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Compactness and connectedness

The two properties that make theorems work: extreme values and intermediate values.

Compact (on the line: closed and bounded) guarantees a maximum is attained; connected guarantees the intermediate value theorem. Picture it: a closed interval versus an open one — the function 1/x has no maximum on (0, 1]. Think it: both are defined by open covers and separations, so they transfer to any space.

Ohatra: critical points of x^3 - 3x

Calculus · Extrema

Critical points of x^3 - 3x

x^{3} - 3 x

Dingana amin'ny dingana

  1. f(x) = x^{3} - 3 x

    Critical points are where f′(x) = 0 or is undefined.

  2. f'(x) = 3 x^{2} - 3

    Differentiate.

  3. x = -1, x = 1

    Solve f′(x) = 0.

  4. f''(x) = 6 x

    Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.

  5. f''(-1) = -6 \Rightarrow (-1, 2) \text{ is a local maximum}

  6. f''(1) = 6 \Rightarrow (1, -2) \text{ is a local minimum}

Asehoy ny valinteny
(-1, 2)\ \text{local maximum},\; (1, -2)\ \text{local minimum}

Symbols used here

\neg,\ \wedge,\ \vee,\ \Rightarrow,\ \Leftrightarrow
not, and, or, implies, iff
Logical connectives.
\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: Compactness and connectedness

  1. Critical points are where f′(x) = 0 or is undefined.
  2. Differentiate.
  3. Solve f′(x) = 0.
  4. Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.

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.

Andramo ny anao manokana

Mbola maro ao Topology

Metric spaces and open setsHomeomorphism and topological invariants