The fields of subsets form the category of $\Cal
L_0^*$-rings simply generated by the discrete algebra $Z(2)$
and the $\sigma$-fields form its epireflective subcategory.
In this framework we study sequential and topological
properties of fields of subsets and their relationshisps to
the foundations of probability. In particular, for each
reduced field $\Bbb A$ of subsets of $X\ne\emptyset$, we
describe a maximal extension of $\Bbb A$ into a reduced
$\sigma$-field $\sigma(\Bbb A^*)$ of subsets of $X^*$,
$X\subseteq X^*$, such that $\Bbb A$ and $\sigma(\Bbb A^*)$
have the same probabilities and in the subcategory of
extensions the morphisms (i.e. sequentially continuous ring
homomorphisms) are induced by point maps. Furthermore,
$X=X^*$ iff the zero-dimensional topology generated on $X$
by $\Bbb A$ is $N$-compact.
