Complex sentence: If we have a preorder declare open any final section (upper set), but any such "fine" topology comes from that because of specialization (pre) order and, between such spaces, a function is continous iff it is monotone !
Split into simpler sentences: If we have a preorder declare open any final section (upper set). Any "fine" topology comes from that because of specialization (pre) order and a function is continous iff it is monotone!