Genetic Submesocompactness Inverse Limit,

In 1986,in the paper [1] Junnila proved the result :A space is hereditarily metacompact iff its every scattered partion has a point finite open expansion .And in the paper [2],by the example 3.2 Zhu Peiyong proved that the hereditarily paracompact spaces have no a similar characterization to Junnila's. Are there similar characterizations for the countable paracompact (mesocompact, metacompact) space and hereditarily mesocompact space?In this paper ,on the basis of the aboves, we obtain some results about them.And the product properties of mesocompact spaces and hereditarily mesocompact spaces have been paid attention .But there is no good result about them.In this paper we obtain a result about the limit of the inverse system of a normal mesocompact space and a hereditarily normal and hereditarily mesocompact space.The paper has four parts .The first chapter, introduction, gives the origin of the problems and our main results.The second chapter proves that countable paracompact(mesocompact,metacompact) spaces have the characterization of Junnila's and that hereditarily mesocompact spaces don't have it. At last,we give the sufficient conditions for a space having the property that its every scattered partition has a compact-finite open expansion. In third chapter ,we study the relation of the expansion of a space's scattered partition and the open refinement of open sets in it .we give the sufficient conditions for a space having the property that its every scattered partition has a open expansion of a given property. And about a given property p which satisfies some conditions ,We give a condition for the open sets in the tychnoff product of two spaces which have a open refinement having the property p .The last chapter obtains a result about the limit of the inverse system of a normal mesocompact space and a hereditarily normal and hereditarily mesocompact space. Let X is the limit of the inverse system .Suppose each projection is an open and ontomap .And X is - paracompact.If each X is normalmesocompact ,then X is mesocompact .Moreover ,we obtain the analogous result for hereditarily normal and hereditarily mesocompact properties.