The Properties Of Generalized Paracompact Spaces

The properties of generalized paracompact spacesPeople begin to research the product properties of paracompact spaces between1940s and 1950s.From 1980s to 1990s,for researching the product properties ofgeneralized paracompact spaces is developing.Some prominent topologists, such asY.Yajima(Japan), G.Gruenhage(America), K.Chiba(Japan)and H.J,K.Junnila(Finland)and so on, have some important results to research the Tychonoff finite productproperties,the Tychonoff countable infinite product properties, inverse limits,∑-products in generalized paracompact spaces.Topologists from inlands and abroadsextremely interest in and actively imerge in the properties of Tychonoff infiniteuncountable product,inverse limits,∑-products of generalized paracompact spaces.In recent years, the research results in three kinds of products have someacheivements, especially in infinite uncountable product properties.However, atpresent, the equivalent characterization of generalized paracompact spaces areshort.This paper uses the mapping and covering methods to preliminary research theTychonoff infinite uncountable product properties, equivalent characterization andother some properties,and has gained the following results.Theorem 1(1) Let X=∏_{τ∈∑}X_Ï„beλ-superparacompact,then it isσ-collectionwise normal if∏_τ∈FX_Ï„isσ-collectionwise normal forevery F∈[∑]^＜ω.(2) For countable paracompact X=∏_i∈ωX_i, the followings are equivalent: Xisσ-collectionwise normal;(?)F=[∑]_＜ω,∏_i∈FX_i isσ-collectionwisenormal;(?)n∈w,∏_i≤nX_i isσ-collectionwise normal.Theorem 2(1) A countably compact,nearly submeta-lindel(?)f T₂-space with countable tightnessis compact.(2)A countably compact, nearly submetacompact of T₃-space with countabletightness is compact.Theorem 3(1) A space X is nearly strongly submetacompact if for every open coveru={U_α:α∈∧} of X there is a dense set D(?)X and a sequence {u_n} of open refinements of u such that for each x∈D.there are n_x∈ωsuch that foreach n≥n_x and (u_n)_x is a finite set familities.(2) A countably compact,nearly strongly submetacompact of T₃-space is compact.(3) A space X is nearly strongly submetacompact if X is nearly discretely stronglysubexpandble and for every open cover u={U_α:α∈∧} of X there is a dense setD(?)X and a sequence n>_n∈ω of open refinements of u such that for eachx∈D.there are n∈ωsuch that for each n≥n_x andα∈∧with x∈U_αand(4) Let X=∏_{σ∈∑}X_σbe|∑|-paracompact, X is nearly strongly submetacompact iff∏_σ∈FX_σis nearly strongly submetacompact for every F∈[∑]^＜ω.(5)Let X=∏_{τ∈ω}X_Ï„be countable paracompact,then the following are equivalent: Xis nearly strongly submetacompact for every F∈[ω]^＜ω,∏_i∈FX_i is nearly stronglysubmetacompact;there is n_x∈ωsuch that for each n≥n_x,∏_i≤nX_i is nearly stronglysubmetacompact.Theorem 4(1) Let X=∏_{σ∈∑}X_σbe |∑|-paracompact, then it is weakly subortho-compact if∏_σ∈FX_σis weakly subortho-compact for every F∈[∑]^＜ω.(2) For countable paracompact X=∏_i∈ωX_i, the followings are equivalent: X isweakly subortho-compact;, (?)F∈[ω]^＜ω,∏_i∈FX_i, is weakly subortho-compact;(?)_n∈ω,∏_i≤nX_i is subortho-compact.