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I was told that the relation $\le$ is a total order on R, it is dense, and it has a least upper bound property. I actually have don't understand those 3 properties... :/

David
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2 Answers2

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Total order means, that the order relation $\leq$ is antisymmetric (if $a\leq b$ and $b\leq a$ then $a=b$), transitive ($a\leq b$ and $b\leq c$ implies $a\leq c$) and total (for all $a,b$ holds $a\leq b$ or $b\leq a$, compare e.g. with the subset relation which satisfies the first two, but not totality).

Dense means that for $a,c$ with $a < c$ there is some $b$ with $a < b$ and $b < c$.

The least upper bound property means that for each set that is bounded from above the supremum exists, e.g. $(-\infty,\sqrt{2})$ has least upper bound $\sqrt{2}$ which does not exist in $\mathbb{Q}$ (which satisfies the two other properties).

$\mathbb{R}$ is determined as the (up to isomorphism) unique field that has a total order and the least upper bound property.

Sebastian Bechtel
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In contrast to a partial order the a total order is defined everywhere. This means that given arbitrary a and b either a<=b or b>=a.

Tom J
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  • In a total order, each pair of elements is comparable (either $x \le y$ or $y \le x$ or both). In a partial order you can have a pair of elements incomparable (neither $x \le y$ nor $y \le x$). – GEdgar Apr 17 '15 at 21:40