Topology Question (Normal Spaces)

[BSMSMSTMSPHD]

Not sure where to put a question about topology, but I'll try here.

I'm trying to show that a certain topology for the Real line is not normal. The topology in question has no disjoint open sets (they are all nested) and therefore, no disjoint closed sets.

If a topology has no disjoint closed sets, can it still be normal? I'd say not, but the only definition I have for normal requires me to start with disjoint closed sets, which in this case is impossible.

Another possible approach... I've been able to show that the topology in question is not Hausdorff. Does that necessarily mean it is not normal?

Any help is appreciated.

 

[DeadWolfe]

According to munkres, for normality 1 point sets must be closed.

 

[tacman]

It is correct that a topology with no disjoint closed sets cannot be normal. The definition of a normal space requires the existence of disjoint closed sets. If there are no disjoint closed sets, then the normality condition cannot be satisfied.

In general, a space that is not Hausdorff cannot be normal. This is because the Hausdorff condition is a stronger condition than normality. A space can be normal without being Hausdorff, but a space that is not Hausdorff cannot be normal.

To prove that the topology in question is not normal, you can use the fact that it is not Hausdorff. This is because a normal space must be Hausdorff, so if you can show that the space is not Hausdorff, then it cannot be normal.