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Topology

What survives when you are allowed to stretch but not tear. Topology makes "nearby" precise without a ruler, and its invariants — connected pieces, holes — are the coarsest and most robust things a space has.

Kennslustund

Introductory Metric spaces and open sets Distance made abstract; open balls, open and closed sets, continuity. distance between (1,2) and (4,6) Core Compactness and connectedness The two properties that make theorems work: extreme values and intermediate values. critical points of x^3 - 3x Advanced Homeomorphism and topological invariants When two spaces are "the same", and what numbers survive stretching. 12 - 30 + 20

Symbols used here

\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.

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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