Does Every Open Set Equal the Interior of Its Closure?

[Quantumpencil]

Homework Statement

Is it true that if U is an open set, then U = Int(closure(U))?

The Attempt at a Solution

I feel like this may be true; I found counter-examples to the general form, Int(U)=(Int(closure(U)), but they all seem to hinge on U being not open (A subset of rationals in the reals, which is neither open nor closed).

However I can't prove this one way or another; I'd like a nudging in the right direction of the proof is this is true, and just a, "it's false" if it's false, so I can keep hunting for a CE.

 

[Dick]

It's true. Where is the proof giving you a problem? You want to show if x is an element of U, then x is an element of int(closure(U)) and vice versa.

 

[Quantumpencil]

lol, I think it being four in the morning was giving me trouble.

So the first direction would be

x in U -> x an interior point and x in Clos(U); Int(U) = all interior points in U(closure), which includes x, so x in Int(Clos(U))

The other direction...

x in Int(clos(U))-> x an interior point of Clos(U) = Int(U) by definition which = U, since U is open.

Does that check out? The only reason I was having difficulty with this, is just intuitively, why is it not possible that closing the set/adding limit points will add points to the interior? I guess that means that if we had any interior point of U, it must have been originally contained in U if U is open. However, not if U isn't open. Why is this?

 

[Dick]

Ooops. Sorry. Actually, it is hard to prove because it's false for exactly the reason you pointed out. Take U to be the union of (0,1) and (1,2). My mistake.

 

[Quantumpencil]

Ok, excellent; then I both understand why it is false and where that BS proof I just made fails.

Thanks a ton.