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Convergence in the Hausdorff metric

  • Thread starter math8
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  • #1
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Let (X,d) be a metric space. Let {An} be a nested family of non empty compact subsets of X. Let A=Intersection of all An.
We have that A is non empty and compact.

We show An converges to A in the Hausdorff metric (D).

I know D(A,B)= Inf {t>or eq.0 : A C B_t and B C A_t} Where A_t is the t-parallel body of A meaning A_t={x in X: d(x,A) < or eq.t}.

But I am not sure how to proceed.
 

Answers and Replies

  • #2
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Haven't thought this through completely, but some thoughts:

Let epsilon > 0.

Show there exists N such that [tex]A_N\subset A_{\epsilon}[/tex].

Suppose not. Get sequence of x_n in A_n but not in A_epsilon, then thin to convergent subsequence.

Obtain contradiction.
 

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