Alexandroff space is T1 if and only if it is discrete
Proof. ,,” It is easy to see, that every discrete space is Alexandroff and .
,,” Recall that topological space is if and only if every subset is equal to the intersection of all its open neighbourhoods. So let . Then the intersection of all open neighbourhoods of is equal to . But since is Alexandroff, then is open and thus points are open. Therefore is discrete.
|Title||Alexandroff space is T1 if and only if it is discrete|
|Date of creation||2013-03-22 18:46:08|
|Last modified on||2013-03-22 18:46:08|
|Last modified by||joking (16130)|