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Meso compact space hereditary class research and product

Author: LiWeiPing
Tutor: ZhuPeiYong
School: University of Electronic Science and Technology
Course: Applied Mathematics
Keywords: strongly hereditarily mesocompact weakly submeso compactness Tychonoff product scattering partition
CLC: O189.11
Type: Master's thesis
Year: 2011
Downloads: 14
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Abstract


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 {Uα:α<γ} of X has a compact finite open refinement V={Vα:α<γ}, for （?）α<γ, has Lα（?）Vα（?）Uα, which Lα= Uα- Uδ<α Uδ.（3） If X=∏σ∈ΣXσis （hereditarily） |Σ|-paracompact space, then X is （hereditarily） weakly submeso compactness if and only if （?） F∈[Σ]<ω,∏σ∈FXσis （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∈ωXi is strongly hereditarily meso compact spaces, the followings are equivalent:1) X is strongly hereditarily mesocompact spaces;2) For （?）a∈[ω]<ω,∏（i∈ω）Xi is strongly hereditarily mesocompact spaces; 3) For （?）n∈ω,∏（i∈ω）Xi is strongly hereditarily mesocompact spaces.（5） If X=∏（i∈ω）Xi is countable paracompact spaces,the followings are equivalent:1) X is weakly submeso compactness spaces;2) For （?）F∈[ω]<ω,∏i∈FXi is weakly submeso compactness spaces;3) For （?）n∈ω,∏i≤nXi is weakly submeso compactness spaces.

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CLC: > Mathematical sciences and chemical > Mathematics > Geometry, topology > Topology ( the situation in geometry ) > General topology > Topological space ( spatial topology )
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