${𝒞}^r$ topologies

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

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

$${𝒰}^r(f,\varphi ,\psi ,\{K_i:i\in I\},\{{ϵ}_i:i\in I\})$$

as the set of $C^r$ mappings $g:M\to N$ such that for all $i\in I$we have $g(K_i)\subset V_i$ and

That is, those maps $g$ that are close to $f$ and have their first $r$derivatives close to the respective first $r$-thderivatives of $f$, in local coordinates.It can be checked that the set of all such neighborhoods forms a basisfor a topology, which we call the Whitney or strong ${𝒞}^r$topology of ${𝒞}^r(M,N)$.

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

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

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