Give an example, or argue that it is impossible

In summary, the conversation is about trying to prove the existence of an open set A contained in the real numbers such that the closure of A is equal to R but R minus A is uncountable. The person asking for help initially thinks it is false, but then realizes that it is possible by using the example of the Cantor set and its complement. The conversation ends with the person thanking the other for their help.
  • #1
davitykale
38
0

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
 
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  • #2
I think it's false and I want to try and argue this. Can anyone help me get started on a proof?
 
  • #3
It's not false. Do you know the Cantor set?
 
  • #4
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?
 
  • #5
Wait...the Cantor set is open?
 
  • #6
davitykale said:
Wait...the Cantor set is open?

Not at all. What set related to the Cantor set is open?
 
  • #7
It's complement? The union of the open intervals that are removed?
 
  • #8
davitykale said:
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.
 
  • #9
That makes a lot of sense. Thanks!
 

Related to Give an example, or argue that it is impossible

1. Can you give an example to support your argument?

Yes, for example, if I argue that it is impossible for humans to travel faster than the speed of light, I can provide evidence from Einstein's theory of relativity and the laws of physics.

2. How do you determine if something is impossible?

Determining if something is impossible requires a thorough examination of scientific principles, laws, and evidence. It also involves considering potential limitations and boundaries within our current understanding of the universe.

3. Is it ever possible to prove that something is impossible?

No, as science is constantly evolving and our understanding of the world is always expanding, it is difficult to definitively prove that something is impossible. However, we can use scientific evidence and reasoning to support the argument that something is highly unlikely or implausible.

4. What if new evidence or technology emerges that challenges the idea of something being impossible?

In science, we are always open to new evidence and ideas that challenge our current understanding. If new evidence or technology emerges, scientists will re-evaluate and potentially revise their arguments about the possibility or impossibility of something.

5. Can something that is currently impossible ever become possible?

Yes, as technology and our understanding of the world advances, things that were once thought to be impossible can become possible. For example, humans once thought it was impossible to fly, but with the invention of airplanes, we can now fly. However, it is important to note that some things may always remain impossible due to fundamental laws of nature.

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