▁to ▁prove ▁directly . ▁ ▁Bo hr ▁compact ification ▁and ▁almost - period icity ▁ ▁One ▁important ▁application ▁of ▁Pon try agin ▁du ality ▁is ▁the ▁following ▁character ization ▁of ▁compact ▁ab elian ▁topological ▁groups : ▁ ▁Theorem . ▁A ▁locally ▁compact ▁ab elian ▁group ▁G ▁is ▁compact ▁if ▁and ▁only ▁if ▁the ▁dual ▁group ▁ ▁is ▁discrete . ▁Con vers ely , ▁G ▁is ▁ ▁discrete ▁if ▁and ▁only ▁if ▁ ▁is ▁compact . ▁ ▁That ▁G ▁being ▁compact ▁implies ▁ ▁is ▁discrete ▁or ▁that ▁G ▁being ▁discrete ▁implies ▁that ▁ ▁is ▁compact ▁is ▁an ▁elementary ▁consequence ▁of ▁the ▁definition ▁of ▁the ▁compact - open ▁topology ▁on ▁ ▁and ▁does ▁not ▁need ▁Pon try agin ▁du ality . ▁One ▁uses ▁Pon try agin ▁du ality ▁to ▁prove ▁the ▁convers es . ▁ ▁The ▁Bo hr ▁compact ification ▁is ▁defined ▁for ▁any ▁topological ▁group ▁G , ▁regardless ▁of ▁whether ▁G ▁is ▁locally ▁compact ▁or ▁ab elian . ▁One ▁use ▁made ▁of ▁Pon try agin ▁du ality ▁between ▁compact ▁ab elian ▁groups ▁and ▁discrete ▁ab elian ▁groups ▁is ▁to ▁character ize ▁the ▁ ▁Bo hr ▁compact ification ▁of ▁an ▁arbitrary ▁ab elian ▁locally ▁compact ▁topological ▁group . ▁The ▁Bo hr ▁compact ification ▁B ( G ) ▁of ▁G ▁is ▁, ▁where ▁H ▁has ▁the ▁group ▁structure ▁, ▁but ▁given ▁the ▁discrete ▁topology . ▁Since ▁the ▁inclusion ▁map ▁▁▁ <0x09> ▁ ▁is ▁continuous ▁and ▁a ▁hom omorphism , ▁the ▁dual ▁morph ism ▁▁▁▁ <0x09> <0x09> <0x09> ▁is ▁a ▁morph ism ▁into ▁a ▁compact ▁group ▁which