超实数域R的拓扑结构

The Topological Structures of the Hyperreal Number Field *R

研究了超实数域上两种有用的拓扑—Ｑ－拓扑和Ｓ－拓扑．证明了以下结果：空间（Ｒ，Ｑ）是完全不连通的；Ｒ的Ｑ－紧子集只有有限集；Ｒ中的每一个银河是（Ｒ，Ｓ）的一个连通分支；Ｒ中的每一个具有有限长度的区间（不必是闭的）都是Ｓ－紧的，同时也纠正了《Ｍａｔｈ．Ｊａｐｏｎｉｃａ》上一篇论文中关于Ｒ上的Ｑ－拓扑的性质的一些错误。

Two kinds of useful topologies, i.e. Q -topology and S -topology on the field of hyperreal numbers are studied. The following results are proved: The space ( tR, Q) is totally disconnected; only finite subsets of * R are Q -compact; Every galaxy of * R is a connected component of ( * R, S) ; Every interval in * R (not necessarily closed) with finite length is S -compact. Some mistakes about the properties of Q -topology in another paper on Math. Japonica are corrected as well.