## Question on the cantor space

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 $2^{\omega}$, 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 $2^{*}$, and their associated set is the set of all infinite sequences that have that string as prefix.

I use the following notation: given $x\in 2^{\omega}$ and $n\in \mathbb{N}$, I write $x\downarrow n$ to denote the string consisting of the first n bits of the infinite sequence x.

Consider the Borel $\sigma$ algebra generated by the basic sets.

Let $S$ be a Borel set.
Now suppose there is a Borel set $T$ such that

1. T is an open set
2. For every $x\in S$, and for every natural number $n$, there exist an $y\in T$ such that $x\downarrow n = y \downarrow n$.

Note that $T\not \subseteq S$.
I believe the point 2) means that (Please correct me if i'm wrong):

1. T is dense in S
2. S is included in the clousure of T.

My question is the following:

is $T \cup S$ open?

If not could you provide a counter example?

Thank you very much,

matteo