s-paracompact

Definition

A topological space X is called s-paracompact if for every open cover U of X, there exists an U-mapping of X onto some semimetrizable 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. s-paracompactness is a generalization of paracompactness from this viewpoint. D-paracompactness is another generalization.

Reference

• C. H. Dowker, An extension of Alexandroff's mapping theorem, Bull. Am. Math. Soc. Vol.54, pp.386-391 (1948).
• H. Brandenburg, Some characterizations of developable spaces, Proc. Amer. Math. 80 (1980), 157-161.
• T. Mizokami, On d-paracompact σ-spaces, TSUKUBA J. MATH. Vol. 15 No. 2 (1991), 425-449.