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 $$f$$.

Also see

 * Image of a Subset