▁prede cess ors ▁( the ▁set ▁of ▁all ▁of ▁its ▁prefix es ). ▁An ▁order - the or etic ▁tree ▁may ▁be ▁represented ▁by ▁an ▁is omorphic ▁tree ▁of ▁sequences ▁if ▁and ▁only ▁if ▁each ▁of ▁its ▁elements ▁has ▁finite ▁height ▁( that ▁is , ▁a ▁finite ▁set ▁of ▁prede cess ors ). ▁ ▁Top ology ▁The ▁set ▁of ▁infinite ▁sequences ▁over ▁ ▁( den oted ▁as ▁) ▁may ▁be ▁given ▁the ▁product ▁topology , ▁tre ating ▁X ▁as ▁a ▁discrete ▁space . ▁In ▁this ▁topology , ▁every ▁closed ▁subset ▁ ▁of ▁ ▁is ▁of ▁the ▁form ▁ ▁for ▁some ▁pr un ed ▁tree ▁. ▁Nam ely , ▁let ▁ ▁consist ▁of ▁the ▁set ▁of ▁finite ▁prefix es ▁of ▁the ▁infinite ▁sequences ▁in ▁. ▁Con vers ely , ▁the ▁body ▁ ▁of ▁every ▁tree ▁ ▁forms ▁a ▁closed ▁set ▁in ▁this ▁topology . ▁ ▁Fre qu ently ▁trees ▁on ▁Cart esian ▁products ▁ ▁are ▁considered . ▁ ▁In ▁this ▁case , ▁by ▁convention , ▁we ▁consider ▁only ▁the ▁subset ▁ ▁of ▁the ▁product ▁space , ▁, ▁containing ▁only ▁sequences ▁whose ▁even ▁elements ▁come ▁from ▁ ▁and ▁odd ▁elements ▁come ▁from ▁ ▁( e . g ., ▁). ▁ ▁Element s ▁in ▁this ▁sub space ▁are ▁identified ▁in ▁the ▁natural ▁way ▁with ▁a ▁subset ▁of ▁the ▁product ▁of ▁two ▁spaces ▁of ▁sequences , ▁ ▁( the ▁subset ▁for ▁which ▁the ▁length ▁of ▁the ▁first ▁sequence ▁is ▁equal ▁to ▁or ▁ 1 ▁more ▁than ▁the ▁length ▁of ▁the ▁second ▁sequence ). ▁In ▁this ▁way ▁we