关于数直线R中分划R的子集

ON SUBSETS OF THE PARTITION R IN THE NUMERICAL STRAIGHT LINE R

本文研究美国学者P.Bankston和R.J.McGovern提出的一个公开问题—数直线R中那些子集分划R?研究表明,R中的每一个0一维子集都分划R;对月中的1一维子集;本文给出了开集、闭集以及具有某种散性的子集分划R的充分必要条件。

A space X partitions a space Y' means that there is a family of embeddings of X into Y, so that the images form a cover of Y by pairwise disjoint sets ( symbol X<<Y ) .In this paper, some partial solutions are given to the question raised by Bankston-McGovern[1] . The main results are as follows: ( 1 ) If M is a nonempty subset of R, o-dimensional, then M<<R; ( 2 ) If U is an open subset of R, U≠φ, then U<<R and only U is an open interval; ( 3 ) If F is a closed subset of R, 1-dimentional, then F<<R, only F has an unbounded domponent C, and ( F/C1 ) is closed.