Denoting the set of initial segments of a binary sequence The index is an infinite, innumerable binary sequence in $\{0,1\}$. $ I= \{f \mid f: \Bbb {N} \to \{0,1\}\} $
Is there a way to get a set $X_i$ from the infinite index number $10110\ldots$
$ X_i = \{1,10,101,1011,10110,\ldots\}$ where $i \in I$ and $i=10110\ldots$
 A: If the elements of $X_f$ are supposed to be decimal numbers, then this would work:
$$X_f = \left\{\sum_{i=0}^n f(i)10^{n-i}\,\middle\vert\, n \in \Bbb N\right\}$$
If the elements of $X_f$ are supposed to be written in binary instead, replace the $10$ by a $2$.

Added: The $\sum$ notation denotes a sum. $f(i)$ is simply the $i$th element of your sequence. In this case, $$\sum_{i=0}^n f(i)10^{n-i} = f(0)10^n + f(1)10^{n-1}+\ldots+f(n)10^0$$
So for $n=0$, we have $f(0)10^0 = 1$. For $n=1$, we have $f(0)10^1+f(1)10^0 = 10+0=10$. For $n=2$, we arrive at $101$ (try it!).
The iteration variable $i$ is called the index (variable). In this example, $\sum\limits_{i=0}^n$ means "sum the results of replacing $i$ by $0, 1,2,\ldots n$ in the expression on the right". A Google search on "sum notation" or "sigma notation" will likely help you further.

On the other hand, if you want $X_f$ to be the set of finite (non-empty) initial segments of $f$ (i.e. the sequences $(1), (1,0), (1,0,1)$ etc.) then this would work:
$$X_f = \left\{f \restriction_{\Bbb N_n}: \Bbb N_n \to \{0,1\} \,\middle\vert\, n \in \Bbb N\right\}$$
where $\Bbb N_n = \{0,1,\ldots n\}$ and $f\restriction_{\Bbb N_n}$ is the function obtained by restricting the domain of $f$ to $\Bbb N_n$.
A: $${ X }_{ a }={ { \left\{ \left\lfloor \frac { a }{ { 10 }^{ n } }  \right\rfloor ,\left\lfloor \frac { a }{ { 10 }^{ n-1} }  \right\rfloor ,...,\left\lfloor \frac { a }{ { 10 }^{ 0 } }  \right\rfloor  \right\}  } }\\n=\left\lfloor \log _{ 10 }{ a }  \right\rfloor $$  This works for only finite sequences but you can divide the sequence and join. For example divide the total sequence such as $n=5$. The first five term would be $${ X }_{ a_{1-5} }={ { \left\{ \left\lfloor \frac { a_{1-5} }{ { 10 }^{ 5 } }  \right\rfloor ,\left\lfloor \frac { a_{1-5} }{ { 10 }^{ 4} }  \right\rfloor ,...,\left\lfloor \frac { a_{1-5} }{ { 10 }^{ 0 } }  \right\rfloor  \right\}  } }$$. The next $5$ term:
$${ X }_{ a_{6-10} }={ { \left\{ { X }_{ a_{1-5} }\left\lfloor \frac { a_{6-10} }{ { 10 }^{ 5 } }  \right\rfloor ,{ X }_{ a_{1-5} }\left\lfloor \frac { a_{6-10} }{ { 10 }^{ 4} }  \right\rfloor ,...,{ X }_{ a_{1-5} }\left\lfloor \frac { a_{6-10} }{ { 10 }^{ 0 } }  \right\rfloor  \right\}  } }$$.
This may not be very practical.
A: Are you wanting something like this?
$$X_n = \{x|x=\lfloor n10^{-a}\rfloor,a\in\Bbb N\cup\{0\}\}\setminus\{0\}$$
A: The set of sequences can be written as
$$
  \{\, (f(i))_{i=1}^n\mid n\in\Bbb N_{>0}\,\},
$$
which is a subset of $\{0,1\}^*$, and is assumes you fixed an infinite sequece $f\in I$. If you want the empty sequence in your set as well, use $\Bbb N$ instead of $\Bbb N_{>0}$.
