# Definition:Image Filter

## Definition

Let $X, Y$ be sets.

Let $\mathcal P \left({X}\right)$ and $\mathcal P \left({Y}\right)$ be the power sets of $X$ and $Y$ respectively.

Let $f: X \to Y$ a mapping.

Let $\mathcal F \subset \mathcal P \left({X}\right)$ be a filter on $X$.

Then

$f \left({\mathcal F}\right) := \left\{{U \subseteq Y: f^{-1} \left({U}\right) \in \mathcal F}\right\}$

is a filter on $Y$, called the image filter of $\mathcal F$ with respect to $f$.