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 $i=10110\ldots$

$${ X }_{ a }={ { \left\{ \left\lfloor \frac { a }{ { 10 }^{ n } }  \right\rfloor ,\left\lfloor \frac { a }{ { 10 }^{ n-1} }  \right\rfloor ,...,\left\lfloor \frac { a }{ { 10 }^{ 0 } }  \right\rfloor  \right\}  } }\\n=\left\lfloor \log _{ 10 }{ a }  \right\rfloor $$  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.
