P(locally-finite)-embedding and rectangular normality of product spaces

Let γ and κ be infinite cardinals and λ a cardinal. A subspace A of a space X is said to be Pγ(locally-finite)-embedded in X if every locally finite partition of unity with cardinality ⩽γ on A can be extended to a locally finite partition of unity on X; this extension property was defined by Dydak recently. In this paper, introducing a space Jγ(κ) and a class of spaces called spaces of type t(γ,κ,λ), we characterize Pγ(locally-finite)-embedding by products with these spaces. We also characterize extendability of locally finite κ+-open covers U of a closed subspace A of a normal space X to those of X by products with these spaces; this extends Przymusińskiʹs result in 1984 of the case |U|=ω. We apply this result to give characterizations of functionally Katětov spaces and Katětov spaces by rectangular normality of products with these spaces.