TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-26

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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

1.1L3L4F6F7

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.

1.2L1L3F1F2F3F4

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 $d_{2} (x, y) \leq n d_{\infty} (x, y) < r$ .

2.1step 1.2L1L2L3F5

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 $Q^{'} = Q_{x}$ .

3.1step 1.1step 2.1L2F7∎

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.

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

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