In Chapters 2-7, we were concerned exclusively with homotopies N (X, X) defined on a cylinder U = U X [0, l] . Thus, all operators HA = H (,., X) (X E [O, l]) had the same domain /. Here we are concerned more generally with homotopies H for which the operators HA may have different domains u x . This situation arises when we look for solutions having a particular property. The idea is to try to follow a branch of solutions to HA (X) = X with the desired property and thus to work on some neighborhood 24 of that branch which avoids all the other solutions. This kind of continuation with solutions having a particular property, will be called selective continuation.