I Dyadic cubes generate Borel sigma algebra of subset

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I know the dyadic cubes in ##\mathbb R^d## generate the Borel ##\sigma##-algebra of ##\mathbb R^d##. Let now ##U\subset \mathbb R^d## be an open subset and the claim is that all dyadic (half-open) cubes whose closure are contained in ##U## generates the Borel ##\sigma##-algebra on ##U##. How?
Dyadic cubes of order ##n\in\mathbb N## are sets of the form $$C=\prod_{j=1}^d (k_j2^{-n},(k_j+1)2^{-n}],\quad k_j\in\mathbb Z.$$ I know that the dyadic cubes in ##\mathbb R^d## induce a generating collection for the Borel ##\sigma##-algebra of ##U## (by intersecting the dyadic cubes in ##\mathbb R^d## with ##U##), however, this induced generating collection may be larger than merely the dyadic cubes whose closure is in ##U##.

If I can somehow show that every open subset of ##U## is a countable union of dyadic cubes whose closure is in ##U##, then I'm done. Note, an open subset of ##U## is of the form ##A\cap U##, where ##A## is open in ##\mathbb R^d## (so the open subsets of ##U## are open in ##\mathbb R^d## as well). I feel stuck on how to proceed.
 
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My 2 ct.

My intuition would be: Consider ##U=(0,1)\subseteq \mathbb{R}.## Then we can exhaust ##U## by dyadic intervals and all can be counted since ##\mathbb{Z}\times \mathbb{N}## is countable.

However, I tend to make mistakes when it comes to topology. Maybe we need Dynkin systems to prove it formally: https://en.wikipedia.org/wiki/Dynkin_system but this could as well be an overreaction and it is as simple as with ##(0,1)##.
 
Cool. After researching a bit on the wide world web, here are my 2 ct. :smile:

If we can show the claim for all closed dyadic cubes that are fully contained in ##U##, then it is also true for all half-open dyadic cubes whose closure is contained ##U##. This is because the closure of a half-open cube is a closed cube and every closed cube can be written as a countable intersection of half-open cubes. Indeed,\begin{align*}C^n&=\prod_{j=1}^d (k_j2^{-n},(k_j+1)2^{-n}],\\ \text{and}\quad\overline{C^n}&=\prod_{j=1}^d [k_j2^{-n},(k_j+1)2^{-n}].\end{align*} But also if we define \begin{align*}C_i^n&=\prod_{j=1}^d (k_j2^{-n}+1/i,(k_j+1)2^{-n}],\quad i\in\mathbb N\setminus\{1,\ldots,N-1\},\\ \text{then}\quad\overline{C^n}&=\bigcap_{i\geq N}C_i.\end{align*}Of course, we choose ##(k_j)_{j=1}^d\subset\mathbb Z## and ##n,N\in\mathbb N## appropriately so that all sets above are in ##U##.

So we proceed by showing all closed dyadic cubes that are fully contained in ##U## generate the Borel ##\sigma##-field of ##U##. Since the dyadic cubes are Borel sets (they are closed), we only need to show that the open subsets of ##U## are contained in ##\sigma(\mathcal C)##, where ##\mathcal C## are the closed dyadic cubes fully contained in ##U##. But ##U## is separable as a subset of separable ##\mathbb R^d##, hence it has a countable base of open balls. So in turn, the problem reduces to showing that every open ball in ##U## can be written as a union of closed dyadic cubes fully contained in the open ball.

This last claim is just this answer.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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