▁and ▁only ▁if ▁the ▁special ization ▁pre order ▁is ▁an ▁equivalence ▁relation . ▁Given ▁any ▁equivalence ▁relation ▁on ▁a ▁finite ▁set ▁X ▁the ▁associated ▁topology ▁is ▁the ▁partition ▁topology ▁on ▁X . ▁The ▁equivalence ▁classes ▁will ▁be ▁the ▁classes ▁of ▁top olog ically ▁ind ist ingu ish able ▁points . ▁Since ▁the ▁partition ▁topology ▁is ▁pseud omet riz able , ▁a ▁finite ▁space ▁is ▁R 0 ▁if ▁and ▁only ▁if ▁it ▁is ▁completely ▁regular . ▁ ▁Non - dis crete ▁finite ▁spaces ▁can ▁also ▁be ▁normal . ▁The ▁ex cluded ▁point ▁topology ▁on ▁any ▁finite ▁set ▁is ▁a ▁completely ▁normal ▁T 0 ▁space ▁which ▁is ▁non - dis crete . ▁ ▁Connect ivity ▁ ▁Connect ivity ▁in ▁a ▁finite ▁space ▁X ▁is ▁best ▁understood ▁by ▁considering ▁the ▁special ization ▁pre order ▁ ≤ ▁on ▁X . ▁We ▁can ▁associate ▁to ▁any ▁pre ordered ▁set ▁X ▁a ▁directed ▁graph ▁ Γ ▁by ▁taking ▁the ▁points ▁of ▁X ▁as ▁vertices ▁and ▁drawing ▁an ▁edge ▁x ▁→ ▁y ▁whenever ▁x ▁ ≤ ▁y . ▁The ▁connect ivity ▁of ▁a ▁finite ▁space ▁X ▁can ▁be ▁understood ▁by ▁considering ▁the ▁connect ivity ▁of ▁the ▁associated ▁graph ▁ Γ . ▁ ▁In ▁any ▁topological ▁space , ▁if ▁x ▁ ≤ ▁y ▁then ▁there ▁is ▁a ▁path ▁from ▁x ▁to ▁y . ▁One ▁can ▁simply ▁take ▁f ( 0 ) ▁= ▁x ▁and ▁f ( t ) ▁= ▁y ▁for ▁t ▁> ▁ 0 . ▁It ▁is ▁easily ▁to ▁verify ▁that ▁f ▁is ▁continuous . ▁It ▁follows ▁that ▁the ▁path