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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.

Eżempju maħdum · distance between (1,2) and (4,6)

Geometry · Coordinate-Geometry

Distance between (1,2) and (4,6)

1,\ 2,\ 4,\ 6

Pass b'pass

  1. \Delta x = 3,\ \Delta y = 4

    Differences of the coordinates.

  2. d = \sqrt{\Delta x^2 + \Delta y^2} = \sqrt{9 + 16} = 5

    Distance formula (Pythagoras).

Jiżvelaw it-tweġiba
d = 5

Issa inti Agħżel problema, jew ittajpja jew iġbed tiegħek stess.Kull pass, stampa, it-tweġiba moħbija sakemm inti tistaqsi.

Żomm tiegħek stess li jaħdmu

Kont b'xejn iżid noti fuq kull lezzjoni, rekord ta' dak li lest, problemi solvuti tiegħek f'post wieħed, u għalliem inti tista' tistaqsi dwar din il-paġna.Il-matematika nnifisha hija miftuħa għal kulħadd, log fil jew le.

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Simboli użati hawnhekk

Taptap kwalunkwe simbolu għad-definizzjoni sħiħa, stampa, u dak li kull ittra fiha tfisser.

Kif għandek tuża: Metric spaces and open sets

  1. Differences of the coordinates.
  2. Distance formula (Pythagoras).

Mistoqsijiet li n-nies jistaqsu

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.

Aktar fil Topology

Compactness and connectednessHomeomorphism and topological invariants