if olds ▁with ▁r ▁= ▁ 0 . ▁An ▁embedded ▁topological ▁sub man if old ▁is ▁not ▁necessarily ▁regular ▁in ▁the ▁sense ▁of ▁the ▁existence ▁of ▁a ▁local ▁chart ▁at ▁each ▁point ▁extending ▁the ▁embedding . ▁C ounter examples ▁include ▁wild ▁ar cs ▁and ▁wild ▁kn ots . ▁ ▁Properties ▁ ▁Given ▁any ▁imm ers ed ▁sub man if old ▁S ▁of ▁M , ▁the ▁tang ent ▁space ▁to ▁a ▁point ▁p ▁in ▁S ▁can ▁naturally ▁be ▁thought ▁of ▁as ▁a ▁linear ▁sub space ▁of ▁the ▁tang ent ▁space ▁to ▁p ▁in ▁M . ▁This ▁follows ▁from ▁the ▁fact ▁that ▁the ▁inclusion ▁map ▁is ▁an ▁imm ersion ▁and ▁provides ▁an ▁injection ▁▁▁ ▁Suppose ▁S ▁is ▁an ▁imm ers ed ▁sub man if old ▁of ▁M . ▁If ▁the ▁inclusion ▁map ▁i ▁: ▁S ▁→ ▁M ▁is ▁closed ▁then ▁S ▁is ▁actually ▁an ▁embedded ▁sub man if old ▁of ▁M . ▁Con vers ely , ▁if ▁S ▁is ▁an ▁embedded ▁sub man if old ▁which ▁is ▁also ▁a ▁closed ▁subset ▁then ▁the ▁inclusion ▁map ▁is ▁closed . ▁The ▁inclusion ▁map ▁i ▁: ▁S ▁→ ▁M ▁is ▁closed ▁if ▁and ▁only ▁if ▁it ▁is ▁a ▁proper ▁map ▁( i . e . ▁inverse ▁images ▁of ▁compact ▁sets ▁are ▁compact ). ▁If ▁i ▁is ▁closed ▁then ▁S ▁is ▁called ▁a ▁closed ▁embedded ▁sub man if old ▁of ▁M . ▁Cl osed ▁embedded ▁sub man if olds ▁form ▁the ▁nic est ▁class ▁of ▁sub man if olds . ▁ ▁Sub man if olds ▁of ▁real ▁coordinate ▁space