▁l att ices ▁( the ▁lattice ▁of ▁open ▁sets ▁of ▁the ▁topology ) ▁and ▁partial ▁orders ▁( the ▁partial ▁order ▁of ▁equivalence ▁classes ▁of ▁the ▁pre order ). ▁This ▁correspond ence ▁also ▁works ▁for ▁a ▁larger ▁class ▁of ▁spaces ▁called ▁finit ely ▁generated ▁spaces . ▁F initely ▁generated ▁spaces ▁can ▁be ▁character ized ▁as ▁the ▁spaces ▁in ▁which ▁an ▁arbitrary ▁intersection ▁of ▁open ▁sets ▁is ▁open . ▁Fin ite ▁topological ▁spaces ▁are ▁a ▁special ▁class ▁of ▁finit ely ▁generated ▁spaces . ▁ ▁Ex amples ▁▁ 0 ▁or ▁ 1 ▁points ▁ ▁There ▁is ▁a ▁unique ▁topology ▁on ▁the ▁empty ▁set ▁ ∅ . ▁The ▁only ▁open ▁set ▁is ▁the ▁empty ▁one . ▁Indeed , ▁this ▁is ▁the ▁only ▁subset ▁of ▁ ∅ . ▁ ▁Like wise , ▁there ▁is ▁a ▁unique ▁topology ▁on ▁a ▁singleton ▁set ▁{ a }. ▁Here ▁the ▁open ▁sets ▁are ▁ ∅ ▁and ▁{ a }. ▁This ▁topology ▁is ▁both ▁discrete ▁and ▁trivial , ▁although ▁in ▁some ▁ways ▁it ▁is ▁better ▁to ▁think ▁of ▁it ▁as ▁a ▁discrete ▁space ▁since ▁it ▁shares ▁more ▁properties ▁with ▁the ▁family ▁of ▁finite ▁discrete ▁spaces . ▁ ▁For ▁any ▁topological ▁space ▁X ▁there ▁is ▁a ▁unique ▁continuous ▁function ▁from ▁ ∅ ▁to ▁X , ▁namely ▁the ▁empty ▁function . ▁There ▁is ▁also ▁a ▁unique ▁continuous ▁function ▁from ▁X ▁to ▁the ▁singleton ▁space ▁{ a }, ▁namely ▁the ▁constant ▁function ▁to ▁a . ▁In ▁the ▁language ▁of ▁category ▁theory ▁the ▁empty ▁space ▁serves ▁as ▁an ▁initial ▁object ▁in ▁the ▁category ▁of ▁topological ▁spaces ▁while