Definition
Let X be a completely regular space and let be its Stone-Cech compactification. X is called weakly superparacompact iff for every compact , there exists a clopen cover of the space X such that , where denotes the closure of W in .
Reference
D. Buhagiar and T. Miwa, On superparacompact and Lindeloef GO-spaces, Houston J. Math. Vol.24, No.3, 1998.