Question about isolated points.

[cragar]

Homework Statement

If I just had the set containing \pi on the real line.
So this is an isolated point. Is this set closed?

The Attempt at a Solution

I think this set is closed because it contains its limit points, because it only has one point.
Am i thinking about this correctly?

 

[NewtonianAlch]

Hmm, that's an interesting question. I guess it would be closed because that's the only point in the set, and there's no option for an e-neighbourhood around it.

 

[Dick]

I think you are thinking about it correctly but that you are even asking this question makes me wonder. Can you explain your doubts further? What's the exact definition of a limit point?

 

[cragar]

thats what I thought too

 

[zooxanthellae]

If we wanted to put things on more definite footing, we could assume that it isn't closed. Then there is some point x1 such that for all E > 0 there is some x in our set such that d(x,x1) < E. However, since x = π in all cases, then the only such point is π, a contradiction.