Let (X, TX) be a topological space and let open(X) be the category associated to (X, TX) in the canonical way. Let R be Ring, i.e. a presheaf of rings over open(X), let for any U, V E open(X) such that V C U, the restriction homomorphism R(U) + R(V) be denoted by $. We consider the Grothendieck category of presheaves of left modules over the fixed presheaf R and we denote it by R-Mod. If IKE R-Mod then e;(M) will stand for the restriction homomorphisms of the presheaf M. In [8], torsion theories in R-Mod have been introduced. The basic theory expounded there allows the formation of Modules of quotients in a way very similar to the usual localization techniques in module categories. A Ring of quotients was constructed in case the torsion theory F is local, i.e. F can be described by giving torsion theories F(U) in R( U)-mod, for every U E open(X), which patch together in a certain way. It has been indicated in [8] that Pointwise results could be derived for arbitrary torsion theories. In this paper we present structural properties for Pointwise localization at contractible torsion theories. Contractibility is a local-global condition which is highly incompatible with locality, what explains why, in order to deal with Ring theoretical aspects of localization, it is necessary to split the theory in a local and a contractible case. In section 3 of this paper we study “Pointwise property (T)” more explicitely.