Give an example, or argue that it is impossible

[davitykale]

Homework Statement

An open set A contained in R (reals) such that the closure of A = R, but R \ A is uncountable

Homework Equations

I guess knowing that for the closure of a set A to be equal to R means that A is dense in R?

The Attempt at a Solution

Every thing I try seems to fail, but I have absolutely no idea how I would go about even trying to prove that this wasn't possible

 

[davitykale]

I think it's false and I want to try and argue this. Can anyone help me get started on a proof?

 

[Dick]

It's not false. Do you know the Cantor set?

 

[davitykale]

Oh, I do know the Cantor's set! That makes a lot of sense. R - Cantor's set is uncountable because they are both uncountable?

 

[davitykale]

Wait...the Cantor set is open?

 

[Dick]

Wait...the Cantor set is open?

Not at all. What set related to the Cantor set is open?

 

[davitykale]

It's complement? The union of the open intervals that are removed?

 

[Dick]

It's complement? The union of the open intervals that are removed?

Right. R-Cantor set is the sort of example you are looking for.

 

[davitykale]

That makes a lot of sense. Thanks!