Cooperative Boolean systems with generically long attractors II

We prove that cooperativity in Boolean networks precludes a strong notion of sensitive dependence on initial conditions. Weaker notions of sensitive dependence are shown to be consistent with cooperativity, but if each regulatory functions is binary AND or binary OR, in N-dimensional networks they impose an upper bound of approximately sqrt(3)^N on the lengths of attractors that can be reached from a fraction p approaching 1 of initial conditions. The upper bound is shown to be sharp

The sizes of relatively compact T1T_1-spaces

summary:The relativization of Gryzlov's theorem about the size of compact $T_1$-spaces with countable pseudocharacter is false