maths.free › Topology
Topology
Open sets, continuity, compactness, connectedness: geometry without distance.
Сургамж
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.
Сургамж
Spaces
Continuity and separation
Compactness and connectedness
Where it leads
Зарим хичээлүүд нь нээлттэй текстээс авсан: Wikipedia. Сургалтын хуудас бүрийн эх үүсвэр, лицензийг тодорхойлно.
Энд ашигласан тэмдэгтүүд
Бүх тодорхойлолт, зураг, үсгийн утгыг харахын тулд ямар ч тэмдэгтийг дарна уу.
Хүмүүс асуудаг асуултууд
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.
Тоглоомын хуудаснаа бичээд эсвэл зур. Бүх алхам, зураг, хариулт нь асуулт асуух хүртэл нуугдаж байдаг. Засварлах →
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