# Connectedness math problem

"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...

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Hurkyl
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)?

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

How can I find a counterexample?

Hurkyl
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.

MathematicalPhysicist
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.

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

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

HallsofIvy