[drdeca]

For each natural number k, f(k) is itself a function, and f(k)(k) means the value of that function at k.

Yes, you could basically think of it as f_{k,k} if you wanted to.

No, this is __not__ meant to be 1 - f(k) . f(k) is a function (or a sequence, if you prefer), not a particular value in {0,1}, f(k)(k) is a particular value in {0,1}.

0.5 is not in the set {0,1}, and therefore if z(k)=0.5 then z is not in {0,1}* .