▁and ▁trans itive . ▁ ▁Given ▁a ▁( not ▁necessarily ▁finite ) ▁topological ▁space ▁X ▁we ▁can ▁define ▁a ▁pre order ▁on ▁X ▁by ▁x ▁ ≤ ▁y ▁if ▁and ▁only ▁if ▁x ▁ ∈ ▁cl { y } ▁where ▁cl { y } ▁denotes ▁the ▁closure ▁of ▁the ▁singleton ▁set ▁{ y }. ▁This ▁pre order ▁is ▁called ▁the ▁special ization ▁pre order ▁on ▁X . ▁Every ▁open ▁set ▁U ▁of ▁X ▁will ▁be ▁an ▁upper ▁set ▁with ▁respect ▁to ▁ ≤ ▁( i . e . ▁if ▁x ▁ ∈ ▁U ▁and ▁x ▁ ≤ ▁y ▁then ▁y ▁ ∈ ▁U ). ▁Now ▁if ▁X ▁is ▁finite , ▁the ▁con verse ▁is ▁also ▁true : ▁every ▁upper ▁set ▁is ▁open ▁in ▁X . ▁So ▁for ▁finite ▁spaces , ▁the ▁topology ▁on ▁X ▁is ▁uniqu ely ▁determined ▁by ▁ ≤ . ▁ ▁Go ing ▁in ▁the ▁other ▁direction , ▁suppose ▁( X , ▁ ≤ ) ▁is ▁a ▁pre ordered ▁set . ▁Define ▁a ▁topology ▁ τ ▁on ▁X ▁by ▁taking ▁the ▁open ▁sets ▁to ▁be ▁the ▁upper ▁sets ▁with ▁respect ▁to ▁ ≤ . ▁Then ▁the ▁relation ▁ ≤ ▁will ▁be ▁the ▁special ization ▁pre order ▁of ▁( X , ▁ τ ). ▁The ▁topology ▁defined ▁in ▁this ▁way ▁is ▁called ▁the ▁Alexand rov ▁topology ▁determined ▁by ▁ ≤ . ▁ ▁The ▁equivalence ▁between ▁pre orders ▁and ▁finite ▁top ologies ▁can ▁be ▁interpreted ▁as ▁a ▁version ▁of ▁B irk hoff ' s ▁representation ▁theorem , ▁an ▁equivalence ▁between ▁finite ▁distrib utive