maths.free › Topology › Compactness and connectedness
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.
مثال عملي · critical points of x^3 - 3x
خطوة خطوة
- 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}
كشفت الإجابة
الآن أنت اختار مشكلة، أو طبعها أو رسمها بنفسك، كل خطوة، صورة، الإجابة مخفية حتى تسأل.
الرموز المستخدمة هنا
لمس أي رمز للحصول على التعريف الكامل، والصورة، وما يعنيه كل حرف في ذلك.
كيف: 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.
أسئلة يطرحها الناس
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.
أكثر في Topology
Metric spaces and open setsHomeomorphism and topological invariants