Editing Diffeomorphism (section)

===Topology===
The diffeomorphism group has two natural [[Topological space|topologies]]: ''weak'' and ''strong'' {{harv|Hirsch|1997}}. When the manifold is [[Compact space|compact]], these two topologies agree.  The weak topology is always [[Metrizable space|metrizable]].  When the manifold is not compact, the strong topology captures the behavior of functions "at infinity" and is not metrizable.  It is, however, still [[Baire space|Baire]].

Fixing a [[Riemannian metric]] on <math>M</math>, the weak topology is the topology induced by the family of metrics
: <math>d_K(f,g) = \sup\nolimits_{x\in K} d(f(x),g(x)) + \sum\nolimits_{1\le p\le r} \sup\nolimits_{x\in K} \left \|D^pf(x) - D^pg(x) \right \|</math>
as <math>K</math> varies over compact subsets of <math>M</math>.  Indeed, since <math>M</math> is <math>\sigma</math>-compact, there is a sequence of compact subsets <math>K_n</math> whose [[Union (set theory)|union]] is <math>M</math>.  Then:
: <math>d(f,g) = \sum\nolimits_n 2^{-n}\frac{d_{K_n}(f,g)}{1+d_{K_n}(f,g)}.</math>

The diffeomorphism group equipped with its weak topology is locally homeomorphic to the space of <math>C^r</math> vector fields {{harv|Leslie|1967}}. Over a compact subset of <math>M</math>, this follows by fixing a Riemannian metric on <math>M</math> and using the [[Exponential map (Riemannian geometry)|exponential map]] for that metric. If <math>r</math> is finite and the manifold is compact, the space of vector fields is a [[Banach space]]. Moreover, the transition maps from one chart of this atlas to another are smooth, making the diffeomorphism group into a [[Banach manifold]] with smooth right translations; left translations and inversion are only continuous. If <math>r=\infty</math>,  the space of vector fields is a [[Fréchet space]]. Moreover, the transition maps are smooth, making the diffeomorphism group into a [[Fréchet manifold]] and even into a [[Convenient vector space#Regular Lie groups|regular Fréchet Lie group]]. If the manifold is <math>\sigma</math>-compact and not compact the full diffeomorphism group is not locally contractible for any of the two topologies. One has to restrict the group by controlling the deviation from the identity near infinity to obtain a diffeomorphism group which is a manifold; see {{harv|Michor|Mumford|2013}}.