cours/distance entre deux parties d'un espace métrique.md
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distance entre deux parties
distance
partie d'un espace métrique
s/maths/topologie
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parsed true

[!definition] Définition Soit (X, d) un espace métrique Soient A, B \subset X deux parties de X On appelle distance entre A et B : d(A, B) := \inf\limits \{ d(a, b) \mid a \in A \wedge b \in B \} ^definition

[!idea] intuition Cette distance représente normalement la plus courte distance entre A et B

$= "![[" + dv.current().file.name + ".svg|800]]"

Propriétés

[!proposition]+ distance entre des parties non disjointes Si A \cap B \neq \emptyset, alors d(A, B) = 0

[!démonstration]- Démonstration Il suffit de prendre x \in A \cap B et d'écrire : d(A, B) \leq d(x, x) = 0 D'où suit, par positivité des distances

Exemples

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