Proving The Hamming Metric: Open Subsets and Basis of X

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Homework Help Overview

The discussion revolves around the Hamming metric in a metric space context, specifically focusing on proving properties related to open subsets and bases of open sets. Participants are tasked with demonstrating that a certain set U(d1,...,dp) is an open subset of X and that it serves as a basis for open sets in the metric space defined by the Hamming metric.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the nature of open balls in the context of the Hamming metric, questioning how to describe these balls and their properties. There are attempts to clarify the relationship between sequences and their sums in relation to the metric.

Discussion Status

The conversation is ongoing, with participants providing hints and guidance on how to approach the problem. There is a focus on understanding the structure of open sets and the implications of certain sequences within the metric space.

Contextual Notes

Participants express uncertainty about the definitions and properties of open sets and the Hamming metric, indicating a need for further exploration of these concepts. There are references to specific sequences and their sums, which are central to the discussion but remain partially unresolved.

  • #61
around an arbitrary element...wouldn't it be similar to before.

ie. (sum,k=1..infinity, (x_k-a_k)/2^k) < (1/2)^n

either x_k is with 1's starting in the kth position and 0's afterwards
and a_k is ..not sureOR 1's in the (K+1)st position

and a_k is not sure
 
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  • #62
Well, the balls around a with radius 1/2 are either:

1) elements which agree with a in it's first coordinate
2) elements which agree with a except possibly in it's first coordinate.

Can you see why?
Can you extend this to balls with radius (1/2)k??
 
  • #63
no...I don't follow
 
  • #64
Can you see why it is true for a=0?
 
  • #65
I think so?
 
  • #66
OK, npw apply the same reasoning to arbitrary a instead of 0...
 
  • #67
...I think I need a hint
 
  • #68
Well, when does

\sum_{k=1}^{+\infty}{\frac{|x_k-a_k|}{2^k}}=\frac{1}{2}

??
 
  • #69
x_k=(1,0,0,0,0...) and a_k=(0,0,0,...)

or

x_k=(0,1,0,0...) and a_k=(0,0,0,...)

or

a_k=(1,0,0,0,0...) and x_k=(0,0,0,...)

a_k=(0,1,0,0...) and x_k=(0,0,0,...)
 
  • #70
No, can you show my how you got that?

If

\sum_{k=1}^{+\infty}{\frac{|x_k-a_k|}{2^k}}=\frac{1}{2},

then what must hold for |x_k-a_k|.
 
  • #71
I have a new question.

How would I show that the metric space defined by the Hamming metric is complete?
 

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