Inverse limits of compact spaces

Abstract

This paper gives conditions under which the inverse limit of a system of compact (but non-Hausdorff) spaces will be non-empty, or compact, or hereditarily compact. The main result (Theorems 3 and 5) is that, if the spaces are compact, T₀ and non-empty and the maps are closed and continuous, then the inverse limit is compact and non-empty (and, trivially, T₀). Simple examples are given to show that the results are reasonably sharp.