order ▁is ▁defined ▁similarly ▁in ▁terms ▁of ▁the ▁down - sets . ▁The ▁pre oder ▁indu cing ▁the ▁upper ▁topology ▁is ▁its ▁special ization ▁pre order , ▁but ▁the ▁special ization ▁pre order ▁of ▁the ▁lower ▁topology ▁is ▁opposite ▁to ▁the ▁indu cing ▁pre order . ▁ ▁The ▁real ▁upper ▁topology ▁is ▁most ▁naturally ▁defined ▁on ▁the ▁upper - ext ended ▁real ▁line ▁ ▁by ▁the ▁system ▁ ▁of ▁open ▁sets . ▁Similarly , ▁the ▁real ▁lower ▁topology ▁ ▁is ▁naturally ▁defined ▁on ▁the ▁lower ▁real ▁line ▁. ▁A ▁real ▁function ▁on ▁a ▁topological ▁space ▁is ▁upper ▁semi - continu ous ▁if ▁and ▁only ▁if ▁it ▁is ▁lower - continu ous , ▁i . e . ▁is ▁continuous ▁with ▁respect ▁to ▁the ▁lower ▁topology ▁on ▁the ▁lower - ext ended ▁line ▁. ▁Similarly , ▁a ▁function ▁into ▁the ▁upper ▁real ▁line ▁is ▁lower ▁semi - continu ous ▁if ▁and ▁only ▁if ▁it ▁is ▁upper - continu ous , ▁i . e . ▁is ▁continuous ▁with ▁respect ▁to ▁the ▁upper ▁topology ▁on ▁. ▁ ▁References ▁▁▁▁▁▁▁ ▁Category : General ▁topology ▁Category : Order ▁theory <0x0A> </s> ▁Ark ad ius z ▁Woj t as ▁( born ▁ 2 9 ▁October ▁ 1 9 7 7 ) ▁is ▁a ▁Polish ▁former ▁cycl ist . ▁ ▁Major ▁results ▁▁ 2 0 0 0 ▁ 1 st ▁Stage ▁ 2 ▁Peace ▁Race ▁ 1 st ▁Over all ▁R inger ike ▁GP ▁ 1 st ▁Stage ▁ 1 ▁ 2 0 0 2 ▁ 3 rd ▁Over all ▁Tour