--- 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 $B$ `$= "![[" + 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 %% # Excalidraw Data ## Text Elements A ^rBDgNyV5 ## Drawing ```compressed-json N4KAkARALgngDgUwgLgAQQQDwMYEMA2AlgCYBOuA7hADTgQBuCpAzoQPYB2KqATLZMzYBXUtiRoIACyhQ4zZAHoFAc0JRJQgEYA6bGwC2CgF7N6hbEcK4OCtptbErHALRY8RMpWdx8Q1TdIEfARcZgRmBShcZQUebQA2bQAOGjoghH0EDihmbgBtcDBQMBKIEm4IbB4AMQBhfABHI2wGoQB2NmwAWQAWAAU22oA5AGV8VJLIWEQKog4kflLMbmce AFYARm0egGYetaSk+J4NtbakjbbFyBgVtcS2ngAGHfiAThfzniS3+OuICgkdTcDY9JLaV5tDZJNprf6SBCEZTSEGbCFJJ5JNZvHaw/7WZTBbhPf7MKCkNgAawQtTY+DYpAqAGJ4m0ehtiG8JqVNLhsJTlBShBxiLT6YyJOTrMw4LhAtluZAAGaEfD4EawIkSQQeRUQMkU6kAdSBkm4fEKAnJVIQGpgWvQOvK/yFyPmzFyaA2/zYsuwaluXqeJMtE EFwjgAEliJ7UHkALr/JXkTLR7gcIRq/6EEVYCq4Hp6oUi92x4qTaDwcS8S0AX1JCAQxBB7w28QxOKS/0YLHYXDQPDaIYrPdYnCGnDE5rWPQ+bR2w9KhGYABF0lAm9wlQQwv9NMIRQBRYKZbKxhP/IRwYi4DfNr3zzZrZ5gvYWitzSnpzP4f70/mbmg274LuoZwGwOY5PklpgAUkwlIukxPDBiYwXB8FbL8Tw4niMFgM4aw7NoGwbDs3wbOsKHXLB eFtG02xtPE9zrNR+FJD02hvG8JwwmsVFoXhoLgpC0K4fB+HPpxbZPKcxx8fBqHwehkyglspw9Dwvy8ax7LaMGGxvIRFHyZMimTMpJRJHEjxYrJLF4fEiQ7IcrKiSZJRmSUFlgKy2hshiPTtmJGFEVZFyUQp1HeRpflvIc5H2eJg4JG8jxMXJ/FKYJTycbsOzYricJ4Sc2hrPc8Q9E8GWRQJ4mIQhKGWp5ED4KEUC0vo+hqHefQQQqaAZlmoZknKU AAEI5o4HDKN+Q0VlkxATSKOYzQNP6klEpBQAAgqQFIUAiuD3qgg2/qGC27fth3HadhT1oU5aQOUEgAPIcAAivQTxDGNK4wBQ721CMtRGvQMBCGwM56tM1YtTmCyhssaA7ERz7Yk8gVPM+8Q7NC/yBqgzighx3GnJpVk4+8fyhoCxDAg+9FvHlBXBZACJIiiXoYnpPzsYV+LTQ6iH6ta1JigyzJVOsazYHqvL8uGwqinSEuSuQHAynKWRQHqKpqna Dr6nSzrDaLCAmnTZoDhthq2pqsNOs2LrCG64Sxt6oa+nyAYgsG/yK1GMb5M1ya4KmN3raGk15hIuBrEWB7EKW3CPVMVbmnWDaAag5xlVx3HUyOTBjv2qBnDs3bF32E4cFOXrQjwuIURp2aruu2fAaBFb7krx4ZNr57NVeN53iCj5k1jQVdqGn6zWdH6dNSx2dwg/zgZB561fB9VgMhNVZeJqxvMRWOBdptEpWlzHuWAnk0YfyXHAuZ+syUaw5c8u MRaZUV4c47wQkIk+OSrFubYkuFpWEmVzLZW0OsAyRlEoYU0sRS4bloFeTwkkIibQ3gbBktfVipFEg4wSjfO+3l8E5UOEcKE59xIXDgTwHg6Vv4eV/gwrY9wej7C/kVcSzlcqgiqmw2+HCMIYzgdiRihC/5EXWOcYyGD74SPBAcOKuxX74VeH5TGOFyHiMmLOTicUMZBX4fBZwuxthnBeEo/eMDxLQgSJVEBci1GKNERQ4q4JsK2L4axImiQDgswM VvFSx8djcTOOFCxkxnD0U+Pg/RyjvI7z3qZJqf42odS6jIJsvVIJzw2qNJaU1VonUjvNEUZSVrFNNqNS6bADohAjnNUoF09rNOunPO6ixHplGOhATQQgGhGgAFrKA2AAVVqO9AAmrgQgAAJJU1RlmSEpPgHY0N04SDmAjCsSNCb5WSM5HiPAwTYVOHEiABN/70RONhJmn9FHOX+LTemqBjgQleFxTYqUma2XhIiZEOsBx+T2HgwyMJZzfHbO+UoB IhY2xtOLCU6AmQ8CVG8BAPC5Z8gFMWZW4oKhSg1rKeUOskyqnVPbCojs9QGhtBbL5iKrS2wNg7Y2TtQyukkMnL0Po/Q+yDMLAO0ZB5JhTAgNMa12lPVzMciAuB4gJyVoK1AqdKwzAHJnYajZjrvDot8TsldeycG4PEYWo5q6TmrA3XY7x9jC2XGuYIo8gI7lXqGHuR4TwD2Dpea8t5s6XHyhPeIJEnjsrhhwL88r56lH/EvLc3q159SgmgCyaTWI ZPYeEko/84go3eIZAyuCwSbFYtZKFCDYXcSOFZRqP8Z45IMHknqmb6kVhGltWp00e0dJqZNOpiaSlbSaS0o6Q7ICdKuq03pJR7olAGc9dApBfrKCGDAAAavHf4MMyVYGpYjFYZM9J7CxExC4cU4r43PacYijxNEsJnFxMEHzTQgiIiRfOmw0H0NKOzMF04BaEmrMLZlYsVYYogFinFeLCx7kJYrEU6KyXq01lS3WtKuUMp5Uys2rKra8FRdSfD2p CPOz8AKt2IJhXe1gL7cVQpA5StDKHcOs6yhKvzG0dVJZ6PjoNWG3BTMcaBXNSXK10m7W12rI8Oirwo3TwrG69uy902+sTn3U8Wa4xDxDZ61A4anzPCYoZHof4cwJsqQqlqi8O7aYrBuTA4L0DbSLJQAAKieioXmkycCgCMQgRhqzPCC9kaoYdVQEw9q5k920iDKFLi1BASpT1Fy2uYAgyWkRpagL6PUehshLPmKQOV9mk2QAZEiHMBA/PuYC3qXA QgisACVwhherOSIQPqPzw2WaCzmpmpF3XAIpFVcA4AalDSnQo0AESZAqDeUgX5FgMEIAgCgY1UPEowxIJkSoTune5JUEQVLIwbn0BqW2h3MXYtxfizb2BLva2uxkPbCsDuwcw9KSl2tztvb2h9m71Q8P0qo7qV773sifdu8R791tFsg6uzdu7NpKOOmo6juHUAEcdZdnRj0DG8eg/hzdl6IrmNithxTgn4Pgsxa6vgeL9P0cZGqMF0L4XzSLguwz hHTWdopbS8ETLwP8cI7m5Orp062lJsF5z/Qh4RRTp6bHeXUuhc3anT5vZG7E7neYNgCkaoAAa3BKrWUYo+CmON2KbdN+b/A8zzTnFKm8wicKolXEW0YNgBgFsjgIP1jOy6Odg4yETjVwmjdK3O4KEgvOIsC+T8QDUCA4DcAsWGUgJAuhsGIAgNXuBNDBC0yBAbkAM8PYGWNOkQzSDKF5AAChOFcXglxqDd67zlNYABKPUXXlCZjlBUFv7fG4kl4A uXvM/e8D+H308nVLMfUmp1APssZTqba4wgLruYC+DrQAMrI5fK/cD6zXyA2AiA57QDf/4HAw69dIP1n07WDnX4/wgVfpQdgAAVggNgDkCMK/nAEXiXmXhXs5tXptnyNvowD5kHuMGfqGEetqOkGAX2CVkIGSAYAbrqtVn+E5lXl3Mmm1NtLgSgWgUumALWOACuhAHrOECnEwbWEAA=== ``` %%