maths.free › Topology
Topology
Open sets, continuity, compactness, connectedness: geometry without distance.
经验教训
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
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.
更多经验教训
Spaces
Continuity and separation
Compactness and connectedness
Where it leads
其中一些经验教训从开放文本中改编: Wikipedia. 每页课目都列出其来源和执照。
此处所用符号
输入完整定义、图片和其中每个字母的含义的符号。
人们对人们提出的问题
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.
输入它或将其输入求解器页面:每个步骤、图片、隐藏的答案直到您询问。 解决 →
其他分支
＋ Arithmetic𝑥 Algebra△ Geometry∿ Trigonometryƒ Precalculusσ Statistics & Probability⊢ Discrete Math & Logic∫ Calculus⊞ Linear Algebraℕ Number Theoryẏ Differential Equations∇ Multivariable Calculusε Real Analysis⋔ Combinatorics & Graph Theory∈ Set Theory & Logic𝔾 Abstract Algebra𝑃 Probability Theory≈ Numerical Methodsℂ Complex Analysisμ Measure Theory‖·‖ Functional Analysisκ Differential Geometryπ₁ Algebraic Topology→ Category Theory∂ Partial Differential Equations≋ Fluid Dynamics∞ Frontiers