By Robert C. Higgins
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Extra resources for Analysis for Financial Management 8th Ed. (McGraw-Hill Irwin Series in Finance, Insurance, and Real Est)
However, this is only very rough and it is useless for many purposes. A further method of determining largeness of sets is in terms of 54 topologies. A first candidate notion for largeness in the topological context is denseness of a set. The possibility to approximate every element of a given set by elements of a certain subset in terms of a topology prevents this subset from being too "small". Clearly, the choice of the topology is crucial. But the attribute "typical" or "usual" should not be given to a subset and to its complement.
2) (cf. 2). 2 In both examples zero of "f is close to f" means that the values of points near f and f and of their derivatives (up to some order) are close. We can define more generally: A differentiable mapping f: JRn + JRm is stable if for any differenf: JRn +JRm sufficiently close to f there are diffeomor- tiable mapping phisms h: JRn + JRm, k: JRm + JRm close to the identity such that the fol- lowing diagram commutes: "Close"is meant with respect to a suitable topology on a space of differentiable mappings.
A one-dimensional differentiable submanifold of P. 5) whenever the full rank assumption is not already fulfilled for gl~. In such a case we have K~(X) = O. But there can be only one direction in I~(X) an infinitesimal move along which leaves the unit normal g~(x) unchanged. 5 After all a look at the possible types of non-convex preference relations shows that the following phenomena may cause non-differentiability of a demand relation 41 1. non-convexity which destroys even uniqueness 2. vanishing Gaussian curvature of indifference hypersurfaces at points where the preference relation is locally strictly convex.
Analysis for Financial Management 8th Ed. (McGraw-Hill Irwin Series in Finance, Insurance, and Real Est) by Robert C. Higgins