section_path: Types of limits/In sequences/Topological space
section_name: Topological space
paragraph_text: Topological space
  
  In some sense the most abstract space in which limits can be defined are topological spaces. If X {\displaystyle X} is a topological space with topology τ {\displaystyle \tau }, and { a n } n ≥ 0 {\displaystyle \{a_{n}\}_{n\geq 0}} is a sequence in X {\displaystyle X}, then the limit (when it exists) of the sequence is a point a ∈ X {\displaystyle a\in X} such that, given a (open) neighborhood U ∈ τ {\displaystyle U\in \tau } of a {\displaystyle a}, there exists an N {\displaystyle N} such that for every n > N {\displaystyle n>N}, a n ∈ U {\displaystyle a_{n}\in U} is satisfied. In this case, the limit (if it exists) may not be unique. However it must be unique if X {\displaystyle X} is a Hausdorff space.