Editing Net (mathematics) (section)

===Function from a well-ordered set to a topological space===

Consider a [[Well-order|well-ordered set]] <math>[0, c]</math> with limit point <math>t</math> and a function <math>f</math> from <math>[0, t)</math> to a topological space <math>X.</math> This function is a net on <math>[0, t).</math>

It is eventually in a subset <math>V</math> of <math>X</math> if there exists an <math>r \in [0, t)</math> such that for every <math>s \in [r, t)</math> the point <math>f(s)</math> is in <math>V.</math>

So <math>\lim_{x \to t} f(x) \to L</math> if and only if for every neighborhood <math>V</math> of <math>L,</math> <math>f</math> is eventually in <math>V.</math>

The net <math>f</math> is frequently in a subset <math>V</math> of <math>X</math> if and only if for every <math>r \in [0, t)</math> there exists some <math>s \in [r, t)</math> such that <math>f(s) \in V.</math>

A point <math>y \in X</math> is a cluster point of the net <math>f</math> if and only if for every neighborhood <math>V</math> of <math>y,</math> the net is frequently in <math>V.</math>

The first example is a special case of this with <math>c = \omega.</math>

See also [[Order topology#Ordinal-indexed sequences|ordinal-indexed sequence]].