K | ▁is ▁the ▁geometric ▁real ization ▁of ▁K . ▁It ▁follows ▁that ▁the ▁hom ot opy ▁groups ▁of ▁| K | ▁and ▁X K ▁are ▁is omorphic . ▁ ▁In ▁fact , ▁the ▁underlying ▁set ▁of ▁X K ▁can ▁be ▁taken ▁to ▁be ▁K ▁itself , ▁with ▁the ▁topology ▁associated ▁to ▁the ▁inclusion ▁partial ▁order . ▁ ▁Number ▁of ▁top ologies ▁on ▁a ▁finite ▁set ▁ ▁As ▁discussed ▁above , ▁top ologies ▁on ▁a ▁finite ▁set ▁are ▁in ▁one - to - one ▁correspond ence ▁with ▁pre orders ▁on ▁the ▁set , ▁and ▁T 0 ▁top ologies ▁are ▁in ▁one - to - one ▁correspond ence ▁with ▁partial ▁orders . ▁Therefore , ▁the ▁number ▁of ▁top ologies ▁on ▁a ▁finite ▁set ▁is ▁equal ▁to ▁the ▁number ▁of ▁pre orders ▁and ▁the ▁number ▁of ▁T 0 ▁top ologies ▁is ▁equal ▁to ▁the ▁number ▁of ▁partial ▁orders . ▁ ▁The ▁table ▁below ▁lists ▁the ▁number ▁of ▁distinct ▁( T 0 ) ▁top ologies ▁on ▁a ▁set ▁with ▁n ▁elements . ▁It ▁also ▁lists ▁the ▁number ▁of ▁in equ ivalent ▁( i . e . ▁non home omorphic ) ▁top ologies . ▁ ▁Let ▁T ( n ) ▁denote ▁the ▁number ▁of ▁distinct ▁top ologies ▁on ▁a ▁set ▁with ▁n ▁points . ▁There ▁is ▁no ▁known ▁simple ▁formula ▁to ▁compute ▁T ( n ) ▁for ▁arbitrary ▁n . ▁The ▁Online ▁Encyclopedia ▁of ▁Integer ▁Se qu ences ▁presently ▁lists ▁T ( n ) ▁for ▁n ▁ ≤ ▁ 1 8 . ▁ ▁The ▁number ▁of ▁distinct