A natural sufficient condition for a finite family of single element extensions of a matroid to be compatible is given. Characterizations of all the finite extensions N of a matroid M (E) are given for which the rank function satisfies ρ N (X)= Min Z⊂ E {ρ M (Z)+| X− Z N|} or equivalently the closure operator satisfies X N= X N⌢ E N⌣ X. The single element extensions and the principal extensions are examples of such matroids. The notion of a sheaf of flats of M. Las Vergnas is used in the proof of a new necessary and sufficient condition for two single element extensions of a matroid to be compatible. An initial announcement of part of these results appeared in R. Cordovil (CR Acad. Sci. Paris. A 284 (1977), 1249–1252).