Editing General topology (section)

===Alternative definitions===
Several [[Characterizations of the category of topological spaces|equivalent definitions for a topological structure]] exist and thus there are several equivalent ways to define a continuous function.

====Neighborhood definition====
Definitions based on preimages are often difficult to use directly. The following criterion expresses continuity in terms of [[neighborhood (topology)|neighborhood]]s: ''f'' is continuous at some point ''x''&nbsp;∈&nbsp;''X'' if and only if for any neighborhood ''V'' of ''f''(''x''), there is a neighborhood ''U'' of ''x'' such that ''f''(''U'')&nbsp;⊆&nbsp;''V''. Intuitively, continuity means no matter how "small" ''V'' becomes, there is always a ''U'' containing ''x'' that maps inside ''V''.

If ''X'' and ''Y'' are metric spaces, it is equivalent to consider the [[neighborhood system]] of [[open ball]]s centered at ''x'' and ''f''(''x'') instead of all neighborhoods. This gives back the above δ-ε definition of continuity in the context of metric spaces. However, in general topological spaces, there is no notion of nearness or distance.

Note, however, that if the target space is [[Hausdorff space|Hausdorff]], it is still true that ''f'' is continuous at ''a'' if and only if the limit of ''f'' as ''x'' approaches ''a'' is ''f''(''a'').  At an [[isolated point]], every function is continuous.

====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.

====Closure operator definition====
Instead of specifying the open subsets of a topological space, the topology can also be determined by a [[Kuratowski closure operator|closure operator]] (denoted cl), which assigns to any subset ''A'' ⊆ ''X'' its [[closure (topology)|closure]], or an [[interior operator]] (denoted int), which assigns to any subset ''A'' of ''X'' its [[interior (topology)|interior]]. In these terms, a function 
:<math>f\colon (X,\mathrm{cl}) \to (X' ,\mathrm{cl}')\, </math>
between topological spaces is continuous in the sense above if and only if for all subsets ''A'' of ''X''
:<math>f(\mathrm{cl}(A)) \subseteq \mathrm{cl}'(f(A)).</math>
That is to say, given any element ''x'' of ''X'' that is in the closure of any subset ''A'', ''f''(''x'') belongs to the closure of ''f''(''A''). This is equivalent to the requirement that for all subsets ''A''<nowiki>'</nowiki> of ''X''<nowiki>'</nowiki>
:<math>f^{-1}(\mathrm{cl}'(A')) \supseteq \mathrm{cl}(f^{-1}(A')).</math>
Moreover, 
:<math>f\colon (X,\mathrm{int}) \to (X' ,\mathrm{int}') \, </math>
is continuous if and only if 
:<math>f^{-1}(\mathrm{int}'(A)) \subseteq \mathrm{int}(f^{-1}(A))</math>
for any subset ''A'' of ''X''.