nLab
restriction

Definition

The restriction of a function $f: X \to Y$ to a subset $U$ of $X$ is simply the composite of $f$ and the inclusion function of $U$:

$$U \stackrel{i_U}\to X \stackrel{f}\to Y .$$

It is often written as $f_{|U}: U \to Y$, or a variation on that theme.

Related concepts

• extension

• lift

• Cauchy principal value