# Characterization of Strictly Increasing Mapping on Woset

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## Theorem

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 statements are equivalent:

- $(1): \quad h$ is strictly increasing and its image is either all of $E$ or an initial segment of $E$

- $(2): \quad \forall \alpha \in J: \map h \alpha = \map \min {E \setminus h \sqbrk {S_\alpha} }$, and $h \sqbrk {S_\alpha} = S_{\map h \alpha}$

where:

- $h \sqbrk {S_\alpha}$ 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 $x,y \in J$:

\(\ds x\) | \(\prec\) | \(\ds y\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds \map h x\) | \(\prec\) | \(\ds \map h y\) | Definition of Strictly Increasing Mapping | ||||||||||

\(\ds h \sqbrk {S_y}\) | \(=\) | \(\ds \set {\map h x \in E: \exists x \in J: \map h x \prec \map h y}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds S_{\map h y}\) | ||||||||||||

\(\ds \map \min {E \setminus h \sqbrk {S_y} }\) | \(=\) | \(\ds \map \min {E \setminus S_{\map h y} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map h y\) | Definition of Smallest Element and Definition of Initial Segment |

$\Box$

### $(2)$ implies $(1)$

Suppose $h$ satisfies:

- $\map h \alpha = \map \min {E \setminus h \sqbrk {S_\alpha} }$

By the Principle of Recursive Definition for Well-Ordered Sets, $h$ is thus uniquely determined.

Then:

\(\ds \map h y\) | \(=\) | \(\ds \map \min {E \setminus h \sqbrk {S_y} }\) | ||||||||||||

\(\ds h \sqbrk {S_y}\) | \(=\) | \(\ds \set {\map h x \in E: \exists x \in J: \map h x = \map \min {E \setminus h \sqbrk {S_y} } }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \set {\map h x \in E: \exists x \in J: \map h x \prec \map h y}\) | ||||||||||||

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

Thus for every $x \in S_y$, we have that $\map h x \in S_{\map h y}$.

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, $h \sqbrk J$ is an initial segment of $E$ or all of $E$.

$\blacksquare$

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## Sources

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