▁a ▁general ization ▁of ▁finite ▁spaces ▁since ▁they ▁share ▁many ▁of ▁the ▁same ▁properties . ▁ ▁Every ▁finite ▁topological ▁space ▁is ▁also ▁second - count able ▁( there ▁are ▁only ▁finit ely ▁many ▁open ▁sets ) ▁and ▁separ able ▁( since ▁the ▁space ▁itself ▁is ▁count able ). ▁ ▁Se par ation ▁ax i oms ▁ ▁If ▁a ▁finite ▁topological ▁space ▁is ▁T 1 ▁( in ▁particular , ▁if ▁it ▁is ▁Haus dor ff ) ▁then ▁it ▁must , ▁in ▁fact , ▁be ▁discrete . ▁This ▁is ▁because ▁the ▁complement ▁of ▁a ▁point ▁is ▁a ▁finite ▁union ▁of ▁closed ▁points ▁and ▁therefore ▁closed . ▁It ▁follows ▁that ▁each ▁point ▁must ▁be ▁open . ▁ ▁Therefore , ▁any ▁finite ▁topological ▁space ▁which ▁is ▁not ▁discrete ▁cannot ▁be ▁T 1 , ▁Haus dor ff , ▁or ▁anything ▁stronger . ▁ ▁However , ▁it ▁is ▁possible ▁for ▁a ▁non - dis crete ▁finite ▁space ▁to ▁be ▁T 0 . ▁In ▁general , ▁two ▁points ▁x ▁and ▁y ▁are ▁top olog ically ▁ind ist ingu ish able ▁if ▁and ▁only ▁if ▁x ▁ ≤ ▁y ▁and ▁y ▁ ≤ ▁x , ▁where ▁ ≤ ▁is ▁the ▁special ization ▁pre order ▁on ▁X . ▁It ▁follows ▁that ▁a ▁space ▁X ▁is ▁T 0 ▁if ▁and ▁only ▁if ▁the ▁special ization ▁pre order ▁ ≤ ▁on ▁X ▁is ▁a ▁partial ▁order . ▁There ▁are ▁numerous ▁partial ▁orders ▁on ▁a ▁finite ▁set . ▁Each ▁defines ▁a ▁unique ▁T 0 ▁topology . ▁ ▁Similarly , ▁a ▁space ▁is ▁R 0 ▁if