▁uniqu ely ▁given ▁the ▁structure ▁of ▁an ▁imm ers ed ▁sub man if old ▁so ▁that ▁f ▁: ▁N ▁→ ▁f ( N ) ▁is ▁a ▁dif fe omorphism . ▁It ▁follows ▁that ▁imm ers ed ▁sub man if olds ▁are ▁precisely ▁the ▁images ▁of ▁inject ive ▁imm ers ions . ▁ ▁The ▁sub man if old ▁topology ▁on ▁an ▁imm ers ed ▁sub man if old ▁need ▁not ▁be ▁the ▁relative ▁topology ▁inherited ▁from ▁M . ▁In ▁general , ▁it ▁will ▁be ▁fin er ▁than ▁the ▁sub space ▁topology ▁( i . e . ▁have ▁more ▁open ▁sets ). ▁ ▁Im mers ed ▁sub man if olds ▁occur ▁in ▁the ▁theory ▁of ▁Lie ▁groups ▁where ▁Lie ▁sub groups ▁are ▁naturally ▁imm ers ed ▁sub man if olds . ▁ ▁Em bed ded ▁sub man if olds ▁ ▁An ▁embedded ▁sub man if old ▁( also ▁called ▁a ▁regular ▁sub man if old ), ▁is ▁an ▁imm ers ed ▁sub man if old ▁for ▁which ▁the ▁inclusion ▁map ▁is ▁a ▁topological ▁embedding . ▁That ▁is , ▁the ▁sub man if old ▁topology ▁on ▁S ▁is ▁the ▁same ▁as ▁the ▁sub space ▁topology . ▁ ▁Given ▁any ▁embedding ▁f ▁: ▁N ▁→ ▁M ▁of ▁a ▁manifold ▁N ▁in ▁M ▁the ▁image ▁f ( N ) ▁naturally ▁has ▁the ▁structure ▁of ▁an ▁embedded ▁sub man if old . ▁That ▁is , ▁embedded ▁sub man if olds ▁are ▁precisely ▁the ▁images ▁of ▁embed d ings . ▁ ▁There ▁is ▁an ▁intr insic ▁definition ▁of ▁an ▁embedded