Obtainable sizes of topologies on finite sets |
| |
Authors: | Ká ri Ragnarsson |
| |
Affiliation: | a Mathematics Institute, Reykjavík University, Kringlunni 1, 103 Reykjavík, Iceland b Department of Mathematical Sciences, DePaul University, 2320 North Kenmore Avenue, Chicago, IL 60614, USA |
| |
Abstract: | 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. |
| |
Keywords: | Finite topology Poset Order ideal Integer sequence |
本文献已被 ScienceDirect 等数据库收录! |
|