Density topology

Let $U\; \backslash subseteq\; \backslash mathbb\{R\}$ be a Lebesgue-measurable set. By the Lebesgue density theorem, almost every point $x$ of $U$ is a density point of $U$, i.e., satisfies

:$\backslash frac\{\backslash lambda(U\; \backslash ,\; \backslash cap\; \backslash ,\; (x-h,x+h))\}\{2h\}\; \backslash \; \backslash underset\{h\; \backslash to\; 0^+\}\{\backslash longrightarrow\}\; \backslash \; 1$

where $\backslash lambda$ is the Lebesgue measure and $(x-h,x+h)$ is the open interval of length $2h$ centered at $x$.

When all points of $U$ are density points of $U$, it is said to be density open.{{harv | Tall | 1976 | loc=definition 2.1 }}

It can be shown that the density open sets of $\backslash mathbb\{R\}$ form a topology (in other words, they are stable under arbitrary unions and finite intersections):{{harv | Tall | 1976 | loc=theorem 2.3 }} this constitutes the density topology.

Every open set in the usual topology of $\backslash mathbb\{R\}$ (i.e., a union of open intervals) is density open, but the converse is not true. For example, the subset $\backslash mathbb\{R\}\; \backslash setminus\; \backslash \{1/n\; :\; n\; >\; 0\backslash \}$ is not open in the usual sense (since every open neighborhood of 0 contains some $1/n$ and is thus not contained in the set), but it is density open (the only problematic point being 0 and the set has density 1 at 0). More generally, any subset of full measure is density open. This includes, for example, the complements of $\backslash mathbb\{Q\}$ and the Cantor set.

Less trivially, and perhaps more instructively, let us show that the set $U\; :=\; \backslash mathbb\{R\}\; \backslash setminus\; \backslash bigcup\_\{n=1\}^\{+\backslash infty\}\; \backslash left[\backslash frac\{1\}\{n\},\; \backslash frac\{1\}\{n\}+\backslash frac\{1\}\{2^n\}\backslash right]$ (which, again, is not open in the usual topology) is density open. Again, at every point $x\backslash in\; U$ 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 $h>0$ and we let $k\; :=\; \backslash left\backslash lfloor\backslash frac\{1\}\{h\}\backslash right\backslash rfloor$, then each interval $\backslash left[\backslash frac\{1\}\{n\},\; \backslash frac\{1\}\{n\}+\backslash frac\{1\}\{2^n\}\backslash right]$ that intersects $(-h,h)$ has $n>k$ so their total measure is $\backslash leq\; 2^\{-k\}$, and $\backslash frac\{\backslash lambda((-h,+h)\; \backslash setminus\; U)\}\{2h\}\; \backslash leq\; \backslash frac\{2^\{-k\}\}\{2h\}\; \backslash leq\; \backslash frac\{2^\{-1/h\}\}\{h\}\; \backslash underset\{h\; \backslash to\; 0^+\}\{\backslash longrightarrow\}\; 0$ proving that 0 is indeed a density point of U.

Let $\backslash mathbb\{R\}\_\{\backslash mathrm\{d\}\}$ denote the real line endowed with the density topology.

• Like $\backslash mathbb\{R\}$ with the usual topology, $\backslash mathbb\{R\}\_\{\backslash mathrm\{d\}\}$ is a Hausdorff (T₂) and Tychonoff (T_{{{frac|3|1|2}}}) topological space but unlike the usual topology, it not normal (T₄).{{harv | Tall | 1976 | loc=theorem 2.4 }}{{harv | Ciesielski | Larson | Ostaszewski | 1994 | loc=theorem 1.2.3(iv) }}

• A subset $Y\; \backslash subseteq\; \backslash mathbb\{R\}\_\{\backslash mathrm\{d\}\}$ is nowhere dense (for the density topology) iff it is meagre (ditto) iff it is closed and discrete (ditto) iff it is a null set (in the sense of Lebesgue measure).{{harv | Tall | 1976 | loc=theorem 2.7 }}{{harv | Ciesielski | Larson | Ostaszewski | 1994 | loc=theorem 1.2.3(ii) }}

• The Borel subsets of $\backslash mathbb\{R\}\_\{\backslash mathrm\{d\}\}$ (for the density topology) are precisely the Lebesgue-measurable sets;{{harv | Tall | 1976 | loc=theorem 2.6(i) }} and the complete Boolean algebra of regular open sets of $\backslash mathbb\{R\}\_\{\backslash mathrm\{d\}\}$ can be identified with the "reduced measure algebra", i.e., the Boolean algebra of Lebesgue-measurable sets modulo null sets.

• Like $\backslash mathbb\{R\}$ with the usual topology, $\backslash mathbb\{R\}\_\{\backslash mathrm\{d\}\}$ is connected.{{harv | Tall | 1976 | loc=theorem 2.10 }}{{harv | Ciesielski | Larson | Ostaszewski | 1994 | loc=theorem 1.2.3(v) }}

• Like $\backslash mathbb\{R\}$ with the usual topology, $\backslash mathbb\{R\}\_\{\backslash mathrm\{d\}\}$ is a Baire space; in fact, unlike the usual topology, it is even hereditarily Baire in the sense that every subspace of $\backslash mathbb\{R\}\_\{\backslash mathrm\{d\}\}$ is a Baire space.{{harv | Tall | 1976 | loc=theorem 2.11 }}

• The compact subspaces of $\backslash mathbb\{R\}\_\{\backslash mathrm\{d\}\}$ are precisely its finite subsets.{{harv | Ciesielski | Larson | Ostaszewski | 1994 | loc=theorem 1.2.3(vi) }}

• The approximately continuous functions $f\backslash colon\; \backslash mathbb\{R\}\backslash to\backslash mathbb\{R\}$ are precisely the continuous functions $f\; :\; \backslash mathbb\{R\}\_\{\backslash mathrm\{d\}\}\backslash to\backslash mathbb\{R\}$ (i.e., placing the density topology at the source but the usual topology at the target).{{harv | Ciesielski | Larson | Ostaszewski | 1994 | loc=§1.3 }}

{{reflist|3}}

• [https://encyclopediaofmath.org/wiki/Density_topology Springer Encyclopedia of Mathematics: article "Density topology"]
• {{Cite journal |last=Tall |first=Franklin D. |title=The Density Topology |journal=Pacific Journal of Mathematics |volume=62 |year=1976 |pages=275–284 |doi=10.2140/pjm.1976.62.275 }}
• {{Cite book | last1=Ciesielski | first1=Krzysztof | last2=Larson | first2=Lee | last3=Ostaszewski | first3=Krzysztof | title=$\backslash mathcal\{I\}$-density Continuous Functions | year=1994 | publisher=American Mathematical Society | isbn=978-0-8218-6238-4 }}

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Category:Measure theory

Category:Real analysis