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Abstract. Let X be an arbitrary set. A topology t on X is said to be useful if every continuous linear preorder on X is representable by a continuous real{valued order preserving function.Continuous linear preorders on X are induced by certain families of open subsets of X that are called (linear) separable systems on X. Therefore, in a rst step useful topologies on X will be characterized by means of (linear) separable systems on X. Then, in a second step particular topologies on X are studied that do not allow the construction of (linear) separable systems on X that correspond to non{representable continuous linear preorders. In this way generalizations of the Eilenberg{Debreu theorems which state that second countable or separable and connected topologies on X are useful and of the theorem of Est evez and Herv es which states that a metrizable topology on X is useful, if and only if it is second countable can be proved.2000 AMS Classi cation: 54F05, 91B16, 06A05.

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Objective functions that are applied in ordinal data analysis must be adequate, i.e. carefully adapted to the structure of the observed data. In addition, any analysis of data that is based upon objective functions must lead to interpretable results. After a general characterization of adequate objective functions in ordinal data analysis, therefore, the particular problems of constructing adequate and interpretable dissimilarity coefficients and correlation coefficients in ordinal data analysis, stress measures (stress functions) in non-metric scaling and generalized stress measures or correlation coefficients in any theory of rank estimation will be discussed.

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