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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.
Kugwira ntchito chitsanzo · distance between (1,2) and (4,6)
Gawo ndi Gawo
- \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).
Kusonyeza yankho
Tsopano inu Sankhani vuto, kapena kulemba kapena kujambula wanu. Pakalipano, yankho limeneli lidzapezeka patsamba lino.
M’malo mwake, maakaunti aulere amawonjezera malemba pazophunzira zonse, mnda wa zomwe mwamaliza, mavuto anu omaliza m’malo limodzi, ndi m’bale amene mungafunse za nkhaniyo. Maphunziro a matekinoloje ndi otsegulira kwa aliyense, olembetsa kapena osalembetsa.
Kulembetsa KulowaZithunzi
Pitani pa dzina lililonse lachifaniziro kuti mudziwe tanthauzo lake, chithunzi chake, ndi zimene limatanthauza.
Momwe mungagwiritsire ntchito: Metric spaces and open sets
- Differences of the coordinates.
- Distance formula (Pythagoras).
Mafunso omwe anthu amafunsa
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.
Zambiri pa Topology
Compactness and connectednessHomeomorphism and topological invariants