Can anyone point me to places in the literature where modern set theory has been applied to say something about the Cech cohomology of connected non-metrizable compacta? I'm looking for something deeper than, e.g., the observation that the Cech cohomology detects that the long loop is a loop.
Most of my work is on non-metrizable compacta, and, most of time, whenever I prove something in the zero-dimensional case, it almost effortlessly generalizes to all compacta. By Stone duality, I'm "really" working on uncountable boolean algebras most of the time. So, on days when I'm particularly interested in connectedness properties, I feel that I'm not asking enough of the right questions.
It seems that just as all questions about zero-dimensional compacta are equivalent to questions about boolean algebras, sometimes interesting and difficult infinitary combinatorial questions, many questions about connected non-metrizable compacta should reduce to questions their Cech cohomologies, presumably including some interesting and difficult infinitary combinatorial questions about uncountable abelian groups. I'm curious as to whether it would be fruitful to pursue this analogy further, and therefore curious as to how the analogy has been pursued in the past.