Limits of Nets
If (xα) is a net from a directed set A into X, and if Y is a subset of X, then we say that (xα) is eventually in Y (or residually in Y) if there exists an α in A so that for every β in A with β ≥ α, the point xβ lies in Y.
If (xα) is a net in the topological space X, and x is an element of X, we say that the net converges towards x or has limit x and write
- lim xα = x
if and only if
- for every neighborhood U of x, (xα) is eventually in U.
Intuitively, this means that the values xα come and stay as close as we want to x for large enough α.
Note that the example net given above on the neighborhood system of a point x does indeed converge to x according to this definition.
Given a base for the topology, in order to prove convergence of a net it is necessary and sufficient to prove that there exists some point x, such that (xα) is eventually in all members of the base containing this putative limit.
Read more about this topic: Net (mathematics)
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