Chu spaces generalize the notion of topological space by dropping the requirements that the set of open sets be closed under union and finite intersection, that the open sets be extensional, and that the membership predicate (of points in open sets) be two-valued. The definition of continuous function remains unchanged other than having to be worded carefully to continue to make sense after these generalizations.
The name is due to Po-Hsiang Chu, who originally constructed a verification of autonomous categories as a graduate student under the direction of Michael Barr in 1979.[1]