With Meso Tight Space The Class Of Hereditary And Product Study

In 1986, Finland topologists named H.J.K.Junlia H first introduce the concept of the scattered partition, using of this property he get a sufficient and necessary condition of heredititarily meta compactness. This conclusion broadened people's new view of the hereditary for topological space, during the subsequent twenty years, pedicted by some closed coverage domestic and overseas topological researchers get the similar result in hereditarily screenable space, hereditarily submeta compactness, and perfect parac- ompact space, and made a series of achievements. It is well known that the meso compact space is between paracompact space and meta compactness space,we naturally asked the following questions :Question: Whether meso compact space has the equivalent characterizations in scattered partition and Tychonoff product?This article is mainly discussed about the strongly hereditarily meso compact spaces and the weakly submeso compactness space. On property of hereditary, we achieved some conclusions as follows:(1) X is strongly hereditarily meso compact if and only if each open subspace of the scattered partition has a compact finite open expansion in X .(2) X is strongly hereditarily meso compact space if and only if every open set {U_α:α<γ} of X has a compact finite open refinement V={V_α:α<γ}, for (?)α<γ, has L_α(?)V_α(?)U_α, which L_α= U_α- U_δ<α U_δ.(3) If X=∏_{σ∈Σ}X_σis (hereditarily) |Σ|-paracompact space, then X is (hereditarily) weakly submeso compactness if and only if (?) F∈[Σ]^<ω,∏_σ∈FX_σis (hereditarily) weakly submeso compactness. At the aspect in Tychonoff product, the strongly hereditarily mesocompact spaces and the weakly submeso compactness has get the conclusions as follows:(4) If X=∏_i∈ωX_i is strongly hereditarily meso compact spaces, the followings are equivalent:1) X is strongly hereditarily mesocompact spaces;2) For (?)a∈[ω]^<ω,∏(i∈ω)X_i is strongly hereditarily mesocompact spaces; 3) For (?)n∈ω,∏(i∈ω)X_i is strongly hereditarily mesocompact spaces.(5) If X=∏(i∈ω)X_i is countable paracompact spaces,the followings are equivalent:1) X is weakly submeso compactness spaces;2) For (?)F∈[ω]^<ω,∏_i∈FX_i is weakly submeso compactness spaces;3) For (?)n∈ω,∏_i≤nX_i is weakly submeso compactness spaces.