Net

Summary

A net in a space $X$ is a family of points of $X$ indexed by a directed set. It generalizes a sequence, which is indexed by $\N$. Convergence of nets is defined just as for sequences: eventually the net stays in every neighbourhood of the limit. In every topological space, nets describe closures, continuity, compactness and the Hausdorff property, which sequences do only in special spaces. Nets are an alternative to the convergence of filters, and the two languages are equivalent.