Topology

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

Μαθήματα

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.

Περισσότερα μαθήματα

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

Ορισμένα από τα μαθήματα αυτά προσαρμόζονται από τα ανοικτά κείμενα: Wikipedia. Κάθε σελίδα μαθημάτων αναφέρει την πηγή και την άδεια.

Σύμβολα που χρησιμοποιούνται εδώ

Χτυπήστε οποιοδήποτε σύμβολο για τον πλήρη ορισμό, μια εικόνα, και τι σημαίνει κάθε γράμμα σε αυτό.

Ερωτήσεις που κάνουν οι άνθρωποι

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.

Έχεις δικό σου πρόβλημα;
Πληκτρολογήστε το ή ζωγραφίστε το στη σελίδα του λύτη: κάθε βήμα, μια εικόνα, η απάντηση κρυμμένη μέχρι να ρωτήσετε. Λύση →

Άλλα κλαδιά

＋ 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