On uniform paracompactness
David Buhagiar, Boris A. Pasynkov · Czechoslovak Mathematical Journal · 1996
We can now formulate the definitions of uniform paracompactness belonging to Rice, Frolik and Borubaev.Definition 0.1.(M.D.Rice [4]) The uniform space (X,°?/) is paracompact (in this situation we will use the term "H-paracompact"), if every open cover of (X, °?/) admits a ^-locally finite open refinement.Rice showed [4], that all It-paracompact uniform spaces are complete.From this follows that all metrizable, by a non complete metric, uniform spaces are not Hparacompact.Hence the class of H-paracompact uniform spaces turns out to be, in our opinion, too narrow.Definition 0.2.(A. A. Borubaev [1]) The uniform space (A", °?/) is paracompact (in this situation we will use the term "H-paracompact"), if for every finitely-additive open cover A of (N, $/) there exists a sequence a n E '?/, nEN such that (*)V x E X 3 n E r\j and L E A with the property a n (x) C L.Note 0.3.It is clear that without loss of generality one can assume that in Definition 0.2 one can add,Although the requirement of finite-additivity of the cover A in definition 0.2 does not seem very natural, L?-paracompactness has a series of good properties.It is worthwhile mentioning that all metrizable and all H-paracompact spaces are Bparacompact [1] and, consequently, the class (B) of all H-paracompact spaces is essentially wider than the class (H) of all H-paracompact spaces.Definition 0.4.(Z.Frolik [3]) The uniform space (X,°?/) is paracompact (in this situation we will use the term "F-paracompact"), if every open cover of (A", °?/) admits a cr-^-discrete (i.e. it is decomposed into a union of a countable number of ^-discrete subsystems) open refinement.The class (F) of F-paracompact uniform spaces is essentially wider than class (H), as it is shown underneath.(In [1] p. 81 is mentioned that the classes (H), (B) and (F) are pairwise distinct.)1. P-PARACOMPACTNESS