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 $X$ if and only if, 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 $t$.

Also see

• Results about towers can be found here.

Sources

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