Is Every Point in a Topological Space Closed?

[ehrenfest]

Homework Statement

Is it true that every point in a topological space is closed? In a metric space?

Homework Equations

The Attempt at a Solution

 

[Hurkyl]

Have you tried constructing a counterexample?

A single point space clearly won't suffice; what about a two-point space?

 

[CompuChip]

Try looking up T₁ spaces.
Then look at Hausdorff spaces and metric spaces and try to prove whether they are in general T₁.

 

[matness]

you can also think about discrete topology, if you have learned before.

 

[ehrenfest]

Have you tried constructing a counterexample?

A single point space clearly won't suffice; what about a two-point space?

So, I found it is only true in a T1 space.

But in a single point space it is also true since the complement of every point is the null set which is open.

 

[HallsofIvy]

Actually, it might be better to think about the indiscreet topology rather than the discreet topology.