Base - Covering Countable Paracompact Space

On the basis of base?cover paracompact spaces and countably paracompact spaces, wedefine a new space called base?cover countably paracompact space. In this thesis, we mainlydiscuss some properties of this new space. The results of this thesis are consist of two parts.The first part introduces the relationship of base?cover countably paracompact spaceand other spaces, we mainly got the following results:1.1 every base?cover paracompact space is base?cover countably paracompact space;1.2 every base?cover countably paracompact space with Lindel¨of property is base?cover paracompact space;1.3 give an example of a Lindel¨of space which is not base?cover countably paracom-pact space, hence, a paracompact space needs not to be a base?cover countably paracompactspace and a countably paracompact space needs not to be a base?cover countably paracom-pact space.The second part discusses the operations on base?cover countably paracompact spaces,including subspaces of the base?cover countably paracompact space and sum spaces andproduct spaces of base?cover countably paracompact spaces, and we mainly got the followingresults:2.1 if X is a base?cover countably paracompact space with Lindel¨of property, thenevery closed subspace of X is base?cover countably paracompact space;2.2 every closed subspace of a base?cover paracompact space is base?cover para-compact space, hence every closed subspace of a base?cover paracompact space is base?covercountably paracompact space;2.3 the sum space X = i∞=1 Xi is a base?cover countably paracompact space iffor every i∈N,the space Xi is base?cover countably paracompact space;2.4 there is a subspace ofω1 + 1 which is not base?cover countably paracompact,but each subspace ofω1 is base?cover countably paracompact;2.5 give an example of the product of a base?cover countably paracompact spaceand a compact space is not a base?cover countably paracompact space, hence, the productof two base?cover countably paracompact spaces needs not to be a base?cover countablyparacompact space.