2008/02/18 by Kari Ragnarsson, Kári Ragnarsson, Ragnarsson, Kari +2
Computer Science · Mathematics · #05A99 (Secondary) #06A07 (Primary) #54A99 #Combinatorics (math.CO) #FOS: Mathematics #General Topology (math.GN) #Interconnection Networks and Systems #math.CO #math.GN #msc:05A99 #msc:06A07 #msc:54A99
paper · pdf · doi:10.48550/arxiv.0802.2550
Final version, to appear in Journal of Combinatorial Theory, Series A
openalex publication_date 2008/02/18 · arxiv created 2009/05/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.