Proving the Connectedness Math Problem in R^n: Is it True?

[kingwinner]

"Let S be a subset of R^n.
If S is connected, then the interior of S is connected."

Is this true or not?

I can't think of a counterexample, but I don't know how to prove it either...

 

[Hurkyl]

"Let S be a subset of R^n.
If S is connected, then the interior of S is connected."

Is this true or not?

I can't think of a counterexample, but I don't know how to prove it either...

It might help to think "backwards" -- rather than thinking of starting with a set S, and then working with its interior, why not start with an open set, and then consider its closure (possibly excluding part of the boundary)?

 

[kingwinner]

But that "backward" one is not equivalent to the original one.

How can I find a counterexample?

 

[Hurkyl]

True, the backward case is just a subset of the possibilities -- but I assert that it covers enough of the possibilities that it should suggest a proof or yield a counterexample.

 

[MathematicalPhysicist]

I think you can take something like
an interval (a,b] or [a,b) (the same should work also for R^n).
obviously if [a,b) is connected, then (a,b) is'nt necessarily.

 

[kingwinner]

Is the claim true or false?
I can't figure it out and I don't know which direction to push my proof towards...

 

[kingwinner]

It seems true with simple examples, but we can't generalize from specifics. How can I start the proof in the general situation?

 

[HallsofIvy]

I think you can take something like
an interval (a,b] or [a,b) (the same should work also for R^n).
obviously if [a,b) is connected, then (a,b) is'nt necessarily.

?? All of [a,b], [a,b), (a,b], (a,b) are connected!