Equivalence of Definitions of Order Complete Set

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Theorem

Let $\struct {S, \preceq}$ be an ordered set.


The following definitions of the concept of Order Complete Set are equivalent:

Definition 1

$\struct {S, \preceq}$ is order complete if and only if:

Each non-empty subset $H \subseteq S$ which has an upper bound admits a supremum.

Definition 2

$\struct {S, \preceq}$ is order complete if and only if:

Each non-empty subset $H \subseteq S$ which has a lower bound admits an infimum.


Proof

Definition 1 implies Definition 2

Let $\struct {S, \preceq}$ be an order complete set by definition 1.

Let $H \subseteq S$ have a lower bound.

Let $K$ be the set of all lower bounds of $H$.

Then $K \ne \O$.

By definition of lower bound:

$\forall x \in K: \forall y \in H: x \le y$

and so all elements of $H$ are upper bounds of $K$.

Thus by hypothesis $K$ admits a supremum.

Let $k = \map \sup K$.

By definition of supremum, $k$ precedes every upper bound of $K$.

In particular, $k$ precedes every element of $H$.

Thus $k$ is a lower bound of $H$.

But we have that $k$ is an upper bound of $K$.

That is, $k$ succeeds every lower bound of $H$.

That is, $k$ is an infimum of $H$.


Thus $\struct {S, \preceq}$ is an order complete set by definition 2.

$\Box$


Definition 2 implies Definition 1

Let $\struct {S, \preceq}$ be an order complete set by definition 2.

Let $H \subseteq S$ have an upper bound.

Let $K$ be the set of all upper bounds of $H$.

Then $K \ne \O$.

By definition of upper bound:

$\forall x \in K: \forall y \in H: y \le x$

and so all elements of $H$ are lower bounds of $K$.

Thus by hypothesis $K$ admits an infimum.

Let $k = \map \inf K$.

By definition of infimum, $k$ succeeds every lower bound of $K$.

In particular, $k$ succeeds every element of $H$.

Thus $k$ is an upper bound of $H$.

But we have that $k$ is a lower bound of $K$.

That is, $k$ precedes every upper bound of $H$.

That is, $k$ is a supremum of $H$.


Thus $\struct {S, \preceq}$ is an order complete set by definition 1.

$\blacksquare$


Sources

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