General Topology/Nets

As are filters, nets are analogues of sequences, which are used to adapt theorems which otherwise would only hold for "nice" spaces to the setting of general topological spaces. The downside is, as with filters, that theorems involving nets often use the axiom of choice. Whether you use nets or filters is a matter of taste, and a matter of selecting that tool which uses the least amount of choice in your given situation.

In order to denote that a sequence $(x_n{)}_{n\in \mathbb{N}}$ converges to a point $x$, we shall resort to the useful notation $x_n\to x$.