Give an example of a linearly ordered set (L,<) and an initial segment S of L which is not of the form {x | x<a}, for any a in L.

Any suggestions?

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- Oct 11th 2009, 09:17 AM #1

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- Oct 11th 2009, 12:12 PM #2

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There is an important difference between $\displaystyle \mathbb{Q}$ and $\displaystyle \mathbb{R}$ that allows you to find initial segments of $\displaystyle \mathbb{Q}$ you cannot write $\displaystyle \{x\in\mathbb{Q}\ ;\ x<a\}$ for any $\displaystyle a\in \mathbb{Q}$.

An example different from the ones you get with the remark: $\displaystyle [-\infty ,0]\subset\mathbb{R}$.