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Show that if $F$ is a closed and connected subset of a metric space $X$ then for every pair of points $a,b\in F$ and each $r>0$ there are points $z_0,z_1,\ldots,z_n$ in $F$ with $z_0=a$, $z_n=b$ and $d(z_{k-1},z_k)<r$ for $0<k<n+1$. Is the hypothesis that $F$ be closed needed? If $F$ is a set which satisfies this property then $F$ is not necessarily connected, even if $F$ is closed. Give an example to illustrate this.

I really don't understand how I will start. Please someone give me some hints.

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First part: Consider a point $a\in F$ and $r>0$ and let $B$ be the set of points $b$ for which such a sequence exists. Show that $B$ is nonempty, open, closed (as subset of $F$).

Second part: Should follow from first part.

Third part: Consider $F=X=\mathbb R\setminus\{0\}$.

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To show B is open , we have to enter a open ball B(b,t) in B where t<=r and this ball is also in F . If F is open then i can do easily. But here F is closed ,plese tell me how i will do this. –  Ajoy Jana Jul 19 '13 at 16:55
    
plese give me the explanation of my above comment.@Hagen Von Eitzen –  Ajoy Jana Aug 5 '13 at 23:02
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