D-paracompact

Definition

A topological space X is called D-paracompact (originally named d-paracompact by Pareek) if for every open cover U of X, there exists an U-mapping of X onto some developable space space M.
Here a map f is a U-mapping if f is continuous and there exists an open cover V of M such that f^{-1}(V) refines U. See D-paracompact, [Dowker1948] and [Nashef1991] for detail.

Remark

Dowker characterized Hausdorff paracompact spaces as those spaces that have the property if U is an open cover of X then there exists a U-mapping f from X onto some metrizable space M. D-paracompactness is a generalization of paracompactness from this viewpoint.

Reference

• C. H. Dowker, An extension of Alexandroff's mapping theorem, Bull. Am. Math. Soc. Vol.54, pp.386-391 (1948).
• C. M. Pareek, Moore spaces, semi-metric spaces and continuous mappings connected with them, Canada. J. Math. 25 (1972), 1033-1042.
• Bassam Al-Nashef, Cover-developements and D-paracompact spaces, Indian J. pure appl. Math. 22(2), pp.135-141, Feb. 1991.