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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.
Çözümlü örnek · critical points of x^3 - 3x
Adım adım
- 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}
Cevabı göster
Şimdi sen Bir problem seçin, ya da kendi probleminizi yazın ya da çizin. Her adım, bir resim, cevap sorulana kadar gizli.
Burada kullanılan sembolleri
Tüm tanım, resim ve her harfin anlamı için herhangi bir sembolü tıklayın.
Nasıl: 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.
Sık sorulan sorular
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.
More in Topology
Metric spaces and open setsHomeomorphism and topological invariants