a space is T1 if and only if every subset A is the intersection of all open sets containing A
Say we have , a -space, and , a subset of . We aim to show that the intersection of all open sets containing equals . By de Morgan’s laws, that would be true if the complement of , , equalled the union of all closed sets in . Let’s call this union of closed sets .
Each set that makes up is contained by , so . If we could show , we’d be done.
Since is , each singleton in is closed (http://planetmath.org/ASpaceIsT1IfAndOnlyIfEverySingletonIsClosed). Their union, a subset of , contains , so we’re through.
Now suppose we know that in some topological space , any subset of is the intersection of all open sets containing . Given , we’re looking for an open set containing but not , to show that is .
by hypothesis. If all open sets containing contained , would be in the intersection; since isn’t in the intersection, must be .
|Title||a space is T1 if and only if every subset A is the intersection of all open sets containing A|
|Date of creation||2013-03-22 14:20:18|
|Last modified on||2013-03-22 14:20:18|
|Last modified by||waj (4416)|