Proof nowhere-dense of a closure's complement

  • Thread starter thekirk
  • Start date
  • #1
2
0

Homework Statement


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


Homework Equations


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


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.
 

Answers and Replies

  • #2
22,089
3,296
First, how did you define "nowhere dense"??

Second, do you know the following formula's:

[tex]cl(X\setminus E)=X\setminus int(E)~\text{and}~int(X\setminus E) = X\setminus cl(E)[/tex]

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

Note: cl means closure and int means interior.
 
  • #3
2
0
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!
 

Related Threads on Proof nowhere-dense of a closure's complement

  • Last Post
Replies
6
Views
8K
Replies
1
Views
2K
  • Last Post
Replies
2
Views
3K
Replies
13
Views
2K
Replies
2
Views
5K
  • Last Post
Replies
4
Views
2K
Replies
2
Views
6K
  • Last Post
Replies
2
Views
7K
Replies
5
Views
3K
Top