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Metric spaces and open sets
Distance made abstract; open balls, open and closed sets, continuity.
A metric is any distance obeying symmetry, positivity and the triangle inequality. Open sets are unions of open balls; a function is continuous when preimages of open sets are open. No ε needed once the open sets are known. Picture it: the open disc around a point, and the same idea with the "taxicab" distance making diamonds. Think it: topology keeps the open sets and throws the metric away.
Contoh yang dikerjakan · distance between (1,2) and (4,6)
Langkah demi langkah
- \Delta x = 3,\ \Delta y = 4
Differences of the coordinates.
- d = \sqrt{\Delta x^2 + \Delta y^2} = \sqrt{9 + 16} = 5
Distance formula (Pythagoras).
Tunjukkan jawapan
Sekarang kau Pilih masalah, atau taip atau lukis sendiri. Setiap langkah, gambar, jawapan tersembunyi sehingga anda tanya.
Akaun percuma menambah nota pada setiap pengajaran, rekod apa yang anda selesaikan, masalah yang anda selesaikan di satu tempat, dan seorang guru yang anda boleh tanya tentang halaman ini. Matematik itu sendiri terbuka kepada semua orang, sama ada anda log masuk atau tidak.
Daftar Masuk Log masukSimbol yang digunakan di sini
Tekan simbol mana-mana untuk definisi penuh, gambar, dan apa makna setiap huruf dalamnya.
Bagaimana untuk: Metric spaces and open sets
- Differences of the coordinates.
- Distance formula (Pythagoras).
Soalan yang orang tanya
Why "a doughnut is a coffee cup"?
Each can be deformed into the other without cutting or gluing: one hole each. Topology studies exactly the properties such deformations preserve.
What is compactness for?
It is the property that makes "every sequence has a convergent subsequence" and "continuous functions attain their maximum" true. On the real line it means closed and bounded.
Lebih dalam Topology
Compactness and connectednessHomeomorphism and topological invariants