# Proof nowhere-dense of a closure's complement

1. Nov 12, 2011

### thekirk

1. The problem statement, all variables and given/known data
A set E in ℝ is nowhere dense if and only if the closure of E's complement(E with a line over it) is dense in ℝ

2. Relevant equations
I need help proving this lemma. I'm not entirely sure where to start.

3. The attempt at a solution
I know we have to proof it two ways, backwards and forwards. For the forward, all I can think to use if the fact that the closure of E complement contains no nonempty open intervals, but I don't know where to go with that.

Any help would be appreciated.

2. Nov 12, 2011

### micromass

Staff Emeritus
First, how did you define "nowhere dense"??

Second, do you know the following formula's:

$$cl(X\setminus E)=X\setminus int(E)~\text{and}~int(X\setminus E) = X\setminus cl(E)$$

These equalities will prove to be handy. Try to prove them!!

Note: cl means closure and int means interior.

3. Nov 12, 2011

### thekirk

micromass-

I have never seen that equation before.

I have nowhere dense defined as: a set E in ℝ is nowhere dense if the closure of E contains no non-empty intervals.

I will look at those equations and see what I can do. Thank you!

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