▁Therefore , ▁one ▁only ▁need ▁check ▁that ▁the ▁union ▁of ▁a ▁finite ▁number ▁of ▁open ▁sets ▁is ▁open . ▁This ▁leads ▁to ▁a ▁simpler ▁description ▁of ▁top ologies ▁on ▁a ▁finite ▁set . ▁ ▁Let ▁X ▁be ▁a ▁finite ▁set . ▁A ▁topology ▁on ▁X ▁is ▁a ▁subset ▁ τ ▁of ▁P ( X ) ▁such ▁that ▁ ∅ ▁ ∈ ▁ τ ▁and ▁X ▁ ∈ ▁ τ ▁if ▁U ▁and ▁V ▁are ▁in ▁ τ ▁then ▁U ▁ ∪ ▁V ▁ ∈ ▁ τ ▁if ▁U ▁and ▁V ▁are ▁in ▁ τ ▁then ▁U ▁ ∩ ▁V ▁ ∈ ▁ τ ▁ ▁A ▁topology ▁on ▁a ▁finite ▁set ▁is ▁therefore ▁nothing ▁more ▁than ▁a ▁sub l attice ▁of ▁( P ( X ), ▁ ⊂ ) ▁which ▁includes ▁both ▁the ▁bottom ▁element ▁( ∅ ) ▁and ▁the ▁top ▁element ▁( X ). ▁ ▁Every ▁finite ▁bounded ▁lattice ▁is ▁complete ▁since ▁the ▁meet ▁or ▁join ▁of ▁any ▁family ▁of ▁elements ▁can ▁always ▁be ▁reduced ▁to ▁a ▁meet ▁or ▁join ▁of ▁two ▁elements . ▁It ▁follows ▁that ▁in ▁a ▁finite ▁topological ▁space ▁the ▁union ▁or ▁intersection ▁of ▁an ▁arbitrary ▁family ▁of ▁open ▁sets ▁( resp . ▁closed ▁sets ) ▁is ▁open ▁( resp . ▁closed ). ▁ ▁Special ization ▁pre order ▁ ▁Top ologies ▁on ▁a ▁finite ▁set ▁X ▁are ▁in ▁one - to - one ▁correspond ence ▁with ▁pre orders ▁on ▁X . ▁Rec all ▁that ▁a ▁pre order ▁on ▁X ▁is ▁a ▁binary ▁relation ▁on ▁X ▁which ▁is ▁ref lex ive