PlanetMath: $\mathcal{C}^r$ topologies

(more info)

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$\mathcal{C}^r$ topologies

(Definition)

The $\mathcal{C}^r$ Whitney (or strong) topology is a topology assigned to the space $\mathcal{C}^r(M,N)$ of mappings from a $\mathcal{C}^r$ manifold to a $\mathcal{C}^r$ manifold having continuous derivatives . It gives a notion of proximity of two $\mathcal{C}^r$ mappings, and it allows us to speak of “robustness” of properties of a mapping. For example, the property of being an embedding is robust: if $f\colon M\to N$ is a $\mathcal{C}^r$ embedding, then there is a strong $\mathcal{C}^r$ neighborhood of in which any $\mathcal{C}^r$ mapping $g\colon M\to N$ is an embedding.

Given a locally finite atlas $\{(U_i, \phi_i):i\in I\}$ and compact sets $K_i\subset U_i$ such that there are charts $\{(V_i,\psi_i) : i\in I\}$ of for which $f(K_i)\subset V_i$ for all $i\in I$ , and given a sequence $\{\epsilon_i>0 : i\in I\}$ , we define the basic neighborhood

$\displaystyle \mathcal{U}^r\left(f,\phi,\psi,\{K_i:i\in I\},\{\epsilon_i:i\in I\}\right)$

as the set of

mappings $g\colon M\to N$ such that for all $i\in I$ we have $g(K_i)\subset V_i$ and

$\displaystyle \sup_{x\in \phi_i(K_i), 0\leq k\leq r} \vert\vert D^k(\psi_if\phi_i^{-1})(x) - D^k(\psi_ig\phi_i^{-1})(x)\vert\vert <\epsilon_i.$

That is, those maps

that are close to

and have their first

derivatives close to the respective first

-th derivatives of

, in local coordinates. It can be checked that the set of all such neighborhoods forms a basis for a topology, which we call the Whitney or strong $\mathcal{C}^r$ topology of $\mathcal{C}^r(M,N)$ .

The weak $\mathcal{C}^r$ topology, or $\mathcal{C}^r$ compact-open topology, is defined in the same fashion but instead of choosing $\{(U_i,\phi_i):i\in I\}$ to be a locally finite atlas for , we require it to be an arbitrary finite family of charts (possibly not covering ).

The space $\mathcal{C}^r(M,N)$ with the weak or strong topologies is denoted by $\mathcal{C}^r_W(M,N)$ and $\mathcal{C}^r_S(M,N)$ , respectively.

We have that $\mathcal{C}^r_W(M,N)$ is always metrizable (with a complete metric) and separable. On the other hand, $\mathcal{C}^r_S(M,N)$ is not even first countable (thus, not metrizable) when is not compact; however, it is a Baire space. When is compact, the weak and strong topologies coincide.