Boolean rings and cohomology

Abstract

This note is a comment on a paper of Franklin Haimo [1]. A Stone space is a compact totally disconnected Hausdorff space. Let B(S) be the Boolean ring of the Stone space S. The multiplication in B(S) is intersection and the addition is symmetric difference. Thus B(S) is isomorphic with the ring of all maps (=continuous functions) 0: S-12, I2 being the integers mod 2 with the discrete topology. Using the Alexander-Kolmogoroff groups (Spanier [2]) it is readily seen that 4CZ0(S), the group of 0-cocycles of S with 12 as coefficient group, if and only if 4 is a map. Using ordinary multiplication of functions in Z0(S) it is at once clear that B(S) -H0(S), since ZO(S) = HO(S). If { Sx, 7rx,} is an inverse system of Stone spaces, then inv lim SA is a Stone space. Using Steenrod's continuity theorem (see [2]) we have