# Connectedness math problem

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...

Staff Emeritus
Gold Member
"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?

Staff Emeritus
Gold Member
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.

Gold Member
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?