Editing Equicontinuity (section)

== Equicontinuity of maps valued in topological groups ==

Suppose that {{mvar|T}} is a topological space and {{mvar|Y}} is an additive [[topological group]] (i.e. a [[Group (algebra)|group]] endowed with a topology making its operations continuous). 
[[Topological vector space]]s are prominent examples of topological groups and every topological group has an associated canonical [[Uniform space|uniformity]].

:'''Definition''':{{sfn | Narici|Beckenstein | 2011 | pp=133-136}} A family {{mvar|H}} of maps from {{mvar|T}} into {{mvar|Y}} is said to be '''equicontinuous at''' {{math|''t'' ∈ ''T''}} if for every neighborhood {{mvar|V}} of {{mvar|0}} in {{mvar|Y}}, there exists some neighborhood {{mvar|U}} of {{mvar|t}} in {{mvar|T}} such that {{math|''h''(''U'') ⊆ ''h''(''t'') + ''V''}} for every {{math|''h'' ∈ ''H''}}. We say that {{mvar|H}} is '''equicontinuous''' if it is equicontinuous at every point of {{mvar|T}}.

Note that if {{mvar|H}} is equicontinuous at a point then every map in {{mvar|H}} is continuous at the point. 
Clearly, every finite set of continuous maps from {{mvar|T}} into {{mvar|Y}} is equicontinuous.