a space is T1 if and only if every singleton is closed
Say is a http://planetmath.org/node/1852 topological space. Let’s show that is closed for every :
The axiom (http://planetmath.org/T1Space) gives us, for every distinct from , an open that contains but not . Since we’re in a topological space, we can take the union of all these open sets to get a new open set,
is the complement of , closed because is open: None of the contain , so doesn’t contain . But any is in , since . That takes care of that.
Now let’s say we have a topological space in which is closed for every . We’d like to show that holds:
Given , we want to find an open set that contains but not . is closed by hypothesis, so its complement is open, and our search is over.
|Title||a space is T1 if and only if every singleton is closed|
|Date of creation||2013-03-22 14:20:15|
|Last modified on||2013-03-22 14:20:15|
|Last modified by||waj (4416)|