= Topological Space =

A '''topological space''' is a [[Analysis/Sets|set]] for which closeness can be defined.

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== Description ==

A topological space is the double ''(X, τ)'' composed of:
 * a given space ''X''.
 * the '''topology''' of ''X'', notated as ''τ''.

The topology of ''X'' is the non-empty collection of '''open''' [[Analysis/Sets|subsets]] of ''X'' satisfying these conditions:

 * Both the empty set and the full set are in the topology; ''Ø, X ∈ τ''
   * In other words, both the empty set and the full set are definitionally open.
 * The topology is closed upon [[Analysis/Cardinality|countable]] unions; ''⋃ X,,i,, ∈ τ'' where members of ''τ'' be indexed as ''X,,i,,'', ''i ∈ N''
   * In other words, any countable union of the collection is open.
 * The topology is closed upon finite intersections; ''⋂ X,,i,, ∈ τ'' for ''i'' from 1 to ''n''
   * In other words, any finite intersection of the collection is open.



=== Alternate Formulation ===

A topological space is a set ''X'' for which a '''neighborhood topology''' [[Analysis/Functions|function]] is defined. Generally speaking, this is a function 𝒩''(x)'' that returns a non-empty subset of ''X'' representing the neighborhood of ''x''. It must satisfy these conditions:
 * a member is always in its own neighborhoods; if ''N ∈'' 𝒩''(x)'' then ''x ∈ N''
 * any superset of a neighborhood of ''x'' is also a neighborhood of ''x''
 * any intersection of two neighborhoods of ''x'' is also a neighborhood of ''x''
 * for all neighborhoods of ''x'' as ''N'', there is another neighborhood of ''x'' as ''M'' that is a subset of ''N''; furthermore ''N'' is a neighborhood for every member of ''M''



=== Comparison to Metric Spaces ===

[[Analysis/MetricSpace|Metric spaces]] are a similar concept. Their main characteristic is the presence of a distance function, which in some ways is a constraint to application, but it does provide a succinct way to define neighborhoods.



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CategoryRicottone