section_path: Uses/Limit points
section_name: Limit points
paragraph_text: Limit points
  
  In topology, limits are used to define limit points of a subset of a topological space, which in turn give a useful characterization of closed sets.
  
  In a topological space X {\displaystyle X}, consider a subset S {\displaystyle S}. A point a {\displaystyle a} is called a limit point if there is a sequence { a n } {\displaystyle \{a_{n}\}} in S ∖ { a } {\displaystyle S\backslash \{a\}} such that a n → a {\displaystyle a_{n}\rightarrow a}.
  
  The reason why { a n } {\displaystyle \{a_{n}\}} is defined to be in S ∖ { a } {\displaystyle S\backslash \{a\}} rather than just S {\displaystyle S} is illustrated by the following example. Take X = R {\displaystyle X=\mathbb {R} } and S = ∪ { 2 } {\displaystyle S=\cup \{2\}}. Then 2 ∈ S {\displaystyle 2\in S}, and therefore is the limit of the constant sequence 2, 2, ⋯ {\displaystyle 2,2,\cdots }. But 2 {\displaystyle 2} is not a limit point of S {\displaystyle S}.
  
  A closed set, which is defined to be the complement of an open set, is equivalently any set C {\displaystyle C} which contains all its limit points.