Tower is Proper Subtower or all of Set

Lemma

Let $X$ be a non-empty set.

Let $\struct {T, \preccurlyeq}$ be a tower in $X$.

Then if $T \ne X$, $\struct {T, \preccurlyeq}$ is a proper subtower in $X$.

Proof

Suppose $T \ne X$.

Then there is an element $x \in X \setminus T$.

The singleton $\set x$ can trivially be given a well-ordering, as there are are no elements in $\set x$ other than $x$ itself.

Consider the order sum on $T \cup \set x$ defined by setting $t \prec x$ for all $t \in T$.

By Order Sum of Totally Ordered Sets is Totally Ordered, this gives a total ordering on $T \cup \set x$.

There is a smallest element of any subset $T \cup \set x$, either the one given by $\preccurlyeq$, or $x$ if the subset considered is the singleton $\set x$.

Extend the choice function $c$ defining $\struct {T, \preccurlyeq}$ by assigning $\map c {X \setminus S_x} = x$.

Then $T$ is a proper subset of $\struct {T \cup \set x, \text {extension of} \preccurlyeq}$.

So $T$ is a proper subtower in $X$.

$\blacksquare$

Sources

• 2000: James R. Munkres: Topology (2nd ed.) $\S 1.11$ Supplementary Exercise $7$