  We characterize the (sequentially) weak and strong closure of planar
diffeomorphisms in the Sobolev topology and we show that they always coincide.
We also provide some sufficient condition for a planar map to be approximable
by diffeomorphisms in terms of the connectedness of its counter-images, in the
spirit of Young's characterisation of monotone functions. We finally show that
the closure of diffeomorphisms in the Sobolev topology is strictly contained in
the class INV introduced by Muller and Spector.
