The Product Properties Of Generalized Paracompact Spaces

People begin to research the product properties between 1940s and 1950s. From 1980s to 1990s, for researching the product properties of generalized paracompact spaces is developing. Some prominent topologists ,such as Y. Yajima(Japan) , G. Gruenhage(America) , K. Chiba(Japan) and so on, have some important results to research the tychonoff finite product properties . the Tychonoff countable infinite product properties , inverse limits, σ-products in generalized paracompact spaces. However, at present, the research results in the tychonoff infinite uncountable product properties, inverse limits, σ-productsof generalized paracompact spaces are short.This paper uses the mapping and covering methods to preliminary research the tychonoff infinite product propert ies , inverse limits, σ-products, and has gained the following results.Theorem1 Let X = Π_{α∈∧} X_α be λ -paracompact and λ is the cardinal number of∧, X is mesocompact (screenable) (?)Π_α∈F X_α is mesocompact (screenable) for every F∈[∧]^<ω.Theorem2 Let X be the limit of an inverse system {X_α, π^α_β , ∧} and λ is the cardinal number of ∧. Suppose each projection π_α : X→ X_α is an open and onto map and X is λ -paracompact, then X is mesocompact (screenbale).Theorem3 Let X =σ { X_α: α∈A}, If each finite subproduct of X is(hereditarily)Lindelof and {X_n , n ∈ω } is an open sets, then X is (hereditarily)Lindelof.Theorem4 Let X = σ { X—α : α∈A },If each finite subproduct of X isMeta-Lindelof and {X_n, n∈ω} is an open sets, then X is Meta-Lindelof and weakly δ(?)-refinable.