Constructing Nested Compact Sets in a Dense Open Set

[kathrynag]

If Gn is a dense open set for every n, then intersection Gn is not
empty.


Well I started by thinking about what dense means.
A set A contained in reals is dense if A closure = reals
So Gn closure = reals


G closure = reals.
Then GUL=reals
Then G contains all of its limit points.
Then G is infinite because otherwise it would contain only its isolated points.
We can write G={x1,x2,...}
We want to construct a sequence of nested compact sets Kn all contained in G

 

[kathrynag]

Is there a way to go about this given my start?

 

[micromass]

Are you working in $$\mathbb{R}$$ or an arbitrary metric space?