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)
Famous quotes containing the words limits of, limits and/or nets:
“I fear chiefly lest my expression may not be extra-vagant enough, may not wander far enough beyond the narrow limits of my daily experience, so as to be adequate to the truth of which I have been convinced.”
—Henry David Thoreau (1817–1862)
“And when discipline is concerned, the parent who has to make it to the end of an eighteen-hour day—who works at a job and then takes on a second shift with the kids every night—is much more likely to adopt the survivor’s motto: “If it works, I’ll use it.” From this perspective, dads who are even slightly less involved and emphasize firm limits or character- building might as well be talking a foreign language. They just don’t get it.”
—Ron Taffel (20th century)
“And Jesus said to them, “Follow me and I will make you fish for people.” And immediately they left their nets and followed him.”
—Bible: New Testament, Mark 1:17,18.