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Summary

A point is a density point of a set if the ratio of the length of its intersection with an interval containing it to that of the interval tends to 1 as the interval shrinks to the point. The classical Lebesgue Density Theorem states that almost all points of a measurable set are density points. Declaring a set open when all its points are density points leads to a topology, the density topology. This is a fine topology – it refines the ordinary (Euclidean) topology, in having more open sets. The density-meagre sets are the Lebesgue-null sets. This result shows how working bitopologically – switching between the Euclidean and density topologies – enables us to switch between the category and measure cases. A list of properties of the line under the density topology is given. Caution is needed: for instance, the line is a topological group under the Euclidean topology, but not (only a paratopological group) under the density topology (as now multiplication is only separately but not jointly continuous).

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7 - Density Topology

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7 - Density Topology

Summary