TOPOLOGIES Flashcards
UNE FICHES DE REVISION POUR MON EXAMEN ET ME PROPOSER DES EXO DE LA MOIN COMPLEXES E LA PLUS DIFFICILE
Qu’est-ce qu’une distance sur un ensemble E?
Une application d : E × E → R+ vérifiant :
* d(x, y) = 0 ⇐⇒ x = y
* d(x, y) = d(y, x) (symétrie)
* d(x, z) ≤ d(x, y) + d(y, z) (inégalité triangulaire)
Ces conditions sont essentielles pour définir une distance.
Comment définit-on un espace métrique?
Un ensemble E muni d’une distance d.
Cela implique que les éléments de E peuvent être mesurés par la distance d.
Quel est un exemple de distance usuelle sur R?
d(x, y) = |x - y|.
Cette distance est communément utilisée dans l’analyse réelle.
Quelle est la définition de la distance discrète?
d(x, y) =
* 1 si x ≠ y
* 0 si x = y.
Cette distance ne mesure que si les points sont égaux ou non.
Qu’est-ce qu’une distance équivalente?
Deux distances d1 et d2 sur un ensemble E sont équivalentes s’il existe α, β ∈ R*+ tels que
* αd1(x, y) ≤ d2(x, y) ≤ βd1(x, y).
Cela signifie que les deux distances mesurent des concepts similaires de proximité.
Qu’est-ce qu’un sous-espace métrique?
Une partie non vide F d’un espace métrique (E, d) avec la restriction dF = d|F × F.
Le sous-espace hérite de la structure métrique de l’espace E.
Comment définit-on une boule ouverte dans un espace métrique?
B(a, r) = {x ∈ E / d(a, x) < r}.
Cela représente tous les points x qui sont à une distance inférieure à r du point a.
Quelle est la définition d’une boule fermée?
Bf(a, r) = {x ∈ E / d(a, x) ≤ r}.
Cette définition inclut le point a et tous les points à une distance r ou moins.
Qu’est-ce qu’une sphère dans un espace métrique?
S(a, r) = {x ∈ E / d(a, x) = r}.
La sphère contient tous les points qui sont exactement à une distance r du point a.
Comment définit-on le diamètre d’une partie A d’un espace métrique?
δ(A) = sup {d(x, y) / (x, y) ∈ A × A}.
Le diamètre mesure la plus grande distance entre deux points dans A.
Quand dit-on qu’une partie A est bornée?
Si δ(A) est fini.
Cela signifie qu’il existe une distance maximale entre les points de A.
Qu’est-ce qu’une application bornée?
Une application f : X → E est bornée si f(X) est une partie bornée de (E, d).
Cela signifie que les images de tous les éléments de X dans E ne s’étendent pas à l’infini.
Quelle est la distance de convergence uniforme entre deux applications f et g?
d∞(f, g) = sup {d(f(x), g(x)) / x ∈ X}.
Cette distance mesure la plus grande différence entre les valeurs des deux fonctions sur l’ensemble X.
What is the distance d∞ on Fb(X, E)?
d∞(f, g) = sup d(f(x), g(x)) for x ∈ X
This distance is called the distance of uniform convergence.
What defines an open set O in a metric space (E, d)?
For every a ∈ O, there exists a ball B(a, r) contained in O for some r > 0
What is the definition of a closed set F in a metric space (E, d)?
The complement of F in E is an open set in (E, d)
What are the open sets in a metric space?
E and ∅ are open sets in (E, d)
What is the proposition regarding open balls in a metric space?
Every open ball B(a, r) in (E, d) is an open set
Fill in the blank: The complement of a closed ball Bf(a, r) is an ______.
open set
What is the relationship between open intervals and open sets in R?
Every open interval in R is an open set
What is the relationship between closed intervals and closed sets in R?
Every closed interval in R is a closed set
True or False: The negation of ‘A is an open set’ is ‘A is a closed set’.
False
Provide an example of a set that is neither open nor closed in R.
A = [0, 1[ is neither open nor closed
What is the proposition about the union of open sets?
The union of any family of open sets is an open set
What is the corollary regarding spheres in a metric space?
The sphere S(a, r) is a closed set in (E, d)
What is true about the intersection of a finite number of open sets?
The intersection of a finite number of open sets is an open set
What is the proposition regarding closed sets?
The intersection of any family of closed sets is a closed set
What are the three fundamental properties of open sets in a metric space?
- E and ∅ are open sets
- Any union of open sets is an open set
- Any finite intersection of open sets is an open set
Define a topology T on a set E.
T is a family of subsets of E satisfying: E and ∅ are in T, any union of elements of T is in T, and any finite intersection of elements of T is in T
What is an example of a discrete topology?
T = (E) is the discrete topology on E
What is a neighborhood of a point a in a metric space?
A set V is a neighborhood of a if there exists r > 0 such that B(a, r) ⊂ V
What is the relationship between neighborhoods and open sets?
Every open set containing a is a neighborhood of a
What properties do neighborhoods of a point a have?
- Every neighborhood of a contains a
- If V ⊂ W and V is a neighborhood of a, then W is also a neighborhood of a
- A finite intersection of neighborhoods of a is a neighborhood of a
What does it mean for a metric space to be separated?
For distinct points a and b, there exist neighborhoods V of a and W of b such that V ∩ W = ∅
What is the definition of a neighborhood of a point a in a metric space?
A neighborhood of a point a is a set B(a, r) for some r > 0.
If W is a neighborhood of a point a, what can be concluded?
B(a, r) ⊆ W.
What can be said about a finite number of neighborhoods V1, …, Vn of a point a?
There exists ri > 0 such that B(a, ri) ⊆ Vi for all i = 1, …, n.
What is the relationship between r and ri when defining a neighborhood of a point a?
r = min(ri) > 0.
What is the conclusion about the neighborhoods V1, …, Vn?
B(a, r) ⊆ ∩_{i=1}^n Vi.
What is the proposition regarding a subset O of a metric space?
O is open if and only if it is a neighborhood of each of its points.
What does it mean for a point x to be interior to a set A?
A is a neighborhood of x.
What is the notation for the interior of a set A?
The interior of A is denoted by A°.
If A is a subset of a metric space, what can be said about its interior?
A ⊆ A°.
What is the definition of adherence of a set A in a metric space?
A point x is adherent to A if B(x, r) ∩ A ≠ ∅ for all r > 0.
What is the notation for the adherence of a set A?
The adherence of A is denoted by A̅.
What is the relationship between a point x and set A for x to belong to A?
x ∈ A if and only if for every neighborhood V of x, V ∩ A ≠ ∅.
What is the definition of a closed set in a metric space?
A set A is closed if A̅ = A.
What is the definition of a dense subset in a metric space?
A subset A is dense in E if A̅ = E.
What is the definition of the boundary of a set A?
Fr(A) = A̅ ∩ ∁A.
What are isolated points in a set A?
A point x ∈ A is isolated if there exists a neighborhood V of x such that V ∩ A = {x}.
What are accumulation points in a set A?
A point x ∈ E is an accumulation point of A if every neighborhood of x contains an infinite number of points of A.
What is the definition of the induced topology on a subset A of a metric space E?
The topology on A is defined as dA = d|A×A.
What is the definition of the product topology on a product of metric spaces?
The topology of the product space E = E1 × … × En is defined using distances δ∞, δ1, or δ2.
What is the definition of a limit in the context of metric spaces?
f(x) tends to l as x tends to x0 if for every neighborhood V of l, there exists a neighborhood U of x0 such that f(U) ⊆ V.
What does it mean for a function f to be continuous at a point x0?
For every ε > 0, there exists η > 0 such that dE(x, x0) < η implies dF(f(x), l) < ε.
What is the definition of the limit of a function f at a point x0?
l = lim f(x) as x approaches x0 if for every neighborhood V of l, there exists a neighborhood U of x0 such that f(U) ⊂ V.
What does it mean for a function f to be continuous at a point x0?
f is continuous at x0 if lim f(x) = f(x0) as x approaches x0.
What is the equivalent condition for continuity of f at x0?
∀ϵ > 0, ∃η > 0 such that ∀x ∈ E, dE(x, x0) < η ⇒ dF(f(x), f(x0)) < ϵ.
What is the unique limit of f at x0 if it approaches two different limits l and l’?
l is unique; if f(x) approaches both l and l’, then l = l’.
What is the notation for the limit of f(x) as x approaches x0?
lim f(x) as x approaches x0.
What is the definition of the restriction of a function f to a subset A?
fA = f|A, where A ⊂ Df.
How is the limit expressed when A excludes a point x0?
lim f(x) as x approaches x0 with x ≠ x0.
True or False: If lim f(x) = l, then lim f(x) = l.
True.
What is the condition for a function to be continuous on the entire domain E?
f is continuous on E if it is continuous at every point x0 ∈ E.
What is the definition of the extension of a function by continuity?
If lim f(x) = l ∈ F as x approaches x0, then g(x) = f(x) if x ≠ x0 and g(x0) = l.
What is the condition for the composition of functions g and f to be continuous?
If lim f(x) = l1 and lim g(y) = l2, then lim (g ◦ f)(x) = l2 as x approaches x0.
What type of function is continuous if it is a polynomial of n real variables?
Every polynomial function of n real variables is continuous on Rn.
What does it mean for two distances d and δ on a set E to be topologically equivalent?
The identity map idE : (E, d) → (E, δ) is a homeomorphism.
What is the condition for f to be a homeomorphism from E to F?
f is a bijection and both f and its inverse f−1 are continuous.
What is the condition for the inverse image of an open set under a function to be open?
f is continuous if for every open set Ω in F, f−1(Ω) is an open set in E.
What is the significance of the projection functions pi in a product space?
The projection functions pi: E → Ei are continuous for each i.
What is the relationship between the continuity of a function f and its component functions fi?
f is continuous if and only if each component function fi is continuous.
What is the consequence if g is continuous at a and g(a) ≠ 0?
1/g is continuous at a.
Fill in the blank: A function is continuous if for every _______ in F, the pre-image under f is open in E.
open set
Fill in the blank: A function is continuous on E if for every _______ in F, the pre-image under f is closed in E.
closed set
Qu’est-ce qu’une application continue en un point ?
Une application f est continue en ai si chacune de ses composantes ψi est continue.
Cela implique que pour toute suite convergente, l’image de cette suite converge vers l’image du point limite.
Vrai ou Faux : La réciproque de la continuité d’une application est toujours vraie.
Faux
Exemple : L’application f définie par f(x,y) = x² + y² si (x,y) ≠ (0,0) et f(0,0) = 0 est continue en (0,0), mais ses applications partielles ne le sont pas.
Définir une application uniformément continue.
Une application f est uniformément continue sur E si pour tout ϵ > 0, il existe η > 0 tel que pour tous (x, y) ∈ E, dE(x, y) < η implique dF(f(x), f(y)) < ϵ.
Cela signifie que la continuité ne dépend pas du point choisi dans E.
Proposition : Quelles implications a une application uniformément continue ?
Toute application uniformément continue est continue.
Cependant, la réciproque est fausse.
Qu’est-ce qu’une application lipschitzienne ?
Une application f est lipschitzienne s’il existe une constante k > 0 telle que pour tous x, y ∈ E, dF(f(x), f(y)) ≤ k * dE(x, y).
On dit alors que f est k-lipschitzienne.
Vrai ou Faux : Toute application lipschitzienne est uniformément continue.
Vrai
Cela découle directement de la définition d’une application lipschitzienne.
Définir la convergence d’une suite dans un espace métrique.
Une suite (xn)n∈N converge vers un point x ∈ E si pour tout voisinage V de x, il existe N ∈ N tel que n ≥ N implique xn ∈ V.
Cela est équivalent à dire que d(x, xn) < ϵ pour tout ϵ > 0.
Proposition : Si une suite converge dans un espace métrique, que peut-on dire de sa limite ?
La limite de la suite est unique.
Cela est dû à la séparation de l’espace métrique.
Qu’est-ce qu’une isométrie ?
Une application f est une isométrie si pour tous x, y ∈ E, dF(f(x), f(y)) = dE(x, y).
Toute isométrie est également lipschitzienne.
Vrai ou Faux : Une valeur d’adhérence d’une suite est unique.
Faux
Une suite peut avoir plusieurs valeurs d’adhérence.
Définir une valeur d’adhérence d’une suite.
Un point a ∈ E est une valeur d’adhérence de la suite (xn)n∈N si pour tout voisinage V de a, l’ensemble {n ∈ N / xn ∈ V} est infini.
Cela implique qu’il existe une infinité de termes de la suite qui se rapprochent de a.
Qu’est-ce qu’une suite de Cauchy ?
Une suite est de Cauchy si pour tout ϵ > 0, il existe N ∈ N tel que pour tous m, n ≥ N, d(xm, xn) < ϵ.
Cela signifie que les éléments de la suite se rapprochent les uns des autres à mesure que l’on progresse dans la suite.
What is the definition of a Cauchy sequence in a metric space (E, d)?
A sequence (xn)n∈N of points in E is a Cauchy sequence if ∀ϵ > 0, ∃N ∈ N such that p, q > N ⇒ d(xp, xq) < ϵ.
True or False: Every convergent sequence in a metric space is also a Cauchy sequence.
True.
What property does every Cauchy sequence possess?
Every Cauchy sequence is bounded.
If a Cauchy sequence (xn)n∈N has an adherence point x, what can be concluded?
It converges to the adherence point x.
What is a complete metric space?
A metric space (E, d) is complete if every Cauchy sequence in (E, d) converges in (E, d).
According to the Bolzano-Weierstrass theorem, what can be extracted from any bounded sequence of real numbers?
A convergent subsequence.
Is the space R with the usual distance complete?
Yes.
What is the result when taking the product of k complete metric spaces?
The product space E = E1 × · · · × Ek is also complete.
What defines a contraction mapping in a metric space (E, d)?
A mapping f: E → E is a contraction if there exists a real number 0 < k < 1 such that ∀x, y ∈ E, d(f(x), f(y)) ≤ kd(x, y).
What can be concluded about the existence and uniqueness of fixed points for contractions in complete metric spaces?
Every contraction has a unique fixed point.
What is a covering of a subset A in a metric space (E, d)?
A family (Ai)i∈I of subsets of E such that A ⊆ ∪i∈I Ai.
What is a compact space in terms of open coverings?
A metric space (E, d) is compact if from every open covering of E, a finite subcovering can be extracted.
True or False: Every finite subset of a metric space is compact.
True.
What is a property of every compact subset of a metric space?
Every compact subset is bounded.
What is the relationship between compact sets and closed sets in metric spaces?
Every compact subset of a metric space is closed.
Qu’est-ce qu’une partie compacte d’un espace métrique?
Une partie compacte d’un espace métrique (E, d) est celle qui est fermée et bornée.
Proposition : Toute partie compacte d’un espace métrique est ______.
fermée.
Quelle est la caractéristique d’une partie ouverte dans le contexte de la compacité?
Si K est compact, alors son complémentaire ∁K est ouvert.
Si E est compact, alors toute partie fermée F de E est ______.
compacte.
Théorème de Borel-Lebesgue : Tout intervalle fermé borné [a, b] de R est ______.
compact.
Quelles sont les caractéristiques d’une partie K de R pour être considérée comme compacte?
K est fermée et bornée.
Quel est le résultat pour les espaces métriques compacts E1, E2, … Ek?
L’espace métrique produit E = E1 × … × Ek est compact.
Vrai ou Faux : Dans un espace métrique compact, si une suite de fermés non vides est décroissante, alors leur intersection est non vide.
Vrai.
Théorème sur les suites : Dans un espace métrique compact (E, d), toute suite (xn)n∈N de points de E admet au moins une ______.
valeur d’adhérence dans E.
Corollaire 1 : Dans un espace m´etrique compact, de toute suite (xn)n∈N de points, on peut extraire une ______.
sous-suite convergente dans E.
Qu’est-ce que le théorème de Bolzano-Weierstrass?
De toute suite bornée de nombres réels, on peut extraire une sous-suite convergente.
Qu’est-ce qu’un espace métrique complet?
Un espace métrique où toute suite de Cauchy converge.
Lemme 1 : Si K est une partie de E et toute suite de points de K admet au moins une valeur d’adhérence dans K, alors pour tout ϵ > 0, il existe un ______.
recouvrement de K par un nombre fini de boules ouvertes de rayon ϵ.
Lemme 2 : Si K est une partie de E et toute suite de points de K admet au moins une valeur d’adhérence dans K, alors il existe ϵ > 0 tel que pour tout x ∈ K, il existe ______.
i ∈ I tel que B(x, ϵ) ⊂ Oi.
Théorème : K est compacte si et seulement si toute suite de points de K admet au moins une ______.
valeur d’adhérence dans K.
En résumé, quelles sont les deux conditions pour qu’une partie K d’un espace métrique soit compacte?
- K est fermée
- K est bornée
Vrai ou Faux : Un intervalle ouvert de R est toujours compact.
Faux.
Quel est le lien entre la compacité et la continuité dans un espace métrique?
Un espace métrique compact garantit que toute fonction continue atteint ses extrema.
What is the definition of a compact set in metric spaces?
A set K is compact if every open cover of K has a finite subcover.
What does it mean for a function f: (E, dE) → (F, dF) to be bounded?
f is bounded if f(E) is a bounded subset of F.
If f is continuous on a compact set A in E, what can be concluded about f(A)?
f(A) is compact in F.
True or False: If E is compact, then f(E) is compact.
True
What is the Heine theorem regarding continuity and compactness?
If f is continuous on a compact subset K of E, then f is uniformly continuous on K.
What is the definition of a connected metric space?
A metric space (E, d) is connected if there is no partition of E into two non-empty open sets.
What are the equivalent assertions that define a connected space?
- E is connected
- The only open and closed subsets of E are E and ∅
- There is no partition of E into two non-empty closed sets.
What is a connected subset of a metric space?
A subset A of a metric space (E, d) is connected if the subspace (A, dA) is a connected metric space.
What is the relationship between connectedness and continuous functions?
If E is connected and f: E → F is continuous, then f(E) is connected.
What is a path in a metric space?
A path joining a and b is a continuous function f: [α, β] → E where f(α) = a and f(β) = b.
What does it mean for a space to be connected by arcs?
A metric space (E, d) is connected by arcs if for all a, b ∈ E, there exists a path joining a and b.
What is the conclusion about intervals in R regarding connectedness?
A subset A of R is connected if and only if A is an interval.
Fill in the blank: A function f is said to be ______ if it attains its bounds on a compact set K.
bounded
True or False: The image of a compact set under a continuous function is always compact.
True
If A is a connected set in a metric space and B contains A, what can be concluded about B?
B is connected.
What is the result of the product of two connected metric spaces?
The product of a finite family of connected metric spaces is connected if each space is connected.
What is a path joining two points a and b in Rn?
f : [0, 1] → Rn, t → (1 − t) a + tb
This function defines a continuous path between points a and b in Rn.
What does it mean for a metric space (E, d) to be arc-connected?
It means every pair of points in E can be joined by a continuous path in E.
True or False: Every arc-connected metric space is connected.
True.
Give an example of an arc-connected space.
Rn is arc-connected.
What is the definition of a normed vector space?
A vector space E equipped with a norm, denoted as (E, .).
What does it mean for an application to be uniformly continuous?
An application is uniformly continuous if it satisfies the Lipschitz condition: |f(x) - f(y)| ≤ k |x - y| for some k > 0.
What are the two types of operations defined for a normed vector space (E, .)?
Addition s : (x, y) ∈ E × E → x + y ∈ E and scalar multiplication p : (λ, x) ∈ K × E → λx ∈ E.
What does it mean for an application to be continuous in a normed vector space?
It means small changes in input result in small changes in output.
Fill in the blank: The norm of uniform convergence is defined as _______.
sup_{x ∈ X} |f(x)|.
What theorem states that if (E, .) is complete, then (Fb(X, E), . ∞) is complete?
Theorem on the completeness of bounded functions.
What is a linear continuous application?
An application f from E to F that is both linear and continuous.
What notation is used for the set of continuous linear applications from E to F?
L(E, F).
What are the equivalent conditions for a linear application f from E to F to be continuous?
- f is continuous on E
- f is continuous at 0E
- There exists a constant k > 0 such that ∀x ∈ E, ||f(x)||F ≤ k ||x||E.
What does it mean for all norms on a finite-dimensional vector space E to be equivalent?
All norms induce the same topology on E.
What is the definition of a multivariable continuous application?
An application f : E1 × E2 × … × En → F that is linear in each variable and continuous.
What is the significance of the closed ball Bf(0E, 1) in a finite-dimensional space?
It is compact.
True or False: Every linear application of Rn into Rp is continuous.
True.
What is the condition for a bilinear application f : E1 × E2 → F to be continuous?
It is continuous if it is continuous at (0E1, 0E2) and there exists a constant k > 0 such that ∀(x1, x2) ∈ E1 × E2, ||f(x1, x2)||F ≤ k ||x1||E1 × ||x2||E2.
What is the definition of a norm on L(E, F)?
For f ∈ L(E, F), ||f|| = sup_{x ∈ E, x ≠ 0} ||f(x)||F / ||x||E.
What does the corollary about composition of linear applications state?
If g ∈ L(F, G) and f ∈ L(E, F), then g ◦ f ∈ L(E, G) and ||g ◦ f|| ≤ ||g|| ||f||.
Comment est défini un espace de Banach ?
Un espace vectoriel normé sur K = R ou C qui est complet
Un espace de Banach réel est un espace de Banach sur R, et un espace de Banach complexe est sur C.
Quelle est la proposition concernant les espaces vectoriels de dimension finie ?
Tout espace vectoriel de dimension finie sur K = R ou C est un espace de Banach.
Quelle est la norme de la convergence uniforme dans le contexte des espaces de Banach ?
La norme de la convergence uniforme est notée ∞.
Qu’est-ce qu’un isomorphisme d’espaces vectoriels normés ?
Une application f : (E, E) → (F, F) qui est linéaire, bijective, continue, et dont la réciproque f⁻¹ est continue.
Quand une application linéaire continue et bijective entre deux espaces de Banach est-elle un isomorphisme ?
Elle est un isomorphisme d’espaces vectoriels normés.
Qu’est-ce que l’inégalité de Cauchy-Schwarz ?
< x, y > | ≤ √< x, x >√< y, y >.
Comment définit-on un espace préhilbertien réel ?
Un espace vectoriel sur R muni d’un produit scalaire.
Qu’est-ce qu’un espace de Hilbert réel ?
Un espace préhilbertien réel qui est complet pour la norme associée au produit scalaire.
Qu’est-ce que la norme associée au produit scalaire dans un espace préhilbertien ?
Pour tout x ∈ H, la norme est définie par x = √< x, x >.
Quelle est la définition d’un espace de Banach complexe ?
Un espace de Banach sur C.
Qu’est-ce que la série exponentielle dans le contexte des espaces de Banach ?
La série Σ (1/n!) fⁿ converge normalement, où f⁰ = IdE et fⁿ = f ∘ … ∘ f (n fois).
Quelle condition doit remplir IdE − u pour être inversible dans L(E, E) ?
Il faut que u < 1.
Qu’est-ce qu’un espace vectoriel normé ?
Un espace vectoriel muni d’une norme.
Pour tous x, y ∈ H, quelle est l’inégalité qui doit être satisfaite par la norme associée ?
x + y ≤ x + y.
Vrai ou Faux : Un espace de Hilbert réel est un espace de Banach réel.
Vrai.
Qu’est-ce que la continuité d’une application f en un point a ?
lim (x→a) f(x) = f(a).
Complétez l’espace vectoriel normé : Un espace vectoriel normé est complet si _______.
[il contient toutes les suites de Cauchy].
Qu’est-ce que l’application : u ∈ Isom(E, F) → u⁻¹ ∈ Isom(F, E) ?
Elle est continue.
