maths.free › Topology › Metric spaces and open sets
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.
Eksempel på arbeid · distance between (1,2) and (4,6)
Steg for trinn
- \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).
Vis svaret
Nå har du Velg et problem, eller skriv inn, eller tegne din egen. Hvert steg, et bilde, svaret skjult til du spør.
Symboler som brukes her
Trykk på et symbol for den fulle definisjonen, et bilde og hva hver bokstav i det betyr.
Hvordan: Metric spaces and open sets
- Differences of the coordinates.
- Distance formula (Pythagoras).
Spørsmål folk stiller
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.
Mer i Topology
Compactness and connectednessHomeomorphism and topological invariants