The weak compact-open C-infinity topology on mapping spaces

Let $M$ and $Q$ be smooth manifolds with $M$ second countable (all smooth manifolds here are Hausdorff and second countable). The weak (compact-open) $C^{\mathrm{\infty }}$ topology on $C^{\mathrm{\infty }}(M,Q)$ is generated by the following subbasis. A basic set is determined by finitely many data: charts $(U_i,{\phi }_i)$ of $M$ and $(V_i,{\psi }_i)$ of $Q$, compact sets $K_i\subseteq U_i$ with $f(K_i)\subseteq V_i$ for the reference map $f$, an integer $r_i\ge 0$ and a tolerance ${\epsilon }_i>0$; it consists of all smooth $g:M\to Q$ with $g(K_i)\subseteq V_i$ and ${\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le r_i}{\sup }_{x\in {\phi }_i(K_i)}\mid D^{\alpha }({\psi }_i\circ g\circ {\phi }_i^{-1})(x)-D^{\alpha }({\psi }_i\circ f\circ {\phi }_i^{-1})(x)\mid <{\epsilon }_i$ for every $i$. The weak topology is the topology generated by all these basic sets, as the reference map $f$ and the finitely many data vary; at each point of a finite intersection, shrinking the tolerances around that point gives a basic neighbourhood contained in the intersection, so these sets form a basis. Empty compact pieces impose no condition and are omitted from the displayed supremum. This is the only topology on $C^{\mathrm{\infty }}(M,Q)$ used on this page. It does not depend on the chosen atlases of $M$ and $Q$ (The weak smooth topology is independent of the chosen atlas ↗), and for compact $M$ the immersion condition is open in it (For compact sources the immersion condition is open in the weak smooth topology). Under ${\mathrm{A}\mathrm{C}}_{\omega }$ the same construction applies to the total spaces $TM$ and $TN$, with their canonical smooth structures (Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure).