Hello,

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:

is open?

If not could you provide a counter example?

Thank you very much,

matteo