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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.
Ví dụ đã làm · distance between (1,2) and (4,6)
Bước một bước
- \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).
Giải đáp
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Làm thế nào: Metric spaces and open sets
- Differences of the coordinates.
- Distance formula (Pythagoras).
Câu hỏi người ta hỏi
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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.
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