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No-Minimum distance problem

  • Thread starter Kreizhn
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  • #1
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Homework Statement



Give an example of a metric space [itex] (X,d) [/itex], and nonempty subsets A,B of X such that both A,B are closed, non-compact, disjoint ([itex] A \cap B = \emptyset [/itex]), and [itex] \forall k>0, \; \exists a \in A, b \in B [/itex] such that d(a,b)<k

The Attempt at a Solution



I've been trying to consider the set of all infinite binary sequences
[tex] X = \left\{ (x^{(1)}, x^{(2)}, \ldots, x^{(n)}, \ldots ) | x^{(i)} \in \{0,1\} \forall i \geq 1 \right\} [/tex]

but I ended up showing that this is a compact metric space and as such all closed subsets are necessarily compact.

So I'm not terribly sure about any other examples that might work...

Edit: X is a compact metric space under

[tex] d(x,y) = \displaystyle \sum_{k=1}^\infty \frac{1}{2^k} | x^{(k)} - y^{(k)} | [/tex]
 
Last edited:

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
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R^2 is not compact. And graphs of continuous functions R->R are closed. Does that suggest anything?
 

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