Sequence of cuboids $\left \{ (x_1, . . . , x_n) ∈ R^n : 2^{−k} j_i ≤ x_i < 2^{−k} (j_i + 1)\right \}$ with $U=\bigcup_{l∈N }Q_l$

Asked Nov 13 '22 at 10:22

Active Nov 13 '22 at 11:14

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I have problems with this exercise. Can someone help me? I have to show that for every open set $U ⊆ R^n$ there is a sequence $(Q_l)_{l∈N}$ of pairwise disjoint semi-open cuboids of the form

$\left \{ (x_1, . . . , x_n) ∈ R^n : 2^{−k} j_i ≤ x_i < 2^{−k} (j_i + 1)\right \}$

with $k ∈ N$ and $j_1, . . . , j_n ∈ \mathbb{Z}$ such that

$U=\bigcup_{l∈N }Q_l$

edited Nov 13 '22 at 11:14

Asaf Karagila

393,674

asked Nov 13 '22 at 10:22

MarcoDJ01

What have you tried ? where are you stuck ? – Surb Nov 13 '22 at 10:51
I don't know how to start – MarcoDJ01 Nov 13 '22 at 10:53
Does this https://math.stackexchange.com/questions/3137124/open-sets-in-mathbbrd-written-as-union-of-partially-open-cubes answer your question? – david_sap Nov 13 '22 at 11:23

Sequence of cuboids $\left \{ (x_1, . . . , x_n) ∈ R^n : 2^{−k} j_i ≤ x_i < 2^{−k} (j_i + 1)\right \}$ with $U=\bigcup_{l∈N }Q_l$

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