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Let $[a,b]$ be an interval in $\mathbb{R}$. Let $P$ be a finite partition of $[a,b]$.

Define $T_n=\{a+\frac{b-a}{n}i \in \mathbb{R}| 0≦i≦n\}$ for each $n\in\mathbb{Z}^+$. (Let's denote $T_n=\{t_0,...,t_n\}$ for convenience)

My questions is, does there exists a finite refinement $Q=\{x_0,...,x_m\}$ of $P$ and $T_N$ satisfy below two properties?

That is;

(i) $i≠j\bigwedge t_i\in [x_k,x_{k+1}]\bigwedge t_j\in [x_l,x_{l+1}] \Rightarrow k≠l$

(ii) $\forall i≦N, \exists k such that $t_i\in[x_k,x_{k+1}]$

(iii) $\forall k such that $t_i\in[x_k,x_{k+1}]$

I believe this is false, but i cannot figure out how to show that..

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    I have missed one more property in my mind.. Now it's edited.2013-01-01

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Hint: There are such sets $Q$. Think about what each of the conditions mean intuitively.

i) No two of the $t_i$ are in the same subinterval of $Q$.

ii) Each $t_i$ is in some subinterval of $Q$ (This seems a slightly unusual conition, surely any partition of $[a,b]$ will satisfy this property for any subset of $[a,b]$?).

iii) Each subinterval of $Q$ has some element of $t_i$ in it.

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    @Katlus I'm not exactly sure what you mean by your comments, but I think that the partition $\{t_0, \frac12(t_0 + t_1), t_1, \frac12(t_1+t_2),\dots, t_n\}$ satisfies all three of the properties in the question.2013-01-02