I have this question about topology, but I'm really not an expert, so please bear with me if I use some not very common notations (Please ask for any clearification!)
Consider the cantor space , i.e., the space of all infinite sequences of 0s and 1s.
Consider its standard topology: the basic open sets are (representable) by finite strings in , and their associated set is the set of all infinite sequences that have that string as prefix.
I use the following notation: given and , I write to denote the string consisting of the first n bits of the infinite sequence x.
Consider the Borel algebra generated by the basic sets.
Let be a Borel set.
Now suppose there is a Borel set such that
- T is an open set
- For every , and for every natural number , there exist an such that .
Note that .
I believe the point 2) means that (Please correct me if i'm wrong):
- T is dense in S
- S is included in the clousure of T.
My question is the following:
If not could you provide a counter example?
Thank you very much,