kathrynag
- 595
- 0
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
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