# Homework Help: Point in a topological space

1. Sep 24, 2007

### ehrenfest

1. The problem statement, all variables and given/known data
Is it true that every point in a topological space is closed? In a metric space?

2. Relevant equations

3. The attempt at a solution

2. Sep 24, 2007

### Hurkyl

Staff Emeritus
Have you tried constructing a counterexample?

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

3. Sep 24, 2007

### CompuChip

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

4. Sep 24, 2007

### matness

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

5. Sep 24, 2007

### ehrenfest

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.

6. Sep 24, 2007

### HallsofIvy

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