Definition
A topological space X is called base paracompact if it has an open base B of cardinality equal to the weight of the space such that every open cover C of X has a locally finite open refinement C' in B.
Reference
S. G. Popvassilev, Base-cover paracompactness, Proc. Amer. Math. Soc. Vol.132, No.10, pp.3121-3130.