▁is ▁c of inal ▁in ▁A — that ▁is , ▁for ▁every ▁α ▁in ▁A ▁there ▁exists ▁a ▁ β ▁in ▁B ▁such ▁that ▁ ▁h ( β ) ▁ ≥ ▁α . ▁ ▁While ▁complicated , ▁the ▁definition ▁does ▁general ize ▁some ▁key ▁the or ems ▁about ▁subsequ ences : ▁ ▁A ▁net ▁( x α ) ▁converges ▁to ▁x ▁if ▁and ▁only ▁if ▁every ▁sub net ▁of ▁( x α ) ▁converges ▁to ▁x . ▁A ▁net ▁( x α ) ▁has ▁a ▁cluster ▁point ▁y ▁if ▁and ▁only ▁if ▁it ▁has ▁a ▁sub net ▁( y β ) ▁that ▁converges ▁to ▁y . ▁A ▁topological ▁space ▁X ▁is ▁compact ▁if ▁and ▁only ▁if ▁every ▁net ▁in ▁X ▁has ▁a ▁conver gent ▁sub net ▁( see ▁net ▁for ▁a ▁proof ). ▁ ▁A ▁seem ingly ▁more ▁natural ▁definition ▁of ▁a ▁sub net ▁would ▁be ▁to ▁require ▁B ▁to ▁be ▁a ▁c of inal ▁subset ▁of ▁A ▁and ▁that ▁h ▁be ▁the ▁identity ▁map . ▁ ▁This ▁concept , ▁known ▁as ▁a ▁c of inal ▁sub net , ▁turns ▁out ▁to ▁be ▁in ade qu ate . ▁ ▁For ▁example , ▁the ▁second ▁theorem ▁above ▁fails ▁for ▁the ▁T ych on off ▁pl ank ▁if ▁we ▁restrict ▁ourselves ▁to ▁c of inal ▁sub net s . ▁ ▁While ▁a ▁sequence ▁is ▁a ▁net , ▁a ▁sequence ▁has ▁sub net s ▁that ▁are ▁not ▁subsequ ences . ▁ ▁For ▁example ▁the ▁net ▁( 1 , ▁ 1 , ▁ 2 , ▁ 3