6431#527unstable-527.xmlProof20231123Lîm Tsú-thuàn
The first direction is Hausdorff implies diagonal map is closed.
Hausdorff means every two different points has a pair of disjoint neighborhoods ,
where and .
therefore, every pair not line on the diagonal has cover them.
The union of all these open sets indeed covers , so the complement is closed.
The second direction is diagonal map is closed implies Hausdorff.
Since
is open, which implies for all the pair .
Since ,
this implies the fact that .
Also, for all and ,
it's natural that ,
since the open set at most cover .
Consider that reversely again, that means for all ,
pair ( will not cover any part of diagonal),
that implies as desired.
Context6433math-002Zmath-002Z.xmlA space is Hausdorff iff its diagonal map is closedTheorem20231123Lîm Tsú-thuànThe proposition says a space is Hausdorff if and only if its diagonal map to product space is closed,
which given an alternative definition.
Notice that close means its complement is open. The definition of is
6435#527unstable-527.xmlProof20231123Lîm Tsú-thuàn
The first direction is Hausdorff implies diagonal map is closed.
Hausdorff means every two different points has a pair of disjoint neighborhoods ,
where and .
therefore, every pair not line on the diagonal has cover them.
The union of all these open sets indeed covers , so the complement is closed.
The second direction is diagonal map is closed implies Hausdorff.
Since
is open, which implies for all the pair .
Since ,
this implies the fact that .
Also, for all and ,
it's natural that ,
since the open set at most cover .
Consider that reversely again, that means for all ,
pair ( will not cover any part of diagonal),
that implies as desired.