Can sigma field be countably infinite?

Answer 1

No. The basic idea of proving why not is this:

1) The underlying space X is at least countably infinite (of course).

2) Use the properties of a sigma field (aka sigma algebra) to find a countable partition of the space, X = disjiont-union( X_i ).

3) Notice that the union(X_i, s in S) is in the sigma algebra for any subset S of natural numbers.

4) Notice that any union(X_i, s in S) is distinct.

5) Conclude, since the set of subsets of natural numbers is uncountable, so too is your sigma algebra.