Geometry
Elements of geometry FS25
Elements of geometry FS25
Fichier Détails
| Cartes-fiches | 89 |
|---|---|
| Langue | English |
| Catégorie | Mathématiques |
| Niveau | Université |
| Crée / Actualisé | 12.06.2025 / 14.06.2025 |
| Lien de web |
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smooth atlas
Let M be a topological n-manifold. A smooth atlas on M is an atlas \(\{(U_α,φ_α)\}_{α∈I}\)such that the following condition is satisfied: For all pairs \(α,β ∈ I\), the composition \(φ_α(U_α ∩U_β) \xrightarrow{φ^{−1}_α}(U_α ∩U_β) \xrightarrow{\varphi_\beta} φ_β(U_α ∩U_β)\) is smooth, i.e., given by an n-tuple of smooth functions on the open subset \(φ_α(U_α∩U_β) ⊂ \mathbb{R}^n\)
atlas
Let M be a topological n-manifold. An atlas for M is a collection of charts \(\{(U_α,φ_α)\}_{α∈I}\) whose domains cover M, i.e., \((U_α)_{α∈I}\) is an open covering of M.
refinement
A refinement of a cover C of a topological space X is a new cover D of X such that every set in D is contained in some set of C.
locally finite
Let X be a topological space. A family \((A_i)_{i∈I }\)of subsets in X is said to be locally finite if every point in X admits to an open neighborhood which meets only finitely many of the \(A_i\)
paracompact
Let X be a topological space. We say that X is paracompact if every open covering of X admits a refinement which is locally finite.
precompact
A subset A of a topological space X is said to be precompact if its closure \(A⊂X\) is compact.
locally connected
A topological space X is said to be locally connected (resp. locally path-connected) if every point of X has a cofinal system of open neighborhood which are connected (resp locally path-connected).
coordinate chart
Let M be a topological manifold. A coordinate chart (or simply chart) on M is a pair \((U,φ)\), where U is an open subset of M and \(φ : U → φ(U)\) is a homeomorphism from \(U\) to an open subset \(φ(U) ⊂ \mathbb{R}^n\)
topological manifold
A topological manifold is a topological space M satisfying the following properties:
1.) M is Hausdorff; For every pair of distinct points \(p,q ∈ M\), there are disjoint open subsets \(U,V ⊂ M\) such that \(p ∈ U\) and \(q ∈ V\).
2.) M is second countable; There is a countable basis for the topology of M.
3.) M is locally euclidean; Every point \(x ∈ M\) admits an open neighborhood which is homeomorphic to an open neighborhood in \(\mathbb{R}^{n_x}\) for some integer \(n_x\). If we can choose the integer \(n_x\) independently of x, we say M has the dimension n
