# 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.