▁the ▁singleton ▁space ▁serves ▁as ▁a ▁terminal ▁object . ▁▁ 2 ▁points ▁ ▁Let ▁X ▁= ▁{ a , b } ▁be ▁a ▁set ▁with ▁ 2 ▁elements . ▁There ▁are ▁four ▁distinct ▁top ologies ▁on ▁X : ▁{ ∅ , ▁{ a , b }} ▁( the ▁trivial ▁topology ) ▁{ ∅ , ▁{ a }, ▁{ a , b }} ▁{ ∅ , ▁{ b }, ▁{ a , b }} ▁{ ∅ , ▁{ a }, ▁{ b }, ▁{ a , b }} ▁( the ▁discrete ▁topology ) ▁ ▁The ▁second ▁and ▁third ▁top ologies ▁above ▁are ▁easily ▁seen ▁to ▁be ▁home omorphic . ▁The ▁function ▁from ▁X ▁to ▁itself ▁which ▁sw aps ▁a ▁and ▁b ▁is ▁a ▁home omorphism . ▁A ▁topological ▁space ▁home omorphic ▁to ▁one ▁of ▁these ▁is ▁called ▁a ▁S ier pi ński ▁space . ▁So , ▁in ▁fact , ▁there ▁are ▁only ▁three ▁in equ ivalent ▁top ologies ▁on ▁a ▁two - point ▁set : ▁the ▁trivial ▁one , ▁the ▁discrete ▁one , ▁and ▁the ▁S ier pi ński ▁topology . ▁ ▁The ▁special ization ▁pre order ▁on ▁the ▁S ier pi ński ▁space ▁{ a , b } ▁with ▁{ b } ▁open ▁is ▁given ▁by : ▁a ▁ ≤ ▁a , ▁b ▁ ≤ ▁b , ▁and ▁a ▁ ≤ ▁b . ▁▁ 3 ▁points ▁ ▁Let ▁X ▁= ▁{ a , b , c } ▁be ▁a ▁set ▁with ▁ 3 ▁elements . ▁There ▁are ▁ 2 9 ▁distinct ▁top ologies