Several Studies Of The Scattered Partition

In this paper ,we mainly study the scattered partition,by restricting the open expansion of the scattered partition,we obtained some groups of equivalent characterizations of the countably paracompact spaces , the countably nearly paracompact ,the countably widly paracompact and the countably nearly widly paracompact which embodied mainly in chapter one and two.these results are abundant of the counterpart results of the paper 2.Coming up next in the chapter three ,the author of the paper use the tool of the scattered partition, accompaning with the perfect hereditarily paracompact space,obtained the equivalent characterization of perfect hereditarily paracompact space which is similar as Junnlia's,the result is the big advanced of the paper [4].In the chapter four ,the author obtained a equivalent characterization of the hereditarily metacompact space by continue using the scattered partition,and showing that there is no such similar characterization as Junnlia's for some hereditarily subparacompact space by an counter example. Simultaneously we obtained a equivalent characterization of the hereditarily strongly paracompact space on the base of the results of the paper 5.Finally in the chapter five we do pilot study of the relation between the scattered partition and partition of unity. and obtained some equivalent characterizations...