I don't seem to get the special properties of Vitali sets which makes them different from the intervals, e.g. [0,1].
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5For one thing, it is not measureable! – Mariano Suárez-Álvarez Aug 17 '12 at 07:02
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I am aware that it is not measurable, but i'm in doubts of the construction of the vitali set itself... it is formed by the collection of representatives of partitions of all real numbers, but why can't the representatives form the interval [0,1] itself? – Matt Aug 17 '12 at 07:10
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2Because no two elements of the Vitali set differ by a rational number: that is precisely its characteristic property. The same is not true of the interval $[0,1]$. – Mariano Suárez-Álvarez Aug 17 '12 at 07:11
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2For another thing, the Vitali set contains only one rational number. – Per Erik Manne Aug 17 '12 at 07:35
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See here for detailed explanations of the construction of Vitali set(s). – leo Aug 18 '12 at 00:29
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The Vitali set can not be an interval. As Mariano Suarez-Alvarez mentioned, a well-known result is that the Vitali set is not measurable, but every interval is measureable.
However, even more elementary than that: A Vitali set contains exactly one element from each coset of $\mathbb{R} / \mathbb{Q}$ (as additive groups). Alternatively, you can define the equivalence relation on $\mathbb{R}$ by $a \sim b$ if and only if $a - b \in \mathbb{Q}$. The Vitali set is then a set containing exactly one element from each equivalence class. $\mathbb{Q}$ is the equivalence class containing $0$. Hence, Vitali set only contain one rational number. The Vitali set can not be a interval since intervals contain more than a single rational number.
Kuba Helsztyński
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William
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Now I notice where I got wrong. The axiom of choice. I was thinking that all elements in the equivalence class Q that's in the interval was also included. Thanks for the clarification.. by the way, sorry if im posting in the answers, I have troubles in signing up – Matt Aug 17 '12 at 08:02
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1@Matt: your accounts have been merged. Please register to avoid such difficulties in the future. – Qiaochu Yuan Aug 17 '12 at 16:51