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Homedense total order

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# dense total order

A total order $(S,<)$ is *dense* if whenever $x<z$ in $S$, there exists at least one element $y$ of $S$ such that $x<y<z$. That is, each nontrivial closed interval has nonempty interior.

A subset $T$ of a total order $S$ is *dense in* $S$ if for every $x,z\in S$ such that $x<z$, there exists some $y\in T$ such that $x<y<z$. Because of this, a dense total order $S$ is sometimes said to be *dense in itself*.

For example, the integers with the usual order are not dense, since there is no integer strictly between $0$ and $1$. On the other hand, the rationals $\mathbb{Q}$ are dense, since whenever $r$ and $s$ are rational numbers, it follows that $(r+s)/2$ is a rational number strictly between $r$ and $s$. Also, both $\mathbb{Q}$ and the irrationals $\mathbb{R}\setminus\mathbb{Q}$ are dense in $\mathbb{R}$.

It is usually convenient to assume that a dense order has at least two elements. This allows one to avoid the trivial cases of the one-point order and the empty order.

## Mathematics Subject Classification

06A05*no label found*

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