Filter limit

Summary

A filter limit is the limit of a function $f \colon S \to X$ along a filter on $S$, where $X$ is a topological space. The function tends to $x$ along the filter when the preimage of every neighbourhood of $x$ belongs to the filter. One notion covers limits of sequences, limits of functions at a point or at infinity, one-sided limits and sums of families, by choosing the filter. It is the same as the convergence of the image filter $f(\mathcal{F})$ to $x$.