Thus, Theorem 3.1 's converse is false. Sequences are of immense usefulness in real analysis and in metric spaces and elsewhere - and their failure to describe general topology adequately is a technical handicap. Not all sets are naturally endowed with so simple an ordering. Note that these last two items generalize the role of sequences in a metric space. Thus, for example, a sequence is an example of a net.
Any real function is a net.
The net which these choices constitute converges to p, as required. This is really just a corollary of the preceding theorem. Proof Not required. The key here is recognizing that many proofs regarding convergence in metric spaces involve constructing sequences of nested open sets about a point. Sometimes these describe the topological structure near the point and other times not.