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A structural framework is presented which allows the systematic determination of all non-isomorphic posets on n points with at least τ ⋅ 2n downsets, or - equivalently - of all T0-topologies on n points with at least τ ⋅ 2n open sets. The framework is developed by defining the type of a special extension of posets and by defining a partial order on these types. It is shown that this partial order...