The $\C^r$ Whitney (or strong) topology is a topology assigned to the space $\C^r(M,N)$ of mappings from a $\C^r$ manifold $M$ to a $\C^r$ manifold $N$ having $r$ continuous derivatives . It gives a notion of proximity of two $\C^r$ 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 $\C^r$ embedding, then there is a strong $\C^r$ neighborhood of $f$ in which any $\C^r$ 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 $N$ 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 $$\mathcal{U}^r\left(f,\phi,\psi,\{K_i:i\in I\},\{\epsilon_i:i\in I\}\right)$$ as the set of $C^r$ mappings $g\colon M\to N$ such that for all $i\in I$ we have $g(K_i)\subset V_i$ and $$\sup_{x\in \phi_i(K_i), 0\leq k\leq r} ||D^k(\psi_if\phi_i^{-1})(x) - D^k(\psi_ig\phi_i^{-1})(x)|| <\epsilon_i.$$ That is, those maps $g$ that are close to $f$ and have their first $r$ derivatives close to the respective first $r$ -th derivatives of $f$ , 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 $\C^r$ topology of $\C^r(M,N)$ .

The weak $\C^r$ topology, or $\C^r$ 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 $M$ , 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 $\C^r_W(M,N)$ and $\C^r_S(M,N)$ , respectively.

We have that $\C^r_W(M,N)$ is always metrizable (with a complete metric) and separable. On the other hand, $\C^r_S(M,N)$ 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.