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In Rudin's "Principles of Mathematical Analysis", there is the following definition of bounded.

Definition: Suppose $S$ is an ordered set, and $E \subset S$. If there exists a $\beta \in S$ such that $x \leq \beta$ for every $x \in E$, we say that $E$ is bounded above, and call $\beta$ an upper bound of $E$. Lower bounds are defined in the same way(with $\geq$ in place of $\leq$).

Let $S = \emptyset$.
Let $E = \emptyset$.
Then, there exists no $\beta \in S$ such that $x \leq \beta$ for every $x \in E$.
So, in this case, I think $E (\subset S)$ is not bounded above.

Am I correct or not?

Isn't $\emptyset$ an orderd set?

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  • $\begingroup$ If $S$ is empty then indeed it has no subsets that are bounded above, But IMV that is no reason to say that $S$ is not an ordered set. There are also non-empty sets endowed with an order that have no subsets that are bounded above. $\endgroup$ – drhab Jan 19 at 10:20
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Being bounded is relative to a given ordered set. If $S$ is empty, then indeed it is not bounded, which is just one more reason to agree that in general the empty set shouldn't be counted as a partial order. But if $S$ is non-empty, then the empty set is bounded, and every element is an upper and lower bound.

There is an analogy here with $(0,1)$. As a subset of itself, it is not bounded. There is no maximal element. But as a subset of $[0,1]$ or $\Bbb R$ it is certainly bounded by $0$ and $1$.

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  • $\begingroup$ Thank you very much, Asaf Karagila. Thank you for the nice example too. $\endgroup$ – tchappy ha Jan 19 at 8:53
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    $\begingroup$ There also good reasons to maintain the empty set as partial order. It serves e.g. as initial object of the category of posets and makes a forgetful functor from sets to posets possible. $\endgroup$ – drhab Jan 19 at 9:57
  • $\begingroup$ @drhab: Yes, or when you consider ordinals. But we are "adult", so we know how to distinguish these situations. $\endgroup$ – Asaf Karagila Jan 19 at 12:35
  • $\begingroup$ Adults do not argue about that. $\endgroup$ – drhab Jan 19 at 16:58

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