f(x)-f(-x)}}}[/$] is continuous and antipode-preserving, contradiction; thus such \(g\) is not well-defined, hence \(f(x)=f(-x)\) for some \(x\)

Question: there exists
Answer: a line that bisects both \(A_i\)
Page: 358
Chapter: 9
Section: 57
Variables: \(A_1,A_2\subseteq\mathbb R^2\) regions in \(\mathbb R^2\)
Index: 4
Kind: Theo
Assuming: \(A_1,A_2\subseteq\mathbb R^2\) polygonal
Name: Bisection theorem

Name: A line bisecting \(A\)
Definition: A line such that the measure of \(A\cap H_1\) and \(A\cap H_2\) are equal, with \(H_2\) the half-plane defined by this line.
Page: 358
Chapter: 9
Section: 57
Variables: \(A\subseteq\mathbb R^2\) a bounded polygonal
Index: 4
Kind: Theo

Smaller0: \(A_i\) are Jordan measurable
Greater0: there exists an hyperplan bisecting each \(A_i\)
Page: 358
Chapter: 9
Section: 57
Variables: \(A_1,\dots,A_{n-1}\subseteq\mathbb R^n\)
Index: 4
Kind: Theo

Smaller0: \(h^1,h^2\) continuous
Greater0: \((h^1_{x_0})_{*}=(h_{x_0}^2)_*\)
Page: 360
Chapter: 9
Section: 58
Variables: \(X_1,X_2\) topologies
\(h^1,h^2:(X_1,x_{1,0})\to (X_2,x_{2,0})\)
Index: 1
Kind: Lem
Smaller1: there is an homotopy from \(h^1\) to \(h^2\) fixing \(x_0\)
Construction: for an homotopy \(H\), and a loop \(f\in\pi_1(X,x_0)\), [$]\defFun{I\times I}{Y}{(a,b)}{H(f(a),b)}[/$] is a homotopy between \(h^1\circ f\) and \(h^2\circ f\)

Smaller0: the fundamental groups of \(S^n\)
Greater0: the fundamental groups of \(\mathbb R^{n+1}\setminus\{0\}\)
Page: 360
Chapter: 9
Section: 58
Variables: \(n\in\mathbb N\)
Index: 2
Kind: Theo
Smaller to greater: is an isomorphic to fundamental group to
Construction: takes as isomorphism the map induced by \(S^n\to\mathbb R^{n+1}\) and [$]\defFun{\mathbb R^{n+1}}{S^n}{x}{\frac x{\norm{x}}}[/$]

Name: A deformation retract of \(X\)
Definition: A subspace \(A\subseteq X\) such that there is a deformation retraction of \(X\) onto \(A\)
Page: 361
Chapter: 9
Section: 58
Variables: \(X\) a topological space

Name: A deformation retraction from \(X\) to \(A\)
Definition: An homotopy \(H:X\times I\to{X}\) such that \(H(x,0)=x\), \(H(x,1)\in A\) and \(H(a,1)=a\) for \(a\in A\)
Page: 361
Chapter: 9
Section: 58
Variables: \(X\) a topological space
\(A\subseteq X\)
Typ: An homotopy from \(1_X\) to [$]\defFun{X}{X}{x}{H(x,1)}[/$]
Applied to3: An homotopy from \(1_X\) to a map \(X\to A\) being identity on \(A\)

Question: [$]\defFun{X}{A}{x}{H(x,1)}[/$] is
Answer: a retraction from \(X\) onto \(A\)
Page: 361
Chapter: 9
Section: 58
Variables: \(X\) a topological space
\(A\subseteq X\)
\(H\) a deformation retraction from \(X\) to \(A\)

Name: Path homotopy equivalence relation
Page: 326
Section: 51
Chapter: 9
Under: product of paths

Question: Difference between homotopy and path homotopy
Answer: In path homotopy, the two bounds of the path remains fixed
Page: 324
Chapter: 9
Section: 51

Definition: The equivalence class of \(f\) in \(\pi_1(B,b_0)\)
Page: 345
Chapter: 9
Section: 54
Variables: \(B\) a topological space
\(b_0\in B\)
\(f\) a loop at \(b_0\)
Notation: \([f]\)
Kind: Def