Topology

Open sets, continuity, compactness, connectedness: geometry without distance.

Fianarana

Introductory Metric spaces and open sets Distance made abstract; open balls, open and closed sets, continuity. 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.

Fianarana marobe

Spaces

Topological space Open set Closed set Metric space Basis (topology)Neighbourhood (mathematics)Interior (topology)Closure (topology)Boundary (topology)Subspace topology Product topology Quotient space (topology)

Continuity and separation

Continuous function Homeomorphism Hausdorff space Separation axiom Regular space Normal space Urysohn's lemma Tietze extension theorem

Compactness and connectedness

Compact space Connected space Path (topology)Heine–Borel theorem Tychonoff's theorem Baire category theorem Sequential compactness Locally compact space

Where it leads

Manifold Euler characteristic Surface (topology)Klein bottle Möbius strip Torus Knot theory Fixed-point theorem

Ny sasany amin'ireo lesona ireo dia nalaina avy amin'ny lahatsoratra misokatra: Wikipedia. Ny pejy tsirairay dia manome ny anaran'ny loharano sy ny fahazoan-dàlana.

Marika ampiasaina eto

Tsindrio ny marika iray mba hahitana ny famaritana feno, sary, ary ny dikan'ny litera tsirairay ao aminy.

Fanontaniana napetrak'ireo olona

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.

Manana olana manokana ve ianao?
Adikao na esory ao amin'ny pejy famahana ny olana: isaky ny dingana, sary iray, ny valinteny dia miafina mandra-pahafantaranao. _Soloy →

Sarana hafa

＋ Arithmetic 𝑥 Algebra △ Geometry ∿ Trigonometry ƒ Precalculus σ Statistics & Probability ⊢ Discrete Math & Logic ∫ Calculus ⊞ Linear Algebra ℕ Number Theory ẏ Differential Equations ∇ Multivariable Calculus ε Real Analysis ⋔ Combinatorics & Graph Theory ∈ Set Theory & Logic 𝔾 Abstract Algebra 𝑃 Probability Theory ≈ Numerical Methods ℂ Complex Analysis μ Measure Theory ‖·‖ Functional Analysis κ Differential Geometry π₁ Algebraic Topology → Category Theory ∂ Partial Differential Equations ≋ Fluid Dynamics ∞ Frontiers