# Definition:Fixed Point

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## Definition

Let $f: S \to T$ be a mapping.

Then a **fixed point** (or **fixed element**) **of $S$ under $f$** is an $x \in S$ such that $\map f x = x$.

## Also defined as

The concept of a **fixed point** is usually encountered in the context of self-maps, that is, where $S = T$.

## Also see

## 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 $4.13$