# Definition:Progressing Mapping

## Definition

Let $C$ be a class.

Let $f: C \to C$ be a mapping from $C$ to $C$.

Then $f$ is a **progressing mapping** if and only if:

- $x \in C \implies x \subseteq \map f x$

That is, if and only if for each $x \in C$, $x$ is a subset of $\map f x$.

## Also known as

A **progressing mapping** is also known as an **inflationary mapping**, which is the term generally used in the context of an arbitrary ordered structure, notably in the field of measure theory.

Some sources use the term **progressive mapping**.

The term **extensive mapping** can also occasionally be seen, but this is not endorsed on $\mathsf{Pr} \infty \mathsf{fWiki}$ because of its possible confusion with the concept of the Axiom of Extensionality.

Sources which prefer the term **function** to **mapping** will tend to use such here: **progressing function**, **inflationary function**, and so on.

## Also see

- Results about
**progressing mappings**can be found**here**.

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 4$ A double induction principle and its applications: Definition $4.4$