AbstractWe investigate some common points between stable structures and weakly small structures and define a structureMto befineif the Cantor-Bendixson rank of the topological space${S_\varphi }\left( {dc{l^{eq}}\left( A \right)} \right)$is an ordinal for every finite subsetAofMand every formula$\varphi \left( {x,y} \right)$wherexis of arity 1. By definition, a theory isfineif all its models are so. Stable theories and small theories are fine, and weakly minimal structures are fine. For any finite subsetAof a fine groupG, the traces on the algebraic closure$acl\left( A \right)$ofAof definable subgroups ofGover$acl\left( A \right)$which are boolean combinations of instances of an arbitrary fixed formula can decrease only finitely many times. An infinite field with a fine theory has no additive nor multiplicative proper definable subgroups of finite index, nor Artin-Schreier extensions.