Cell decomposition and dimension function in the theory of closed ordered differential fields

In this paper we develop a differential analogue of o-minimal cell decomposition for the theory C O D F of closed ordered differential fields. Thanks to this differential cell decomposition we define a well-behaving dimension function on the class of definable sets in C O D F . We conclude this paper by proving that this dimension (called δ -dimension) is closely related to both the usual differential transcendence degree and the topological dimension associated, in this case, with a natural differential topology on ordered differential fields.