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)
“Caring for children is a dance between setting appropriate limits as caretakers and avoiding unnecessary power struggles that result in unhappiness.”
—Charlotte Davis Kasl (20th century)
“He’s leaving Germany by special request of the Nazi government. First he sends a dispatch about Danzig and how 10,000 German tourists are pouring into the city every day with butterfly nets in their hands and submachine guns in their knapsacks. They warn him right then. What does he do next? Goes to a reception at von Ribbentropf’s and keeps yelling for gefilte fish!”
—Billy Wilder (b. 1906)