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 $7$