Characterization of Strictly Increasing Mapping on Woset
Please fix formatting and $\LaTeX$ errors and inconsistencies.
It may also need to be brought up to our standard house style.
If you have done this or know it has been done, please remove {{Tidy}} from the code.
Lemma
Let $J$ and $E$ be well-ordered sets.
Let $h: J \to E$ be a mapping.
Let $S_\alpha$ denote an initial segment determined by $\alpha$.
The following are equivalent:
- $(1):\quad$ $h$ is strictly increasing and its image is either all of $E$ or an initial segment of $E$
- $(2):\quad$ $\forall \alpha \in J: h\left({\alpha}\right) = \min \left({E\setminus h\left[{S_\alpha}\right]}\right)$, and $h[S_\alpha] = S_{h(\alpha)}$
where:
- $h\left[{S_\alpha}\right]$ denotes the image of $S_\alpha$ under $h$
- $\min$ denotes the smallest element of the set.
Proof
$(1)$ implies $(2)$
Suppose $h$ satisfies:
- $h$ is strictly increasing and its image is either all of $E$ or an initial segment of $E$
Then for any $x,y \in J$:
| \(\displaystyle x\) | \(\prec\) | \(\displaystyle y\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \implies \ \ \) | \(\displaystyle h(x)\) | \(\prec\) | \(\displaystyle h(y)\) | $\quad$ Definition of strictly increasing | $\quad$ | ||||||||
| \(\displaystyle h[S_y]\) | \(=\) | \(\displaystyle \left\{ { h(x) \in E: \exists x \in J: h(x) \prec h(y) } \right\}\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle S_{h(y)}\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \min\left({E \setminus h\left[{S_y}\right] }\right)\) | \(=\) | \(\displaystyle \min\left({E \setminus S_{h(y)} }\right)\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle h(y)\) | $\quad$ Definition of smallest and of initial segment | $\quad$ |
$\Box$
$(2)$ implies $(1)$
Suppose $h$ satisfies:
- $h(\alpha) = \min\left({E \setminus h\left[{S_\alpha}\right] }\right)$
By the Principle of Recursive Definition for Well-Ordered Sets, $h$ is thus uniquely determined.
Then:
| \(\displaystyle h(y)\) | \(=\) | \(\displaystyle \min\left({E \setminus h\left[{S_y}\right] }\right)\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle h[S_y]\) | \(=\) | \(\displaystyle \left\{ { h(x) \in E: \exists x \in J: h(x) = \min\left({E \setminus h\left[{S_y }\right] }\right)} \right\}\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \left\{ { h(x) \in E: \exists x \in J: h(x) \prec h(y)} \right\}\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle S_{h(y)}\) | $\quad$ Definition of initial segment | $\quad$ |
Thus for every $x \in S_y$, we have that $h(x) \in S_{h(y)}$.
Therefore $h$ is an strictly increasing mapping.
Furthermore, the image set of $h$ is the union of initial segments in $E$.
By Union of Initial Segments is Initial Segment or All of Woset, $h[J]$ is an initial segment of $E$ or all of $E$.
$\blacksquare$
If you are fairly certain that there are none, please remove {{proofread}} from the code.
Sources
- 2000: James R. Munkres: Topology (2nd ed.): Supplementary Exercise $1.2$
