induced Alexandroff topology on a poset
Let be a poset. For any define following subset:
Proposition 1. is a , Alexandroff space.
Proof. Let be such that . Note that this implies that or (because is antisymmetric). Therefore or . Thus is .
Now in order to show that is Alexandroff it is enough to show that an arbitrary intersection of base sets is open. So assume that is a subset of such that
and let . Then (since is transitive) it is clear that
and therefore the intersection is open, which completes the proof.
Proof. ,,” Assume that preserves order and let be an open base set. We wish to show that is open in . So take any . Now if , then (since preserves order) and thus . Therefore . Since was arbitrary we obtain that for any we have and thus
which implies that is open.
,,” Assume that is continuous and let for some . Assume that . Let . Therefore , but is open, so is open (because is continuous). Thus . But , so . But this implies that . Contradiction.
|Title||induced Alexandroff topology on a poset|
|Date of creation||2013-03-22 18:46:01|
|Last modified on||2013-03-22 18:46:01|
|Last modified by||joking (16130)|