# Topology on function space between smooth manifolds

I'm having trouble reconciling my intuitive notion of "functions close to $f$" with the seemingly very technical definition of the weak and strong topologies on $C^r(M,N)$ in Hirsch's Differential Topology.

Hirsch's definition of the weak topology:

Let $f \in C^r(M,N)$. Let $\varphi: U \to \mathbf R^m$ and $\psi: V \to \mathbf R^n$ be charts on $M$ and $N$, respectively. Let $K$ be a compact subset of $U$ such that $f(K) \subseteq V$. Let $\varepsilon \in (0,\infty]$. Define a weak subbasic neighborhood $$\mathcal N^r(f;(\varphi,U), (\psi,V), K, \varepsilon)$$ to be $$\{ g \in C^r(M,N) \mid g(K) \subseteq V \text{ and } \|D^k(\psi f \varphi^{-1})(x) - D^k(\psi g \varphi^{-1})(x) \| < \varepsilon \text{ for all } x \in \varphi(K) \text{ and all } k = 0, \ldots, r\}.$$ The weak topology on $C^r(M,N)$ is the one generated by all such sets.

I won't bother typing out the definition of the strong topology. These definitions look pretty opaque to me. I have no idea how anyone came up with this definition, I have a very vague picture of what an "$\varepsilon$-neighborhood of $f$" looks like, and I'm not even sure my picture is right.

A picture that is clear to me is the following special case. Let's pretend we don't know about the weak/strong topologies. Certainly in any reasonable universe we would like to define a path of homeomorphisms (resp., diffeomorphisms), i.e, a map $[0,1] \to \operatorname{Homeo}(M)$ (resp., $(-\varepsilon,1+\varepsilon) \to \operatorname{Diff}(M)$), to be continuous (resp., smooth) iff the induced map $M \times [0,1] \to M$ is continuous (resp., $M \times (-\varepsilon,1+\varepsilon) \to M$ is smooth).

Let's call this the "isotopy topology." More precisely, let $I := (-\varepsilon,1+\varepsilon)$, let $M$ be a $C^r$-manifold, and consider a set map $I \to \operatorname{Diff}^r(M)$. Declare such a map to be "continuous" iff the induced map $M \times I \to M$ is $C^r$. Then endow $\operatorname{Diff}^r(M)$ with the finest topology that makes all "continuous" maps continuous (see final topology).

Question 1: $\operatorname{Diff}^r(M)$ inherits the weak/strong topology as a subspace of $C^r(M)$. How does the isotopy topology compare to either of these topologies? (One would certainly hope that the isotopy topology is finer than both the weak and the strong topologies.)

Question 2: Is there a reference for this kind of stuff?

Soft question: Why would anyone not define this to be the natural topology on $\operatorname{Diff}^r(M)$? This makes it clear (at least to me) that an $\varepsilon$-neighborhood of a diffeomorphism is the set of diffeomorphisms that are "a small isotopy away," that "tangent vectors" are vector fields, etc.

Not-really-a-question: I'm sure this generalizes easily to maps in $C^r(M,N)$ that are not necessarily homeomorphisms/diffeomorphisms (can define $\varepsilon$-neighborhood to be functions that are "a small homotopy away," etc.) I just haven't given it too much thought.