A definition of germ of \(C^k\) functions at point, \(C^k_p (M)\)
Topics
About: \(C^\infty\) manifold with boundary
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) manifold with boundary.
- The reader knows a definition of neighborhood of point.
- The reader knows a definition of function over \(C^\infty\) manifold with boundary.
- The reader knows a definition of map between arbitrary subsets of \(C^\infty\) manifolds with boundary \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
Target Context
- The reader will have a definition of germ of \(C^k\) functions at point, \(C^k_p (T)\).
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Definition
For any \(C^\infty\) manifold with (possibly empty) boundary, \(M\), and any point, \(p \in M\), the equivalence class of \(\{U_{p, \alpha}, f_\alpha\}\) where \(U_{p, \alpha}\) is any neighborhood of \(p\) and \(f_\alpha\) is any \(C^k\) function, \(f_\alpha: U_{p, \alpha} \to \mathbb {R}\), such that \((U_{p, \alpha_1}, f_{\alpha_1}) \sim (U_{p, \alpha_2}, f_{\alpha_2})\) if and only if there is a neighborhood, \(U_{p, \alpha_3}\), such that \(U_{p, \alpha_3} \subseteq U_{p, \alpha_1} \cap U_{p, \alpha_2}\) and \(f_{\alpha_1} = f_{\alpha_2}\) on \(U_{p, \alpha_3}\)
2: Note
Strictly speaking, \(C^\infty\) manifold with boundary is not required for \(k \neq \infty\), but just topological manifold with boundary does not suffice, because \(C^k\)-ness is not defined there.
