Proper Subtower is Initial Segment
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Theorem
Let $\struct {T_1, \preccurlyeq}$ be a proper subtower of $\struct {T_2, \preccurlyeq}$.
Then $\struct {T_1, \preccurlyeq}$ is an initial segment of $\struct {T_2, \preccurlyeq}$.
This article, or a section of it, needs explaining. In particular: What sort of object is $\struct {T_2, \preccurlyeq}$ -- an ordered set, a totally ordered set, a well-ordered set, an ordinal, what? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
Proof
Define the set:
- $Y = \set {y \in T_1: S_y \text { is an initial segment of } \struct {T_2, \preccurlyeq} }$.
Then:
\(\ds \map {S_x} {T_1}\) | \(=\) | \(\ds \set {b \in T_1, x \in T_1: b \prec x}\) | Definition of Initial Segment | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {b \in T_2, x \in T_2: b \prec x}\) | Definition of Proper Subtower in Set, as $T_1 \subseteq T_2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {S_x} {T_2}\) | Definition of Initial Segment |
By Induction on Well-Ordered Set, $Y = T_1$.
That is, $\struct {T_1, \preccurlyeq}$ is an initial segment in $\struct {T_2, \preccurlyeq}$.
$\blacksquare$
Sources
- 2000: James R. Munkres: Topology (2nd ed.): Supplementary Exercises $1.7$