Editing Net (mathematics) (section)

=== Open sets and characterizations of topologies ===

{{See also|Axiomatic foundations of topological spaces#Definition via convergence of nets}}

A subset <math>S \subseteq X</math> is open if and only if no net in <math>X \setminus S</math> converges to a point of <math>S.</math>{{sfn|Howes|1995|pp=83–92}} Also, subset <math>S \subseteq X</math> is open if and only if every net converging to an element of <math>S</math> is eventually contained in <math>S.</math> 
It is these characterizations of "open subset" that allow nets to characterize [[Topology (structure)|topologies]]. 
Topologies can also be characterized by closed subsets since a set is open if and only if its complement is closed. So the characterizations of "[[closed set]]" in terms of nets can also be used to characterize topologies.