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单词 HausdorffMetricInheritsCompleteness
释义

Hausdorff metric inherits completeness


Theorem 1.

If (X,d) is a complete metric space, then the Hausdorff metric induced by d is also completePlanetmathPlanetmathPlanetmathPlanetmath.

Proof.

Suppose (An) is a Cauchy sequencePlanetmathPlanetmath with respect to the Hausdorff metric. By selecting a subsequence if necessary, we may assume that An and An+1 are within 2-n of each other, that is, that AnK(An+1,2-n) and An+1K(An,2-n).Now for any natural numberMathworldPlanetmath N, there is a sequence (xn)nN in Xsuch that xnAn and d(xn,xn+1)<2-n. Any such sequence is Cauchywith respect to d and thus convergesPlanetmathPlanetmath to some xX. By applying the triangle inequalityMathworldMathworldPlanetmath, we see that for any nN, d(xn,x)<2-n+1.

Define A to be the set of all x such that x is the limit of a sequence(xn)n0 with xnAn and d(xn,xn+1)<2-n.Then A is nonempty.Furthermore, for any n, if xA,then there is some xnAn such that d(xn,x)<2-n+1, and soAK(An,2-n+1). Consequently, the set A¯ isnonempty, closed and boundedPlanetmathPlanetmathPlanetmath.

Suppose ϵ>0. Thus ϵ>2-N>0 for some N. Let nN+1.Then by applying the claim in the first paragraph, we have that for anyxnAn, there is some xX with d(xn,x)<2-n+1. HenceAnK(A¯,2-n+1).Hence the sequence (An) converges to A in the Hausdorff metric.∎

This proof is based on a sketch given in an exercise in [1].An exercise for the reader: is the set A constructed above closed?

References

  • 1 J. Munkres, TopologyMathworldPlanetmathPlanetmath (2nd edition), Prentice Hall, 1999.
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