Closure-preserving covers by small sets

AbstractIn 1971 H. Tamano asked the question: Is a space X paracompact if X has a closure-preserving cover by compact closed sets? From this came many results concerning spaces with various types of closure-preserving covers as well as new questions about spaces having these properties. In this paper we generalize many known results by considering spaces with closure-preserving J-covers, where J is any ideal of closed subsets. Several characterization theorems are also obtained linking the properties of J-scattered spaces, hereditarily metacompact spaces, spaces with special closure-preserving J-covers, and spaces defined by certain topological games