On the dimension of limits of inverse systems

Abstract:

We say that the limit of an inverse system $X = \underleftarrow {\lim }\left \{ {{X_\lambda },\pi _\mu ^\lambda ,\Lambda } \right \}$ is cylindrical if each finite cozero cover of $X$ has a $\sigma$-locally finite refinement consisting of sets of the form $\pi _\lambda ^{ - 1}(U)$, where $U$ is a cozero-set in ${X_\lambda }$ and ${\pi _\lambda }:X \to {X_\lambda }$ is the projection. We prove that if $X$ is cylindrical, then $\dim X = \sup \left \{ {\dim {X_\lambda }:\lambda \in \Lambda } \right \}$.