Editing General topology (section)

====Sequences and nets {{anchor|Heine definition of continuity}}====
In several contexts, the topology of a space is conveniently  specified in terms of [[limit points]].  In many instances, this is accomplished by specifying when a point is the [[limit of a sequence]], but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a [[directed set]], known as [[net (mathematics)|nets]].<ref>{{Cite journal | doi = 10.2307/2370388 | last1 = Moore | first1 = E. H. | last2 = Smith | first2 = H. L. | author1-link = E. H. Moore | author2-link = Herman L. Smith | year = 1922 | title = A General Theory of Limits | journal = American Journal of Mathematics | volume = 44 | issue = 2 | pages = 102&ndash;121 | jstor = 2370388}}</ref>  A function is continuous only if it takes limits of sequences to limits of sequences.  In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.

In detail, a function ''f'': ''X'' → ''Y'' is '''sequentially continuous''' if whenever a sequence (''x''<sub>''n''</sub>) in ''X'' converges to a limit ''x'', the sequence (''f''(''x''<sub>''n''</sub>)) converges to ''f''(''x'').<ref>{{cite journal | last1 = Heine | first1 = E. | year = 1872 | title = Die Elemente der Functionenlehre.. | url = http://eudml.org/doc/148175 | journal = Journal für die reine und angewandte Mathematik | volume = 74 | pages = 172–188 }}</ref>  Thus sequentially continuous functions "preserve sequential limits".  Every continuous function is sequentially continuous.  If ''X'' is a [[first-countable space]] and [[Axiom of countable choice|countable choice]] holds, then the converse also holds: any function preserving sequential limits is continuous.  In particular, if ''X'' is a metric space, sequential continuity and continuity are equivalent.  For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called [[sequential space]]s.) This motivates the consideration of nets instead of sequences in general topological spaces.  Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions.