Page "Net (mathematics)" Paragraph 9

Another important example is as follows.

Given a point x in a topological space, let N < sub > x </ sub > denote the set of all neighbourhoods containing x.

Then N < sub > x </ sub > 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 N < sub > x </ sub >, let x < sub > S </ sub > be a point in S. Then ( x < sub > S </ sub >) is a net.

As S increases with respect to ≥, the points x < sub > S </ sub > in the net are constrained to lie in decreasing neighbourhoods of x, so intuitively speaking, we are led to the idea that x < sub > S </ sub > must tend towards x in some sense.

We can make this limiting concept precise.