Definition
A topological space X is said to be almost paracompact if for every open cover U, there exists a family of open subsets such that V is a partial refinement? of U and that the closure of V covers X.
Here "the closure of V" means the family of the closures of the member in V.
Reference
M. K. Singal and Shashi Prabha Arya, On M-Paracompact Spaces, Math. Ann. 181, 119--133 (1969).