This　paper　addresses　Gabor　analysis　on　a　discrete　periodic　set.　Such　a　scenario　can　potentially　find　its　applications　in　signal　processing　where　signals　may　present　on　a　union　of　disconnected　discrete　index　sets.　We　focus　on　the　Gabor　systems　generated　by　characteristic　functions.　A　sufficient　and　necessary　condition　for　a　set　to　be　a　tight　Gabor　set　in　discrete　periodic　sets　is　obtained;　discrete　periodic　sets　admitting　a　tight　Gabor　set　are　also　characterized;　the　perturbation　of　tight　Gabor　sets　is　investigated;　an　algorithm　to　determine　whether　a　set　is　a　tight　Gabor　set　is　presented.　Furthermore,　we　prove　that　an　arbitrary　Gabor　frame　set　can　be　represented　as　the　union　of　a　tight　Gabor　set　and　a　Gabor　Bessel　set.