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Every open subset of $R^{n}$ is the union of a countable pairwise disjoint family of dyadic cubes

Statement

Let $n \geq 1$ and let $U \subseteq R^{n}$ be open in the metric topology of $(R^{n}, d_{2})$ (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, $R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it). Then there is an at most countable family $M$ of pairwise disjoint dyadic cubes (Dyadic cubes of generation $k$ in $R^{n}$ , Finite, countably infinite, countable, uncountable) with

$U = ⋃ M .$

For $U = \emptyset$ the family is empty. No choice principle is used: the cube attached to a point is the one of least generation that fits inside $U$ , and least is a definition.

Facts & Assumptions

Given: A natural number $n \geq 1$ and an open subset $U \subseteq R^{n}$ .

[L1]

Every $x \in R^{n}$ lies in exactly one dyadic cube of generation $k$ (For each generation, the dyadic cubes of that generation are pairwise disjoint and cover $R^{n}$ ).

[L2]

If $Q$ and $Q^{'}$ are dyadic cubes of generations $k \leq k^{'}$ and $Q \cap Q^{'} \neq \emptyset$ , then $Q^{'} \subseteq Q$ (Two dyadic cubes are either disjoint or one contains the other).

[L3]

$Q_{k, m} = {x \in R^{n} : m_{i} 2^{- k} < x_{i} \leq (m_{i} + 1) 2^{- k} for every i < n}$ , and every dyadic cube is nonempty (Dyadic cubes of generation $k$ in $R^{n}$ , Integer powers $a^{m}$ ).

[L4]

A nonempty box determines its parameter pair, since $B$ determines $a_{i}$ and $b_{i}$ for every $i$ (Half-open boxes in $R^{n}$ and their volume).

[F1]

A subset $U \subseteq X$ is open in $(X, d)$ if for every $x \in U$ there is a real $r > 0$ with $B (x, r) \subseteq U$ , where $B (x, r) : = {y \in X : d (x, y) < r}$ (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Open ball, closed ball and sphere in a metric space).

[F2]

$d_{2} (x, y) : = \sqrt{\sum_{k < n} (x_{k} - y_{k})^{2}}$ and $d_{\infty} (x, y) : = \max {∣ x_{k} - y_{k} ∣ : k < n}$ are metrics on $R^{n}$ for $n \geq 1$ ( $R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it).

[F3]

For every $x \in R^{n}$ , $∥ x ∥_{2} \leq ∥ x ∥_{1}$ and $∥ x ∥_{1} \leq n ∥ x ∥_{\infty}$ , $n$ being the canonical natural of $R$ ; and $∥ x - y ∥_{2} = d_{2} (x, y)$ , $∥ x - y ∥_{\infty} = d_{\infty} (x, y)$ (The finite and reverse triangle inequalities for a norm; and for $n \geq 1$ every norm $N$ on $R^{n}$ satisfies $N (x) \leq C ∥ x ∥_{1}$ and is Lipschitz, hence continuous, for $d_{2}$ , claim 3; Each $∥ \cdot ∥_{p}$ is a norm on $R^{n}$ , and the induced metrics are exactly $d_{1}$ , $d_{2}$ and $d_{\infty}$ of the published metric-spaces page, claim 3; The $p$ -norms $∥ x ∥_{p}$ for rational $p \geq 1$ , and $∥ x ∥_{\infty}$ ).

[F4]

If $∣ r ∣ < 1$ then $(r^{k})$ is null, that is $r^{k} \to 0$ (For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ , claim 1; Limits and Cauchy sequences of reals).

[F5]

Every nonempty subset $S \subseteq N$ has a least element (The well-ordering principle).

[F6]

The set $Q^{n}$ is countable and dense in $R^{n}$ ( $Q^{n}$ is a countable dense subset of $R^{n}$ , and rational open boxes form a countable basis).

[F7]

If $A$ and $B$ are at most countable then so is $A \times B$ (A product of two at most countable sets is at most countable); and if $A$ is at most countable and $B \subseteq A$ then $B$ is at most countable (Every subset of an at most countable set is at most countable, Finite, countably infinite, countable, uncountable).

Proof

technique · direct

1.1

The set $D$ of all dyadic cubes is at most countable: each of its members is nonempty and so determines its parameter pair, whose entries $m_{i} 2^{- k}$ and $(m_{i} + 1) 2^{- k}$ are rational, so the assignment of a cube to that pair is an injection of $D$ into $Q^{n} \times Q^{n}$ , a countable set, and $D$ is therefore equinumerous with an at most countable subset of it.

L3L4F6F7

1.2

For every $x \in U$ there is a natural number $k$ such that the generation- $k$ dyadic cube containing $x$ is a subset of $U$ : openness supplies a real $r > 0$ with $B (x, r) \subseteq U$ ; since $(2^{- k})_{k \in N}$ is null there is $k$ with $2^{- k} < r / n$ ; and every $y$ in the generation- $k$ cube containing $x$ has $∣ y_{i} - x_{i} ∣ < 2^{- k}$ in each coordinate, because $x_{i}$ and $y_{i}$ lie in one parameter interval of length $2^{- k}$ , so $d_{\infty} (x, y) < 2^{- k}$ and $d_{2} (x, y) \leq n d_{\infty} (x, y) < r$ .

L1L3F1F2F3F4

2.1

For $x \in U$ let $k (x)$ be the least natural number provided by step 1.2 and let $Q_{x}$ be the generation- $k (x)$ dyadic cube containing $x$ , which is unique; then $x \in Q_{x} \subseteq U$ , and $Q_{x}$ is maximal among the dyadic cubes contained in $U$ , for if $Q_{x} \subseteq Q^{'} \subseteq U$ with $Q^{'}$ of generation $k^{'}$ , then $Q^{'}$ contains $x$ and is therefore the generation- $k^{'}$ cube containing $x$ , so $k^{'} \geq k (x)$ by minimality, and then $Q_{x} \cap Q^{'} \neq \emptyset$ with $k (x) \leq k^{'}$ gives $Q^{'} \subseteq Q_{x}$ and hence $Q^{'} = Q_{x}$ .

step 1.2L1L2L3F5

3.1

Put $M : = {Q_{x} : x \in U}$ : its members are dyadic cubes contained in $U$ and every $x \in U$ lies in one of them, so $⋃ M = U$ ; two members meeting each other are nested by [L2], and each being maximal in $U$ they are equal, so the members are pairwise disjoint; and $M \subseteq D$ is at most countable by step 1.1.

step 1.1step 2.1L2F7∎

Depends on

Used by

Dependency tree · two levels

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Sources

T. Tao, An Introduction to Measure Theory (GSM 126), Lemma 1.2.11 (standard reference, not scraped)
John K. Hunter, Measure Theory (UC Davis lecture notes), Proposition 2.20 (standard reference, not scraped)

Every open subset of RnRn is the union of a countable pairwise disjoint family of dyadic cubes