Including Functions
Definition
Let’s denote $X \subset Y$. A function that satisfies the following is called an inclusion function.
$$ i : X \to Y, \quad \text{and} \quad i(x) = x,\quad \forall x\in X $$
Explanation
Simply put, it’s an identity function whose codomain could be larger than its domain.
$i : X \hookrightarrow Y$1 or $i : X \subset Y$2 notation is also used.