Editing Bounded set (section)

== Definition in a metric space ==

A [[subset]] {{mvar|S}} of a [[metric space]] {{math|(''M'', ''d'')}} is '''bounded''' if there exists {{math|''r'' > 0}} such that for all {{mvar|s}} and {{mvar|t}} in {{mvar|S}}, we have {{math|''d''(''s'', ''t'') < ''r''}}. The metric space {{math|(''M'', ''d'')}} is a ''bounded'' metric space (or {{mvar|d}} is a ''bounded'' metric) if {{mvar|M}} is bounded as a subset of itself.

*[[Total boundedness]] implies boundedness. For subsets of {{math|'''R'''{{sup|''n''}}}} the two are equivalent.
*A metric space is [[compact space|compact]] if and only if it is [[Complete metric space|complete]] and totally bounded.
*A subset of [[Euclidean space]] {{math|'''R'''{{sup|''n''}}}} is compact if and only if it is [[closed set|closed]] and bounded. This is also called the [[Heine-Borel theorem]].