(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Are closures and interiors of connected sets always connected?

2. Relevant equations

3. The attempt at a solution

I think this is true for closures and I know it's false for interiors (Found a Counterexample.) I'm having trouble proving it for closures though; I've tried a arguments along the following lines...

Assume X is connected but X(closure) is not. Then X(closure) = AuB, for some open separated sets A and B. So, x in X(closure), then x in A or B. x is in X or a limit point of X; so if I could show x cannot be a limit point of x, then X=AuB and I'd have the result I want. So assume x in X'. Then all N(x) contain some y in X. x is also in A or B, and A and B are open, so lets say x in A. Then x is an interior point of A; and some N(x) is contained in A. We therefore know that X intersect A is not empty. So X(closure) intersect A(closure) is also not empty. But this is not a useful result; it just says A(closure) intersect (AuB) is not empty... lol. I don't think this will work, because there is no reason (I can think of), that A and X can't intersect (which is what this contradiction would depend on.), and I also see no reason a limit point of X couldn't be an interior point of A, assuming that they can intersect... sad ; (. (I later tried a small modification of this to try and show that B was empty... but it also failed miserably)

So... now I'm trying to show from A intersect B(closure) = Null and B intersect A(closure = Null; that X must be the null set. But how else can I express X(closure) so that I can work with it and extract information about X from it, other than as XuX'?

Intuitively, I'm feeling this is probably true because if X is connected, then "adding" the "boundary" points (roughly speaking) isn't going to make it "unconnected." So I feel like I should be able to derive the contradiction, "If X(closure) is not connected, then X is not connected, but we said it was!) However I am just awful at proving things, or am not seeing the trick here.

Ugh. The adjustment to proof based math is not treating me well...

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# Homework Help: Baby Rudin, Chapter 2, Problem #20.

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