Examples of Nets
Every non-empty totally ordered set is directed. Therefore every function on such a set is a net. In particular, the natural numbers with the usual order form such a set, and a sequence is a function on the natural numbers, so every sequence is a net.
Another important example is as follows. Given a point x in a topological space, let Nx denote the set of all neighbourhoods containing x. Then Nx is a directed set, where the direction is given by reverse inclusion, so that S ≥ T if and only if S is contained in T. For S in Nx, let xS be a point in S. Then (xS) is a net. As S increases with respect to ≥, the points xS in the net are constrained to lie in decreasing neighbourhoods of x, so intuitively speaking, we are led to the idea that xS must tend towards x in some sense. We can make this limiting concept precise.
Read more about this topic: Net (mathematics)
Famous quotes containing the words examples of, examples and/or nets:
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“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.