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For any 𝑝 ∈ ℕ and any 𝑑1 , 𝑑2 , … , 𝑑𝑝 ∈ {0, 1} denote
𝑈𝑑1 ,𝑑2 ,… ,𝑑𝑝 = {𝑥 ∈ 𝑋: 𝑥 𝑖 = 𝑑𝑖 , 𝑖 = 1, … , 𝑝}.
We need to prove that the sets of this form are open in the weighted Hamming distance
and that they form a basis for its open sets.
For the first we need to prove that any point of any such set has an open ball inside the set
(centered at that point). For the second it is enough to find a set of this form inside any
open ball (ask me why). So we need to analyze open balls in 𝑋 (with small radiuses
mostly).
1
The maximum distance is ∑∞
𝑖 =1 2𝑖 = 1. The difference between the first coordinates, if
nonzero (i.e. 1), contributes
1
2
to the distance an...
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