almost (m,n)-paracompact

Definition

m and n denote cardinalities.
A space X is said to be almost (m, n)-paracompact if for each open covering U of X with cardinality no more than m, there exists a locally-n family V of open subsets of X which refines U and the union of V is dense in X.

Reference

M. K. Singal, Some Generalizations of Paracompactness, Proceedings of the Kanpur topological conference, 1968. Academia Publishing House of the Czechoslovak Academy of Sciences, Praha, 1971. pp. 245-263.