The simple examples of such spaces which we use as `model spaces', are the subspaces of Euclidean space. That is,
If n=1 then there are only two cases, namely
We use the latter, more usual, notation. The main point I want to emphasise today is that the other models are just products of these one dimensional cases. Clearly
Now, consider any (relatively) open subset 
There is a simple way to choose a the space of smooth functions on X
turning it into a manifold with corners according to the defintion
above. Namely we can double the space across each boundary. Set
where the choice of all k signs is available.
Exercise Show that 
is open if
is (relatively) open
Then we just define 
Exercise Check that with this definition X becomes a manifold with corners according to the definition above.