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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · distance between (1,2) and (4,6)

Geometry · Coordinate-Geometry

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

1,\ 2,\ 4,\ 6

Àwọn ìṣàmúlò-ètò

\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).

Fi àwọn àgbèwọlé hàn

d = 5

Nígbà yìí Yan àwọn àkóónú kan, tàbí tẹ̀ àtí kọ́ àwọn àkóónú rẹ̀. Àwọn ìṣàmúlò-ètò gbogbo, àwòrán, àwọn à

Fi iṣẹ́ rẹ̀ pamọ́

A free account adds notes on every lesson, a record of what you have finished, your solved problems in one place, and a tutor you can ask about this page. The maths itself is open to everyone, signed in or not.

Fikún Àwọn Ìṣàmúlò-ètò

Àwọn àmì-ìwé tí a lo níbẹ̀

Tẹ àwọn àmì-ìwé fún ìṣàfarawé fún àwọn ìṣàmúlò-ètò, àwòrán, àti àwọn àmì-ìwé àwọn àmì-ìwé.

Bii o ṣe le: Metric spaces and open sets

Differences of the coordinates.
Distance formula (Pythagoras).

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Topology

Compactness and connectedness Homeomorphism and topological invariants