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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.

Radni primjer: distance between (1,2) and (4,6)

Geometry · Coordinate-Geometry

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

1,\ 2,\ 4,\ 6

Korak po korak

  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).

Otkrij odgovor
d = 5

Symbols used here

\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
\lim_{x \to a} f(x)
limit
The value f(x) approaches as x approaches a.
x \in A,\ A \subseteq B
element of, subset
x belongs to A; every element of A is in B.
A \cup B,\ A \cap B,\ A \setminus B
union, intersection, difference
In either; in both; in A but not B.
\forall,\ \exists
for all, there exists
Quantifiers: every x; at least one x.
\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
\varepsilon,\ \delta
epsilon, delta
Small positive tolerances in the definition of a limit.
B(x, r),\ \overline{A},\ \partial A
open ball, closure, boundary
Points within r of x; A plus its limit points; the edge of A.
\pi_1(X),\ H_n(X),\ \chi
fundamental group, homology, Euler characteristic
Loops up to deformation; holes in each dimension; V − E + F.

How to: Metric spaces and open sets

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

Questions people ask

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.

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Compactness and connectednessHomeomorphism and topological invariants