区间数系的内蕴拓扑及度量表示

INTRINSIC TOPOLOGIES OF THE INTERVAL NUMBER SYSTEM AND THEIR METRIC REPRESENTATIONS

单位区间Ｉ上的区间数系ＳＩ在自然序下是一个完全分配格，其上的区间拓扑是连通的紧可度量拓扑，并具有不动点性质．一般地，实数集Ｒ上的区间数系ＳＲ在自然序下是局部完全分配格，其上的双Ｓｃｏｔ拓扑是第二可数的局部紧连通可度量拓扑，该拓扑是Ｒ上通常序拓扑的自然推广，还是道路连通的．其实，ＳＲ这一空间可嵌入到Ｒ２中．当考虑代数运算时，ＳＲ和ＳＩ都是拓扑格。

With a natural order, the interval number system S I on the unit interval is a completely distributive lattice. And it′s a connected compact metrizable topological space under the interval topology. This topological space has the fixed point property. Generally, with a natural order, the interval number system S R on the real number set is a locally completely distributive lattice. And it′s a second countable, connected, locally compact, metrizable topological space under the two sided Scott topology which is a generalization of the usual order topology on R. In fact, S R is also path connected and can be embedded in R 2. When algebraic operations are involved, S R and S I are also topological lattices, and S R is a topological group and a topological ring.