Density topology

In mathematics, the density topology on the real numbers is a topology on the real line that is different, but in some ways analogous, to the usual topology. It is sometimes used in real analysis to express or relate properties of the Lebesgue measure in topological terms.

Definition

Let be a Lebesgue-measurable set. By the Lebesgue density theorem, almost every point of is a density point of, i.e., satisfies
where is the Lebesgue measure and is the open interval of length centered at.
When all points of are density points of, it is said to be density open.
It can be shown that the density open sets of form a topology : this constitutes the density topology.

Examples

Every open set in the usual topology of is density open, but the converse is not true. For example, the subset is not open in the usual sense, but it is density open. More generally, any subset of full measure is density open. This includes, for example, the complements of and the Cantor set.
Less trivially, and perhaps more instructively, let us show that the set is density open. Again, at every point other than 0 this is clear because it is even neighborhood of x for the usual topology, so the only point to consider is 0. But if and we let, then each interval that intersects has so their total measure is, and proving that 0 is indeed a density point of U.

Properties

Let denote the real line endowed with the density topology.

Like with the usual topology, is a Hausdorff and Tychonoff topological space but unlike the usual topology, it not normal.
A subset is nowhere dense iff it is meagre iff it is closed and discrete iff it is a null set.
The Borel subsets of are precisely the Lebesgue-measurable sets; and the complete Boolean algebra of regular open sets of can be identified with the "reduced measure algebra", i.e., the Boolean algebra of Lebesgue-measurable sets modulo null sets.
Like with the usual topology, is connected.
Like with the usual topology, is a Baire space; in fact, unlike the usual topology, it is even hereditarily Baire in the sense that every subspace of is a Baire space.
The compact subspaces of are precisely its finite subsets.
The approximately continuous functions are precisely the continuous functions .