A net is a family indexed by a directed ordered set: every pair of indices has a common upper bound. It converges to when it is eventually in every neighbourhood of . Unlike sequences, nets characterize closure in every topological space: a point is in the closure of a set precisely when some net in that set converges to it.
A subnet is obtained from a net by an order-preserving cofinal map from another directed index set. It preserves any limit of the original net, and eventual properties still hold along it. Every net in a compact set has a convergent subnet. This supplies compactness arguments when sequential compactness is unavailable.
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In mathematics, the term "net" can refer to several different concepts, depending on the context. Here are two of the most common interpretations: 1. **Net in Topology**: In topology, a net is a generalization of a sequence that allows the indexing of elements by a directed set. While a sequence is indexed by the natural numbers, a net can be indexed by any directed set, which gives it more flexibility.