As the authors say ``the Hausdorff distance... works well as long as the sets lie in a bounded region. In many applications one has to deal with unbounded sets or with collections of bounded sets which are not uniformly bounded''. To deal with these situations this paper builds on earlier work of two of the authors and others and is a comprehensive study of the topology generated by the \(\rho\)-Hausdorff distances on spaces of subsets in a normed linear space. For \(\rho\geq 0\) \(\rho B\) is the ball centred at 0 and radius \(\rho\), \(C_ \rho=C\cap \rho B\) and \(\text{haus}_ \rho(C,D)=\max\{e(C_ \rho,D), e(D_ \rho,C)\}\), where \(e(C,D)=\sup\{d(x,D): x\in C\}\). The section headings are: completeness, compactness, connectedness and separability.