82 lines
5.1 KiB
Markdown
82 lines
5.1 KiB
Markdown
---
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aliases:
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- distance entre deux parties
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up:
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- "[[distance]]"
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- "[[partie d'un espace métrique]]"
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tags:
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- s/maths/topologie
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- excalidraw
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excalidraw-plugin: parsed
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excalidraw-open-md: true
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---
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> [!definition] Définition
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> Soit $(X, d)$ un [[espace métrique]]
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> Soient $A, B \subset X$ deux parties de $X$
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> On appelle distance entre $A$ et $B$ :
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> $d(A, B) := \inf\limits \{ d(a, b) \mid a \in A \wedge b \in B \}$
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^definition
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> [!idea] intuition
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> Cette distance représente normalement la plus courte distance entre $A$ et $B$
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> ![[distance entre deux parties d'un espace métrique.svg|300]]
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# Propriétés
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> [!proposition]+ distance entre des parties non disjointes
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> Si $A \cap B \neq \emptyset$, alors $d(A, B) = 0$
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> > [!démonstration]- Démonstration
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> > Il suffit de prendre $x \in A \cap B$ et d'écrire :
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> > $d(A, B) \leq d(x, x) = 0$
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> > D'où suit, par positivité des distances
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# Exemples
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%%
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# Excalidraw Data
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## Text Elements
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A ^rBDgNyV5
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B ^kynULWDT
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## Drawing
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```compressed-json
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```
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%% |