We shall prove that if is a Hausdorff paracompactum with at least one nontrivial quasicomponent, then none of its Čech systems can be commutative and none can support the structure of an approximate inverse system.
"Quasicomponents, Čech Systems, and approximate Systems." Rocky Mountain J. Math. 51 (2) 657 - 667, April 2021. https://doi.org/10.1216/rmj.2021.51.657