Proposition 8

Let $(E_\iota)_{\iota\in I}$ be a family of ordered sets. Let A be a subset of the product ordered set $E = \prod_{\iota\in I} E_\iota$, and let $A_\iota = \mathrm{pr}_\iota A$ for each $\iota \in I$. For A to have a least upper bound in E it is necessary and sufficient that, for each $\iota \in I$, $A_\iota$ should have a least upper bound in $E_\iota$, and then we have

Suppose that, for each $\iota \in I$, $A_\iota$ has a least upper bound $b_\iota$ in $E_\iota$. To say that $c = (c_\iota)$ is an upper bound of A then means that $c_\iota \geqslant b_\iota$ for each $\iota \in I$, hence $(b_\iota)_{\iota\in I}$ is an upper bound of A. Conversely, suppose that A has a least upper bound $a = (a_\iota)_{\iota\in I}$; for each $\varkappa \in I$, $a_\varkappa$ is an upper bound of $A_\varkappa$, because if $x_\varkappa \in A_\varkappa$, there exists $x \in A$ such that $\mathrm{pr}_\varkappa x = x_\varkappa$, by the definition of $A_\varkappa$; on the other hand, if $a'_\varkappa$ is an upper bound of $A_\varkappa$ in $E_\varkappa$, the element $c' = (c'_\iota)_{\iota\in I}$ for which

is an upper bound of A; consequently $c' \geqslant a$ and therefore $a'_\varkappa \geqslant a_\varkappa$; hence $a_\varkappa$ is the least upper bound of $A_\varkappa$ in $E_\varkappa$.

☡

¶ Let F be a subset of an ordered set E, and let A be a subset of F. It can happen that one of the two elements $\sup_E A$, $\sup_F A$ exists but the other does not, or that both exist but are unequal.

Examples