Cauchy sequences of finite sets

Question

Consider the metric space $\bf R$ with the standard Euclidean metric $d$ and let $F(\bf R)$ denote the collection of all finite subsets of $\bf R$. Endow $F(\bf R)$ with the Hausdorff metric $d_H$. See Wikipedia for the definition of $d_H$.

Is $F(\bf R)$ complete w.r.t. $d_H$?

Closely related: http://math.stackexchange.com/questions/123789/ — t.b., Mar 25 '12 at 21:42

score 3 · Answer 1 · answered Mar 25 '12 at 20:54

3

No. Take a dense subset $\{d_1,d_2,d_3,\ldots\} $ of $[0,1]$. For each $n$, let $F_n=\{d_1,\ldots, d_n\}$. Then $(F_n)$ converges to $[0,1]$ and is therefore a Cauchy-sequence. But the limit $[0,1]$ is obviously not a finite set.

answered Mar 25 '12 at 20:54

Michael Greinecker

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Cauchy sequences of finite sets

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