--- aliases: - distance entre deux parties up: - "[[distance]]" - "[[partie d'un espace métrique]]" tags: - s/maths/topologie - excalidraw excalidraw-plugin: parsed excalidraw-open-md: 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 $B$ > ![[distance entre deux parties d'un espace métrique.svg|300]] # 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 %% # Excalidraw Data ## Text Elements A ^rBDgNyV5 B ^kynULWDT ## Drawing ```compressed-json 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