Definition:Image Filter

Let $$X, Y$$ be sets, $$f : X \rightarrow Y$$ a mapping and $$\mathcal{F} \subset \mathcal{P}(X)$$ a filter on $$X$$. Then $$f(\mathcal{F}) := \{ U \subseteq Y: f^{-1}(U) \in \mathcal{F} \}$$ is a filter on $$Y$$, called the image filter of $$\mathcal{F}$$ with respect to $$f$$.