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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.

Gumagana halimbawa: distance between (1,2) and (4,6)

Geometry · Coordinate-Geometry

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

1,\ 2,\ 4,\ 6

Hakbang-hakbang

  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).

Ipahayag ang sagot
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.

Subukan ang iyong sarili

Higit pa sa Topology

Compactness and connectednessHomeomorphism and topological invariants