有限集上拓扑的绝对尺寸
Obtainable Sizes of Topologies on Finite Sets
摘要：
本文研究了具有k个开集的拓扑空间中的点的最小可能个数。等价地，这是具有k阶理想的偏序集中元素的最小可能数目。使用有效的算法构造一个拓扑结构与规定的大小，我们表明，这个数字有一个对数上界。我们推出，存在一个拓扑上的n点有k个开集，所有的k在一个区间是指数大的n。的建设算法可以修改，以产生拓扑结构的最小邻域的每个点具有最小的大小，我们给这样的拓扑结构的可获得的大小的范围。
英文摘要：
We study the smallest possible number of points in a topological space having k open sets. Equivalently, this is the smallest possible number of elements in a poset having k order ideals. Using efficient algorithms for constructing a topology with a prescribed size, we show that this number has a logarithmic upper bound. We deduce that there exists a topology on n points having k open sets, for all k in an interval which is exponentially large in n. The construction algorithms can be modified to produce topologies where the smallest neighborhood of each point has a minimal size, and we give a range of obtainable sizes for such topologies.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
