Definition:Tower in Set

Definition
Let $X$ be a set.

Let $T$ be any non-empty subset of $X$

Let $c$ be a fixed choice function on the non-empty subsets $T$ of $X$.

Let $\preccurlyeq$ be a well-ordering on $T$.

The well-ordered set $\struct {T, \preccurlyeq}$ is a tower in $X$, for all $t \in T$:


 * $t = \map c {X \setminus \map {S_t} T}$

where $\map {S_t} T$ is the initial segment of $T$ determined by $t$.