# Characterization of Strictly Increasing Mapping on Woset

Jump to navigation
Jump to search

## 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 $x,y \in J$:

\(\displaystyle x\) | \(\prec\) | \(\displaystyle y\) | |||||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle h(x)\) | \(\prec\) | \(\displaystyle h(y)\) | Definition of strictly increasing | |||||||||

\(\displaystyle h[S_y]\) | \(=\) | \(\displaystyle \left\{ { h(x) \in E: \exists x \in J: h(x) \prec h(y) } \right\}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle S_{h(y)}\) | |||||||||||

\(\displaystyle \min\left({E \setminus h\left[{S_y}\right] }\right)\) | \(=\) | \(\displaystyle \min\left({E \setminus S_{h(y)} }\right)\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle h(y)\) | Definition of smallest and of initial segment |

$\Box$

### $(2)$ 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:

\(\displaystyle h(y)\) | \(=\) | \(\displaystyle \min\left({E \setminus h\left[{S_y}\right] }\right)\) | |||||||||||

\(\displaystyle h[S_y]\) | \(=\) | \(\displaystyle \left\{ { h(x) \in E: \exists x \in J: h(x) = \min\left({E \setminus h\left[{S_y }\right] }\right)} \right\}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left\{ { h(x) \in E: \exists x \in J: h(x) \prec h(y)} \right\}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle S_{h(y)}\) | Definition of initial segment |

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 $E$.

$\blacksquare$

## Sources

- 2000: James R. Munkres:
*Topology*(2nd ed.): Supplementary Exercise $1.2$