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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.
דוגמה עובדת · critical points of x^3 - 3x
צעד אחר צעד
- f(x) = x^{3} - 3 x
Critical points are where f′(x) = 0 or is undefined.
- f'(x) = 3 x^{2} - 3
Differentiate.
- x = -1, x = 1
Solve f′(x) = 0.
- f''(x) = 6 x
Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.
- f''(-1) = -6 \Rightarrow (-1, 2) \text{ is a local maximum}
- f''(1) = 6 \Rightarrow (1, -2) \text{ is a local minimum}
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איך: Compactness and connectedness
- Critical points are where f′(x) = 0 or is undefined.
- Differentiate.
- Solve f′(x) = 0.
- Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.
שאלות שאנשים שואלים
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.
יותר בפנים. Topology
Metric spaces and open setsHomeomorphism and topological invariants